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Présentations Arun Ram

Department of Mathematics and Statistics University of Melbourne Parkville VIC 3010 Australia aram@unimelb.edu.au

I have, too slowly, been coming to the realisation that what I understand in mathematics is far ahead of what I can write up in a polished form and that much of my contribution to the progress of mathematics comes from my lectures. Hopefully, putting as many of these lectures on the web as I can manage will help mathematics move along as quickly as possible. (This is not a complete list. There are some years where I seem to have no records at all, and others for which I have records only of colloquium and invited conference talks.)

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Talks of Arun Ram in 2024

1. Groups over local fields: Step one towards Γ\G/K, Number Theory Seminar, Melbourne University, 3 September 2024.
   Abstract: I want to try to make a basic path to these coset spaces, using the kinds of tools we use in first year Linear algebra and 2nd year GTLA. The key idea is to consider infinite n-periodic matrices In place of the n×n matrices that are used in first and second year. The hope is that, once one sets up infinite n-periodic matrices cleanly, then all that remains is to redo what is done in first and second year in this setting.
   
   
2. Master Class: How to write a thesis, Master Classes for Mathematics Masters students, Melbourne University, 3 September 2024.
   Abstract: This was a presentation discussion on how to organize time and tasks for completing a Masters or PhD thesis in good order. Much of the session was run in a question-answer format.
   
   
3. Tribute to Ian Macdonald Ian G. Macdonald: Works of Art, FPSAC 2024 Bochum, Germany, 22 July 2024. YouTube
   Abstract: Ian Macdonald's works changed our perspective on so many parts of algebraic combinatorics and formal power series. This talk will display some selected works of the art of Ian Macdonald, representative of different periods of his oeuvre, and analyze how they resonate, both for the past development of our subject and for its future.
   
   
4. Lusztig varieties and Macdonald polynomials, Representation Theory and Related Geometry: Progress and Progress in honor of Daniel K. Nakano, University of Georgia, 27-31 May 2024. You Tube
   Abstract: In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibers and nilpotent Lusztig varieties. This points the way to a generalisation of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalisation of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).
   
   
5. Proof Machine, a performance work with Sofia Carbonara, New Music Studio Concert Series, University of Melbourne, 22 May 2024. You Tube
   Abstract: Proof Machine presents interweaving stories from percussionist Sofia Carbonara and mathematician Arun Ram. The pair connect over a decades-long history, rooted in Carbonara's upbringing by a mathematician (their father) who pursued his doctorate alongside Ram in the 80's. Today, Ram is a lifetime lover of sound who integrates music into his lectures, while Carbonara is an experimental musician utilizing performance to unravel the embodied aesthetics of reason. This program features a world premiere by Carbonara and Ram in which they explore lessons from mathematics, music, personal history, and Ram's "proof machine" - a recipe for logical argumentation - through contemporary music.
   
   
6. ETA Modified Macdonald polynomials are not Macdonald polynomials, The UC Berkeley Combinatorics Seminar, 30 April 2024.
   Abstract: I will give a new combinatorial mechanics for computing the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials in type A. This new point of view points to a generalisation to all Lie types of modified Macdonald polynomials, integral form Macdonald polynomials and the plethyistic transformation that relates them. Except in type A, there is no immediate relationship to the Macdonald polynomials for all Lie types defined by Macdonald.
   
   
7. A new definition of integral form Macdonald polynomials, CART seminar (KIAS), 26 April 2024.
   Abstract: Mellit explained the Modified Macdonald polynomials can be viewed as a generating function for the number of points in parabolic affine Springer fibers. In this talk we explain that the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials has an interpretation in terms of traces on the Hecke algebra. This leads to an interpretation of the integral form Macdonald polynomials as generating functions for the number of points Lusztig varieties.
   
   
8. Generalised divided differences and Monk rules, Schubert seminar, 12 February 2024.
   Abstract: Generalised divided differences are a tool for building Macdonald polynomials, analogous to the way that classical divided differences are a tool for building Schubert polynomials. The calculus of generalised divided differences is called the double affine Hecke algebra, analogous to the way that the calculus of classical divided differences is called the nil Hecke algebra. In the first half of the talk I will review the analogies. In the second half I will explain how we use the calculus of generalised divided differences to compute Monk rules for Macdonald polynomials (joint work with Tom Halverson).
   
   
9. A survey of Macdonald's paper on symmetric products of an algebraic curve, Pure Math Student seminar, University of Melbourne, 19 January 2023.
   Abstract: Before starting university IG Macdonald wrote a paper about intersecting lines and circles (Clifford's chain and Longchamps chain, which generalise the Wallace point and the Miquel circle). After university, this exploration grew into a study of the cohomology of the symmetric product of a curve, and his computation of the cohomology gave a proof of the Weil conjectures for this case.
   
   
10. Macdonald's paper on the n-line, Vacation Scholar working seminar, University of Melbourne, 17 January 2023.
    Abstract: This was a survey talk about a portion of the material in Macdonald's unpublished paper The four-line, five-line and n-line.
    
    

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Talks of Arun Ram in 2023

1. A 5 lecture minimcourse on Representations of quantum affine algebras, at Algebraic and Combinatorial Methods in Representation Theory, ICTS-TIFR Bengaluru 13-18 November 2023.
   
   It was a pleasure to deliver these lectures in honor of Professor Vyjayanthi Chari, whose contribution has been fundamental in the development of the subject.
   
   The goal of the lecture series was to provide a view of the integrable representations of quantum affine algebras. The lectures were largely be based on section 2 of the following survey paper where further references can be found.
   • Positive level, negative level and level zero, (with Finn McGlade and Yaping Yang), appeared in the special volume in conjuction with the conference International Festival in Schubert Calculus, Sun-Yat Sen University, Guangzhou China, 5-11 November 2017, Schubert calculus and its applications in combinatorics and representation theory, Springer Proc. Math. Stat. 332, Springer, Singapore (2020) 153-194, MR4167516, arXiv1907.11796.
   
   
2. The Secrets for a Successful Career around Mathematics, Pure Math Student Seminar, University of Melbourne, 4 August 2023.
   
   
3. Introduction to Hessenberg varieties and relations to polynomials from combinatorics, Representation Theory working seminar, University of Melbourne, 14 September 2023.
   Abstract: I will define Hessenberg varieties and give some examples. Then I will explain some of the “combinatorial" results (Shareshian-Wachs, Brosnan-Chow, Abe-Horiguschi-et al, Precup-Sommers) about the cohomology of Hessenberg varieties coming from b-submodules of the adjoint representation. I will endeavour to explain the structural/conceptual relationship between this “non-interesting” setting and the “interesting” setting that Yau Wing has introduced us to. As Kari explained, the “interesting" setting is when the motive of the Hessenberg variety shows features that do not appear in the Springer resolution and the Grothendieck simultaneous resolution. For this talk I will move away from the Springer resolution and the Grothendieck simultaneous resolution but I will stick to the very fascinating and amazing "non-interesting" setting.
   
   
4. The Secrets for a Successful Career around Mathematics, Pure Math Student Seminar, University of Melbourne, 4 August 2023.
   
   
5. Lusztig varieties and Macdonald polynomials, IMJ Groupes, Répresentations et Géométrie seminar, Institut de Mathématiques de Jussieu - Paris Rive Gauche, 2 June 2023.
   Abstract: In recent works Abreu-Nigro and Xuhua He have introduced the term Lusztig variety. I like this term, as Lusztig has many papers about these varieties. In 1997 Halverson and I computed the number of points of Type A nilpotent Lusztig varieties over finite fields in connection to characters of Hecke algebras. Recently, my study of Macdonald polynomials and central elements in Hecke algebras have led me to look at these computations again.
   
   
6. Boson-Fermion correspondence for Macdonald polynomials, Algebra and Representation Theory seminar, Roma "Tor Vergata", 26 May 2023.
   Abstract: In its simplest form, this correspondence is the map from symmetric functions to skew-symmetric functions given by multiplication by the Weyl denominator (the Vandermonde determinant). A generalization produces the motivating shadow of "geometric Satake", a diagram which contains the Satake isomorphism, the center of the affine Hecke algebra and the Casselman-Shalika formula. In a miracle that I wish I understood better, the whole diagram generalizes to the case of Macdonald polynomials and sends the bosonic Macdonald polynomial to the fermionic Macdonald polynomial. Does this suggest an "elliptic version" of geometric Langlands? This talk is based upon arXiv2212.03312, joint with Laura Colmenarejo.
   
   
7. Lusztig varieties and Macdonald polynomials, Representation Theory seminar, University of Melbourne, 18 May 2023.
   Abstract: In recent works Abreu-Nigro and Xuhua He have introduced the term Lusztig variety. I like this term, as Lusztig has many papers about these varieties. In 1997 Halverson and I computed the number of points of Type A nilpotent Lusztig varieties over finite fields in connection to characters of Hecke algebras. Recently, my study of Macdonald polynomials and central elements in Hecke algebras have led me to look at these computations again.
   
   
8. Row reduction and flag varieties , Pure Mathematics Student seminar, University of Melbourne, 28 April 2023.
   Abstract: All cohomology computations that I know are done by row reduction for matrices (as in first year linear algebra). I will review row reduction for matrices and explain how to use it to make the computation of the cohomology of the flag varieties trivial. Hopefully, I will also be able to use the same method to derive the cohomology of Hessenberg varieties. I'll do a bit of "Ask me a question", and a bit of "This is what I would like to ask the Oracle".
   
   
9. Cosets and Hecke algebras , Number Theory seminar, University of Melbourne, 6 April 2023.
   Abstract: The goal is to describe what the Hecke operators are and how they arise from cosets and double cosets. The groups that play a role are primarily GL_2(ℚ) and SL_2(ℤ).
   
   

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Talks of Arun Ram in 2022

1. Murphys, Casimirs, Transvections and Hecke algebras, invited talk, 13 December 2022 at the Workshop on representation theory of symmetric groups and related algebras 12-16 December 2002, as part of the program on Representation Theory, Combinatorics and Geometry, National University of Singapore, 12 December 2022-07 January 2023.
   Abstract: Abstract: One way to discover Murphy elements in the group algebra of the symmetric group is to push Casimir elements across the Schur-Weyl duality. I will review this construction and then explain how a similar construction allows one to push the conjugacy class of transvections in GL_n(F_q) across a type of Schur-Weyl duality to obtain "Murphy elements" in Hecke algebras. In fact, the construction works for all Lie types and for conjugacy classes generalising the conjugacy class of transvections. These elements have been used to analyse a Markov chain on the symmetric group coming from double cosets. This is a report on joint works with Persi Diaconis, Mackenzie Simper and James Parkinson.
   
   
2. Introduction to Hessenberg varieties , Representation Theory seminar, University of Melbourne, 10 November 2022.
   Abstract: Abstract: It useful, and sensible, to view Hessenberg varieties as 'the baby case' of affine Springer fibres. In this talk I will review results of Martha Precup and Eric Sommers which analyse the affine paving and the equivariant cohomology of Hessenberg varieties. The primary references are arxiv:1205.3976 and arxiv:2201.13346.
   
   
3. c-functions and Macdonald polynomials , invited talk at Integrability, combinatorics and representation theory, MATRIX/RIMS tandem workshop, 26-30 September 2022, Creswick, Victoria, Australia.
   Abstract: Abstract: S. Helgason has a paper entitled "Harish-Chandra's c-function. A Mathematical Jewel". In his work on spherical functions on p-adic group Macdonald pointed to an analogue of the c-function for p-adic groups. In Lusztig's work on affine Hecke algebras this version of the c-function for p-adic groups appears in the formula for the action of the Demazure-Lusztig operators. In this talk we will explain how the c-function enters into (and simplifies) formulas for Macdonald polynomials: expansions, principal specializations, and norm formulas.
   
   
4. The modular semigroup , NOVAMath seminar, Universidade Nova de Lisboa, 20 June 2022.
   Abstract: Abstract: Actions of the modular group SL₂(ℤ) seem to be ubiquitous in many parts of mathematics (modular forms, modular tensor categories, elliptic cohomology, symmetries of Macdonald polynomials). I have two little scraps of paper in my wallet that I carry everywhere. The first says "find a normal form". The second says "Is that the best choice of generators?". In an effort to try to understand actions of SL₂(Z) I was staring at a picture in Serre's book on trees which, from the correct angle, indicates that SL₂(ℤ_≥0) is a free semigroup on two generators. My gut tells me that its structural importance is akin to the structural importance of the free semigroup on 1 generator ℤ_≥0.
   
   
5. Is there a Kac-Moody-like presentation of toroidal algebras? , Representation Theory Seminar, University of Melbourne, 9 June 2022.
   Abstract: Ion-Sahi have pointed to a Coxeter like presentation of the double affine Artin group (DAArt). I will explain how this presentation could be discovered from a matrix representation of the double affine Weyl group (DAWG) which naturally exhibits the action of SL2(Z) (acting on the DAWG) by automorphisms. The position of the Heisenberg group inside the DAWG is clearly visible in this representation. The Coxeter-like presentation uses three affine Dynkin diagrams of the same type glued together along the common finite Dynkin diagram and a single additional "superglue" relation. I wonder if these results could be extended to provide a Kac-Moody-like presentation of quantum toroidal algebras.
   
   
6. 12 lectures on Macdonald polynomials , Grad Studies A, University of Melbourne, February-May 2022.
   Abstract: The goal of this lecture series is to provide a review of the theory of Macdonald polynomials with a focus on examples. It was mostly a work through of the content and examples of the results of the following texts. My style differs, so it does not look the same, but the main results are mostly the same -- I added details, explicit examples and specific computations. Several of the proofs that were presented are different. and there were a few additional combinatorial formulas. The texts below are written in the setting of general affine root systems, but the lectures focused on type GL_n.
   • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki 1996 Vol. 1994/1995, Astérisque No. 237, Exp. No. 797, 4, 189-207.
   • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press 2003 ISBN: 9780511542824 DOI: https://doi.org/10.1017/CBO9780511542824.
     
   
   
   

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Talks of Arun Ram in 2021

1. Reflection groups and the KZ functor, Student Summer Representation Research Seminar, December - January 2021-2022.
   Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5,
   Abstract: This series of lectures will be focused on working through Section 3.1, the first three sentences of 5.1.1, and section 5.2.2 of the paper by Ginzburg-Guay-Opdam-Rouquier arXiv:math/0212036 entitled "On the category O for rational Cherednik algebras”. The goal is to do the examples of the KZ-functor for cyclic groups, dihedral groups and symmetric groups. The plan is for the lectures to be suitable for a GroupTheory and Linear Algebra student to build on their knowledge of what a group, vector space, and an eigenvalue are.
   
   
2. Transvections and Hecke algebras, Groups and Geometries, Online MATRIX Workshop, 29 November - 3 Becember 2021.
   Abstract: This is joint work with Persi Diaconis and Mackenzie Simper. Let V be an n dimensional vector space over the finite field F(q). We study the walk on permutations which arises from repeated transvections acting on maximal flags of subspaces of V (the permutation is the type of the flag with respect to the standard flag). We show that, up to addition of a multiple of the identity (i.e. up to holding), the walk is equivalent to repeated mutliplication by the sum of he Hecke algebra basis elements corresponding the transpositions of the symmetric group. We provide an explicit formula which also determines the holding. The transvections walk on GL(n) is studied by Hildebrand who shows that n + c steps are necessary and sufficient for convergence to stationarity. The walk on permutations is shown to mix faster.
   
   
3. Specializations of Macdonald polynomials, Solvable Lattice Models Seminar, Stanford University, 12 October 2021.
   Abstract: Macdonald often included in his talks the "specialization square" for symmetric Macdonald polynomials, which has monomial symmetric functions on the top edge, elementary symmetric functions on the right edge, Hall-Littlewood polynomials on the left edge, Schur functions on the diagonal and Jack polynomials in the upper right corner. This talk will explore the analogous "specialization square" for non-symmetric Macdonald polynomials. The primary tools are the monomial expansion formulas, the E-expansion formula, and the creation formula.
   
   
4. The quantum group is a fake, Working seminar, University of Melbourne, 30 August 2021.
   Abstract: This talk was an attempt to organize the results of Drinfeld, proving that the quantum group is isomorphic to the nonquantum group and that this isomorphism (up to the twist that corrects the R-matrix to be symmetric) takes the trivial associator to the KZ associator and the R-matrix to the exponential of the Casimir. The focus was on the sl₂ case where, hopfully, everything can be written out explicitly.
   
   
5. Principal specializatons of Macdonald polynomials, Workshop on Macdonald polynomials, Indian Institute of Science, Bangalore, 30 June 2021. YouTube
   Abstract: The principal specialization of a Schur function (or Weyl character) has a striking factorization as a product, and specializing the parameter gives the Weyl dimension formula. In type GL_n this gives the hook formula for the number of column strict tableaux. The nonspecialized formula is often called the quantum dimension. Amazingly the Macdonald polynomials have similar formulas for their principal specializations, which might be thought of as elliptic Weyl dimension formulas.
   
   
6. Five (q,t) analogues of Kostka numbers, Workshop on Macdonald polynomials, Indian Institute of Science, Bangalore, 28 June 2021. YouTube
   Abstract: The Kostka numbers arise as the coefficients in the monomial expansion of Schur functions. In the Macdonald polynomial case I know 5 different analogues of these numbers coming from
   1. The monomial expansion of the P_λ
   2. The expansion of e_μ in terms of the P_λ
   3. The expansion of g_μ in terms of the P_λ
   4. The expansion of P_μ(q,qt) in terms of the P_λ(q,t)
   5. The expansion of J_μ in terms of the big Schurs S_λ
   In the first three cases I know formulas for these as weighted sums of column strict tableaux. I do not understand the relationships between these 5 different analogues of Kostka numbers.
   
   
7. The (q,t) Weyl character formula and the (q,t) Boson-Fermion correspondence, Workshop on Macdonald polynomials, Indian Institute of Science, Bangalore, 21 June 2021. YouTube
   Abstract: One of the favorite formulas for the Schur function is as a quotient of two determinants. I will explain the corresponding formula for Macdonald polynomials. The boson-fermion correspondence is the correspondence between symmetric functions and skew symmetric functions which is given by multiplying by the Vandermonde determinant. I will explain the (q,t) version of this correspondence and how the Weyl character formula fits into this correspondence.
   
   
8. Level 0 modules of affine Lie algebras, Solvable Lattice Models Seminar, Stanford University, 16 June 2021. YouTube
   Abstract: The favourite R-matrices and transfer matrices (which give the 6 vertex model) arise from the evaluation representation of the standard representation of the quantum group of sl(n). This is a level 0 representation of quantum affine sl(n) and the lattice models are based on tensor powers of this representation. With Finn McGlade and Yaping Yang we have written a survey about the classification of these modules (by dominant weights for extremal weight modules and by Drinfeld polynomials for finite dimensional modules), their characters (which are q-Whittaker functions) and their crystals. I will try to sketch how I think this category (of level 0 modules) is the controlling structure for vertex operators, Fock spaces (Kyoto path model) and the Algebraic Bethe ansatz.
   
   
9. Representations of affine Hecke algebras IV: Where do Hecke algebras come from?, Representation Theory seminar, University of Melbourne, 15 June 2021.
   Abstract: This will be a talk about the origins of the game:
   1. double centralisers and correspondence between some representations of A and representations of the centralizer
   2. Convolution, Inducing from the trivial representation, double coset algebras, and correspondence between some representations of the group and the representations of the double coset algebra
   3. Generators and relations for p-adic groups, cosets and double cosets
   4. Computation of the Iwahori-Hecke algebras and correspondence between some representations of the p-adic groups and representations of the Hecke algebra.
   
   
   
10. Representations of affine Hecke algebras III: Standard modules, Representation Theory seminar, University of Melbourne, 8 June 2021.
    Abstract: Most standard modules are built by induction. I'll describe principal series modules and the basic theorems about their simplicity and their composition factors. Then I'll define tempered and square integrable modules, and explain how the standard modules corespond to generalized Springer fibers and the square integrable modules correspond to cuspidal nilpotent elements. This provides an indexing of irreducible representations of affine Hecke algebras by Deligne-Langlands parameters.
    
    
11. Representations of affine Hecke algebras II: Central characters, weight spaces and intertwiners, Representation Theory seminar, University of Melbourne, 1 June 2021.
    Abstract: I will try to explain how the representation theory of the affine Hecke algebra is coded by local regions in a hyperplane arrangement, sometimes called the Shi arrangement. I'll introduce central characters and explain how to study irreducible representations of affine Hecke algebras by weight spaces and intertwiners. I'll explain the indexing of calibrated simple H-modules and the type GL_n indexing of simple H-modules by multisegments.
    
    
12. Representations of affine Hecke algebras I: Isogeny and lifting representations of cyclotomic Hecke algebras, Representation Theory seminar, University of Melbourne, 25 May 2021.
    Abstract: In this first talk, I'll define the isogeneous affine Hecke algebras and explain how to compare their representations. Then I'll explain how to get the representations of finite Hecke algebras of the finite complex reflection groups G(r,p,n) from the affine Hecke algebra of type GL_n.
    
    
13. Decomposition numbers for standard objects in categories O, Representation Theory seminar, University of Melbourne, 11 May 2021.
    Abstract: The Verma modules are indexed by their highest weight. They have a simple quotient and so the irreducibles are indexed by their highest weight. The composition factors of a Verma module must all lie in the same orbit of the Weyl group. In the affine case the orbits take three different shapes depending on whether it is positive level, negative level, or critical level. In each case there is a different family of Kazhdan-Lusztig type polynomials that describes the multiplicity of the irreducible in the layers of the Jantzen filtration of the Verma module. I'll try to explain what these affine Weyl group orbits and Kazhdan-Lusztig polynomials are.
    
    
14. Open boundary Hecke and Temperley-Lieb algebras, Mathematical Physics seminar, University of Melbourne, 4 May 2021.
    Abstract: In 2005 Vladimir Rittenberg explained me the idea of two boundary spin chains. In my discussions with him, I outlined how I would like to think about this question, in terms of the affine Hecke algebra of a p-adic symplectic group. Then one can use the Jucys-Murphy type elements and the description of the center that one gets from Bernstein-Lusztig 1980, and the classification of irreducible representations that one gets from Kazhdan-Lusztig 1987. In joint work with Zajj Daugherty we have, over several years, been working out the details for this sketch. One part of our work has been to explain the conversion between this method to the classification of irreducible open boundary Temperley-Lieb representations that was done by De Gier-Nichols in 2007.
    
    
15. The universal Pieri rule and Pieri/difference operator duality, Macdonald polynomial reading seminar, University of Melbourne, 21 March 2021.
    Abstract: This talk was a survey of material in Chapter 6 Section 6 of Macdonald's book Symmetric functions and Hall polynomials. I explained how Macdonald's Pieri rule for Macdonald polynomial is given by evaluations of the coefficients of the expansion of Macdonald's difference operator and then argued that this (along with symmetry) is a forecasting of "duality" on the double affine Hecke algebra. This is done by introducing the double affine Weyl group, where there is an obvious symmetry between the Xs and the Ys, and then look at its polynomial representation.
    
    
16. The elliptic hook length formula, Macdonald polynomial reading seminar, University of Melbourne, 14 March 2021.
    Abstract: This talk was a survey of material in Chapter 6 Section 6 of Macdonald's book Symmetric functions and Hall polynomials. I explained how the Weyl dimension formula (for GL_n) corresponds to a specialization of Schur functions, how the principal specialization of a Schur function is the quantum dimension, and argued that perhaps the pricipal specialization of the Macdonald polynomial can be considered as an elliptic dimesion. Then I gave a combinatorial/pictorial explanation of Macdonald's proof of the hook formula for this elliptic dimension.
    
    

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Talks of Arun Ram in 2020

1. Examples in affine combinatorial representation theory, Minicourse at DMRT2020, Discussion Meeting on Representation Theory 2020 Indian Institute of Science, Bengaluru, 10-12 December 2020.
   
   Although the theory applies to all quantum affine algebras we shall focus on some illustrative examples in type GL_n. The goal is to highlight some amazing connections between combinatorics, representations, mathematical physics and probability.
   
   Lecture 1: Examples of Macdonald polynomials
   Abstract: We'll study examples of GL_n Macdonald polynomials. The basic tool is the affine Weyl group. This lecture is motivated by the papers of Haglund-Haiman-Loehr 2006 and Lenart 2008. An amazing connection is between tableau formulas for Macdonald polynomials and sequences of elements in the affine Weyl group.
   
   Lecture 2: Examples of level zero extremal weight modules
   Abstract: We'll construct the standard and simple level 0 modules corresponding to skew shape Young diagrams (for the quantum affine algebra of type GL_n). This lecture is motivated by the papers of Drinfeld 1986, Cherednik 1987 and Nazarov-Tarasov 1998. An amazing result is that the irreducible modules bases indexed by tableaux and that the characters of the standard modules are specialisations of Macdonald polynomials.
   
   Lecture 3: Examples connecting to probability and conformal field theory
   Abstract: V = ℂⁿ[t,t^-1] and its relation to R-matrices, transfer matrices, and Macdonald polynomials. This lecture is motivated by the papers of Takhtajan-Faddeev 1979, Kashiwara-Miwa-Stern 1995 and Borodin-Wheeler 2018. An amazing result is that an eigenvector of the transfer matrix is related to the stationary distribution of the ASEP (asymmetric exclusion process) and has coefficients which are specialisations of Macdonald polynomials.
   
   
2. Extremal weight modules, global Weyl modules and local Weyl modules , Special session in Representation Theory, Australian Mathematical Society Meeting, 8-10 December 2020.
   Abstract: This talk will be a brief review of the indexing and structure of level 0 standard modules for affine Lie algebras. Basically, the extremal weight modules are shaped like infinite tubes, and the local Weyl modules have the shape of the torus obtain by gluing the ends of the tube together. The Heisenberg subalgebra inside the affine Lie algebra moves vectors up and down the tube.
   
   
3. Periodic permutations and Macdonald polynomials, Combinatorics seminar, Univ. of Southern California, 11 November 2020.
   Abstract: Haglund-Haiman-Loehr gave formulas for Macdonald polynomials as sums over nonattacking fillings (of boxes) counted with statistics (arm, leg, maj and coinv). The goal of this work is to try to make the Haglund-Haiman-Loehr statistics “fall out" of an analysis of reduced words for periodic permutations. At the root of this relationship is the intertwiner construction of the (nonsymmetric) Macdonald polynomials.
   
   
4. Flags, crystals, and orthogonal polynomials, Colloquium, University of Talca, 15 October 2020.
   Abstract: When nonmathematicians ask me what research I do, I say symmetry. From symmetry of polyhedra, to symmetries of universes like spheres and tori, now we are fascinated by "paths" of symmetries, and this is the source of loop groups. It turns out that loop groups capture amazing geometry, combinatorics, and representation theory. I will endeavour to explain what the integrable representations for loop groups look like (paraboloids, mountains, craters, and tubes) and how these shapes are a reflection of the corresponding geometry (of an infinite dimensional flag variety). And then, miraculously, the characters of these modules turn out to be (specialised) Macdonald polynomials!
   
   
5. Curves of symmetries and categories of modules, Colloquium, Macquarie University, 9 October 2020.
   Abstract: When nonmathematicians ask me what research I do, I say symmetry. From symmetry of polyhedra, to symmetries of universes like spheres and tori, now were are fascinated by "paths" of symmetries, and this is the source of loop groups. It turns out that loop groups capture amazing geometry, combinatorics, and extremely structured tensor categories. I will endeavour to explain what the integrable modules for loop groups look like: paraboloids, mountains, valleys, and tubes.
   
   
6. Integrable modules for affine Lie algebras, Algebra seminar, University of Georgia Athens, 6 October 2020. YouTube
   Abstract: These modules naturally divide themselves into three categories: positive level, negative level and level 0. The positive level modules are highest weight, the negative level ones are lowest weight, and the level 0 ones are neither. But all three classes of modules have some nice character formulas, a good crystal theory in the sense of Kashiwara-Lusztig-Littelmann, and Borel-Weil-Bott type geometric constructions. The geometric constructions use, respectively, the thin affine flag variety (for positive level), the thick affine flag variety (for negative level), and the semi-infinite flag variety (for level 0).
   
   
7. Presenting your research: Six points for giving talks, Univ. of Melbourne Mathematics and Statistics MSc Masterclasses, University of Melbourne, 9 September 2020.
   Abstract: This talk covered: Audience, Preparation, Cutting material, Time management, Presentation tools, Explain it, "Talk machine", differences between talks, developing confidence, finding the right conference, Finding funding opportunities, and technology for online presentations.
   
   
8. Teaching Math in The Next Life, University of Western Sydney Abend Seminars , 30 July 2020.
   Abstract: For many years I've been thinking about how to teach mathematics with honesty and inspiration. This has resulted in ideas like "Reality teaching", "Proof machine", "Marking apocalypse", and "Just do it". And then a virus came, and the new life began, online, on Zoom. This will be a talk about the adventures of the past life and the preparations for the next.
   
   
9. Nonsymmetric Macdonald polynomials: Part I and Part II, Séminaire Corteel, UC Berkeley, 12 and 19 May 2020.
   Abstract: Following on from the statement that the Key polynomials and Demazure atoms are specialisations of nonsymmetric Macdonald polynomials, I will explain how I like to construct the nonsymmetric Macdonald polynomials. Depending on which "flavor" of the construction is used, one obtains the HHL formula or the alcove walk formula. For this talk I will stick to type A (more precisely, type GLn).
   
   
10. Presenting your research: Six points for giving talks, Univ. of Melbourne Mathematics and Statistics MSc Masterclasses, University of Melbourne, 23 April 2020.
    Abstract: This talk covered: Audience, Preparation, Cutting material, Time management, Presentation tools, Explain it, "Talk machine", differences between talks, developing confidence, finding the right conference, Finding funding opportunities, and technology for online presentations.
    
    
11. Formulas for Macdonald polynomials, Representation Theory seminar, University of Melbourne, 27 March 2020.
    Abstract: I will review/compare and contrast some of the formulas for nonysmmetric (and symmetric) Macdonald polynomials including the Haglund-Haiman-Loehr formula, the Ram-Yip formula, and the recent formulas of de Gier-Cantini-Wheeler and Corteel-Williams-Mandelshtam. One result I'd like to highlight provides the specialisations of the Ram-Yip formula for q and t taking values 0 or infinity. I may also make some comments about the Macdonald polynomials for type (C^∨, C), which are called Koornwinder polynomials, and Macdonald polynomials for other classical (unitary, orthogonal and symplectic) types.
    
    
12. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written and performed by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), at Tempo Rubato, Brunswick, Victoria Australia, 12 March 2020.
    Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
    
    
13. Level 0 and the Bethe ansatz, Pure mathematics seminar, University of Melbourne, 21 February 2020.
    Abstract: First I will review the mechanics for finding eigenvalues and eigenvectors of the Murphy elements in the group algebra of the symmetric group as a model method for finding the eigenvalues and eigenvectors of the transfer matrices that appear in the Algebraic Bethe ansatz. Second I will review the connection between transfer matrices and pseudoquasitriangular Hopf (psqtH) algebras. Third I will visit the 3 favourite families of pqtH-algebras and hint at their relation to affine Lie algebras. Finally I will explain, via an illustrative (9-dimensional) example for sl₃, the construction of the eigenvalues and eigenvectors in a level 0 representation.
    
    
14. Introductory preseminar for Level 0 and the Bethe ansatz, Introductory seminar for the Pure mathematics seminar, University of Melbourne, 21 February 2020.
    Abstract: This was an introductory preseminar for the Pure Mathematics seminar that same afternoon.
    
    
15. Level 0 and the Bethe ansatz, invited speaker at Talking Across Fields, a conference in Honor of Persi Diaconis, Stanford University, 31 Jan.-2 February 2020.
    Abstract: This is about the relation between representations of affine Lie algebras, the energy (Hamiltonian) for the Bethe ansatz and the combinatorics of Macdonald polynomials. Although I probably won't get to talk about the shuffles in the background (sorry Persi), I probably will talk about Young tableaux, quantum groups, and Heisenberg spin chains.
    
    
16. Level 0 and the Bethe ansatz, Mathematical Physics seminar, University of Melbourne, 21 January 2020.
    Abstract: First I will review the mechanics for finding eigenvalues and eigenvectors of the Murphy elements in the group algebra of the symmetric group as a model method for finding the eigenvalues and eigenvectors of transfer matrices. Second I will review the connection between transfer matrices and pseudoquasitriangular Hopf (psqtH) algebras. Third I will visit the 3 favourite families of pqtH-algebras and explain the special case that gives the classical XXX, XXZ, and XYZ Hamiltonians. Finally I will explain, via an illustrative (9-dimensional) example for sl₃, the construction of the eigenvalues and eigenvectors.
    
    

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Talks of Arun Ram in 2019

1. Tits buildings and the structure of Chevalley groups, Representation Theory seminar, University of Melbourne, 15 October 2019.
   Abstract: This was an introductory talk covering philosophy motivating the definition of a Tits building, how to picture a building in cases where it is a Coxeter complex, where it is the building of a finite Chevalley group, and in cases where it is an affine building. We ended with the connection between certain additive root subgroups and the corresponding pictorial point of view that motivates the Moy-Prasad filtrations for groups over local fields.
   
   
2. The être of Michèle Vergne, a short talk about Michèle Vergne for the Women in Maths Day at the University of Melbourne, 29 August 2019.
   
   
3. Presenting your research: 6 points to think about for giving talks, MasterClass program for Master Students, University of Melbourne, 27 August 2019.
   
   
4. Maybe I could be a mathematician: A story of growing up alongside vinyl, CD, MP3 and YouTubeRed, A lecture performance presented by National Science Week and the Faculty of Science at University of Melbourne at the Bendigo Discovery Science and Technology Centre, 22 August 2019.
   Abstract: In this unique presentation Professor Arun Ram tells a series of stories, interweaving mathematics and music. Humorous, educational, personal, often all at once, these collected stories illuminate the remarkable journey of an enquiring mind who became a mathematician. A mathematical mixtape from a lifelong road trip.
   
   
5. The Music of Mathematics, Castlemaine Primary School, Castlemaind Victoria Australia, 23 August 2019.
   Abstract: This was a presentation for school students, including an excerpt from the Glass Bead Game on the emotion of mathematics leading to Maria Callas and Alexander Grothendieck, and excerpts from Maybe I could be a Mathematician, (45 minutes, 9:00-9:45, 56 students in grade 5 and 6; This visit was covered by the Castlemaine Mail).
   
   
6. The Music of Mathematics, Girton Grammar School, Bendigo Victoria Australia, 22 August 2019.
   Abstract: This was a presentation for school students, including an excerpt from the Glass Bead Game on the emotion of mathematics leading to Maria Callas and Alexander Grothendieck, an excerpt from the Mendelssohn Salon, and an excerpt from Maybe I could be a Nathematician, a juggling analysis with Anthony Mays, a poem of Sofia Kovalevsky and Carl Gauss with Anita Ponsiang and Anthony Mays, a portrait of Persi Diaconis and a card trick performed by Anthony Mays. (75 minutes, 9:30-10:45, with students from Girton Grammar as well students from Holy Rosary Primary School, White Hills Primary School, Marist College, Cursoe Secondary College, a total of 150 students for 75min, from grades 5,6,7,8, 9; This visit was covered by the Bendigo advertiser)
   
   
7. The musical sensation of mathematics, A lecture performance at ExtraSensory, presented by National Science Week and the Royal Society of Victoria, at Parliament of Victoria,hMelbourne 10 August 2019.
   Abstract: It's the pivotal scene of your favourite movie, the soundtrack builds, the tension rises. You experience a wave of sound and roller coaster of emotions. Music makes us feel things, but how could this possibly relate to mathematics?
   With a collage of musical excerpts and mathematical ideas, join us for an emotional ride where music evokes the drama and excitement of major mathematical advances. Hear Bach's music illuminate Euler's circle formula and Maria Callas' incomparable voice capture the expanse of Alexander Grothendieck's highways of geometry.
   
   
8. The triumvirate of affine flag varieties, Invited speaker at the conference Flags, Galleries and Reflections at University of Sydney, 5-8 August 2019.
   Abstract: This will be a survey talk about the positive level (thin) affine flag variety, the negative level affine flag variety (thick) affine flag variety, and the level 0 (semi-infinite) affine flag variety. The first part will be a discussion of what I know (gallery models for points) and the second part will be a discussion of things I don't know but would like to know (a universal Borel-Weil-Bott theorem covering all three cases simultaneously and the K-theory of Springer fibres as DAHA-modules).
   
   
9. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), performed by Joshua Hooke and Arun Ram (University of Melbourne), Federation Hall, Southbank, Melbourne, 4 August July 2019.
   Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
   
   
10. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), performed by Joshua Hooke and Arun Ram (University of Melbourne), The Women's College, University of Sydney, 6 August 2019.
    Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
    
    
11. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), performed by Joshua Hooke and Arun Ram (University of Melbourne), Wonthaggi Baptist Church, 31 July 2019.
    Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
    
    
12. The Music of Mathematics, Wonthaggi Secondary College, Wonthaggi Gippsland Victoria Australia, 31 July 2019.
    Abstract: This was a presentation for school students, including an excpert from the Glass Beadn Game on the emotion of mathematics leading to Maria Callas and Alexander Grothendieck, an excerpt from the Mendelssohn Salon, and an excerpt from Maybe I could be a Nathematician. (13:15-14:00 for about 160-200 students in years 7,8 9, this school visit was covered by WIN News Goppsland)
    
    
13. The Music of Mathematics, Mary McKillop Catholic Regional College, Leongatha Gippsland Victoria Australia, 31 July 2019.
    Abstract: This was a presentation for school students, including an excpert from the Glass Beadn Game on the emotion of mathematics leading to Maria Callas and Alexander Grothendieck, an excerpt from the Mendelssohn Salon, and an excerpt from Maybe I could be a Nathematician. (9:00-9:45 for about 84 students in Year 10).
    
    
14. The Music of Mathematics, Chairo Christian College, Leongatha Gippsland Victoria Australia, 31 July 2019.
    Abstract: This was a presentation for school students, including an excpert from the Glass Beadn Game on the emotion of mathematics leading to Maria Callas and Alexander Grothendieck, an excerpt from the Mendelssohn Salon, and an excerpt from Maybe I could be a Nathematician. (12:00-12:45 for about 54 students years 7-10).
    
    
15. Combinatorics in level 0, Representation Theory seminar, University of Melbourne 19 February 2019.
    Abstract: I gave an introduction to level 0 representations of the affine Lie algebra with lots of pictures: including their indexing, and their character theory via crystals (via LS paths following Naito-Sagaki).
    
    
16. A point of view on Conformal Field Theory, invited lecture at the conference Subfactors in Sydney, University of New South Wales, 7 February 2019.
    Abstract: I will give a proposal for studying conformal field theory by combinatorics of crystals (in the sense of Kashiwara and Lusztig), more precisely, via crystals of level 0 integrable representations of affine Lie algebras.
    
    
17. The door is ajar, Concluding speaker, Strengthening Engagement and Achievement in Mathematics and Science (SEAMS) Senior Residential Camp, 21-23 January 2019, Melbourne University.
    Abstract: The door is ajar for the next generation of Mathematicians and Scientists.
    
    

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Talks of Arun Ram in 2018

1. Two boundary Temperley-Lieb and the exotic nilcone, speaker at Geometric and Categorical Representation Theory, MATRIX, Creswick, Australia, 20 December 2019, 14 December 2019
   Abstract: This talk is based on work with Zajj Daugherty, Iva Halacheva and Arik Wilbert. We describe the representations of the two boundary Temperley-Lieb algebra combinatorially, diagrammatically and geometrically. The diagrammatic approach follows de Gier-Nichols and Green-Martin-Parker. The geometric approach follows S. Kato and uses the exotic nilcone to construct representations of the two boundary Hecke algebra which are then identified as representations of the two boundary Temperley-Lieb algebra. The combinatorial approach uses the theory of multisegments and standard Young tableaux.
   
   
2. Schubert calculus on semi-infinite flag varieties, Representation Theory Seminar, University of Melbourne, 9 October 2018
   Abstract: I will endeavour to explain the main facets of the way that I think about Schubert calculus, focusing on the case of the semi-infinite flag variety. I will review the definition of the semi-infinite flag variety, the Schubert classes, the action of polynomials, the moment graph description, the push-pull operators and the Pieri-Chevalley formula. None of this is my work except, perhaps, a certain point of view on the subject.
   
   
3. Representations of two boundary Hecke and Temperley-Lieb algebras (TBHA and TBTL), Representation Theory Seminar, University of Melbourne, 14 August 2018
   Abstract: I will discuss classifications, constructions and combinatorics of irreducible and standard modules of TBHA and TBTL. The TBTA is the affine Hecke algebra of type C with arbitrary "unequal" parameters. The TBTL is a quotient of the TBHA by local idempotents (for rank 2 sub root systems). The TBTL has been of interest in statistical mechanics: Heisenberg spin chains with boundaries (de Gier-Nichols). The geometry construction of TBHA-modules (Kato) for unequal parameters is via the exotic nilpotent cone.
   
   
4. Maybe I could be a mathematician: A story of growing up alongside vinyl, CD, MP3 and YouTubeRed, A lecture performance presented by The Institute for Enquiring Minds, RMIT Cinema Theatre Building 80, Melbourne, 31 July 2018
   Abstract: In this unique presentation Professor Arun Ram tells a series of stories, interweaving mathematics and music. Humorous, educational, personal, often all at once, these collected stories illuminate the remarkable journey of an enquiring mind who became a mathematician. A mathematical mixtape from a lifelong road trip.
   
   
5. The door is ajar, Concluding speaker, Strengthening Engagement and Achievement in Mathematics and Science (SEAMS) Senior Residential Camp, 3-5 July 2018, Melbourne University.
   Abstract: The door is ajar for the next generation of Mathematicians and Scientists.
   
   
6. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written and performed by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), at the Fergg Salon, Munich, Germany, 16 March 2018.
   Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
   
   
7. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written and performed by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne), at the Clemann-Zimansky Salon, Basel, Switzerland, 11 March 2018.
   Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
   
   
8. Flag varieties and Schubert Calculus, 4 lectures at the NZMRI Summer school, Nelson New Zealand, 7-13 January 2018.
   Lecture 1, Lecture 2, Lecture 3, Lecture 4. Abstract: Schubert calculus is the study of the structure of flag varieties. An example of a flag variety is the set of k-dimensional subspaces of a vector space (when k=1, this gives us projective space). The structure of flag varieties is intimately connected to the combinatorics of symmetric functions, and Schubert calculus is the mechanics for doing computations in flag varieties with polynomials. This course will be a survey of how we play this game.

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Talks of Arun Ram in 2017

1. Are there Symmetric group crystals? , invited speaker at the conference Representation Theory of Symmetric Groups and Related Algebras, Institute for Mathematical Sciences, National University of Singpore, 11-20 Dec 2017. (video on YouTube) (pdf slides)
   (talk notes short version) (talk notes with diagrams) (Towards B⊗B⊗B) (B⊗B⊗B computation) (badnews proof and computations)
   Abstract: Several recent conversations about the Kronecker problem (positive combinatorial expressions for the decomposition of tensor products of irreducible symmetric group representation in characteristic 0) made me feel that it would be useful to think about the possibility of a theory of crystals for symmetric group representations (a graph based categorification of the character ring of the symmetric group). Initial explorations have been fascinating. I will give a summary of what I have learned about what such a theory would need to look like.
   
   
2. Combinatorics of level 0 representations, invited speaker at the conference Future Directions in Representation Theory, University of Sydney, 5-8 December 2017.
   Abstract: Recent work of Kato-Naito-Sagaki-Feigin-Makyedonskyi has provided an improved understanding of the combinatorics of integrable level 0 representations of the affine Lie algebra. In particular, there is a connection to the Schubert calculus of the semi-infinite flag variety from which Kato-Naito-Sagaki prove an analogue of the Pieri-Chevalley formula for the semi-infinite flag variety (line bundle multiplied with a Schubert class). The connection to crystals in this new formula (analogous to that in the Pieri-Chevalley formula of Pittie-Ram) provides a geometric interpretation of (a part of) the path model which was used by Ram-Yip to give a formula for Macdonald polynomials.
   
   
3. Combinatorics of level 0 representations, Algebra and Topology seminar, Australian National University, 28 November 2017.
   Abstract: Recent work of Kato and Kato-Naito-Sagaki has provided an improved understanding of the combinatorics of integrable level 0 representations of the affine Lie algebra. In particular, there is a connection to the Schubert calculus of the semi-infinite flag variety from which Kato-Naito-Sagaki prove an analogue of the Pieri-Chevalley formula for the semi-infinite flag variety (line bundle multiplied with a Schubert class). The connection to crystals in this new formula (analogous to that in the Pieri-Chevalley formula of Pittie-Ram) provides a geometric interpretation of (a part of) the path model which was used by Ram-Yip to give a formula for Macdonald polynomials.
   
   
4. Combinatorics of semi-infinite flag varieties, invited speaker at the conference International Festival in Schubert Calculus, Sun-Yat Sen University, Guangzhou China, 5-11 November 2017. See http://math.sysu.edu.cn/Files/ArticleFiles/20171103/02a46408-0f1d-4d73-a492-24afa953f412.pdf
   Abstract: I will discuss the path model/alcove walk model for the semi-infinite flag variety. This model has connections to the path model formula for Macdonald polynomials and to the Littelmann path model for representations of affine Lie algebras at level 0.
   
   
5. TOYKAMP, short presentation for the School Review, School of Mathematics and Statistics, University of Melbourne, 1 November 2017.
   Abstract: TOYKAMP is a pure math playgroup that meets weekly on Thursday afternoons. We do weaving (Fine Springer Fibres to make Higgs Bundles) and box building (black boxes to put the fibres and bundles in to make Modules). The participants are usually local, but we have some Commuting Operators who come in to run the black box modules and navigate Spectral Curves with them. Our systems are often Integrable into a larger mathematics department (if the bundle is sufficiently ample). Everyone is welcome.
   
   
6. Fusion following Kazhdan-Lusztig, contributed talk at the conference Tensor Categories and Field Theory, University of Melbourne 5-9 June 2017.
   Abstract: In recent discussions with S. Kanade, K. Kawasetsu and D. Ridout we have been trying to understand what fusion products are and how to compute them, following work of Gaberdiel, Kausch, Nahm, Huang, Lepowsky, Zhang, Li, Tsuchiya, Miyamoto and many others. I will outline the rigorous approach of Kazhdan-Lusztig from Tensor structures arising from affine Lie algebras I, J. Amer. Math. Soc. 1993.
   
   
7. Mendelssohn Salon 1828; Elliptic functions - Kosmos - Beethoven Sonata in A flat Major Opus 110, written and performed by Michael Leslie (Richard-Strauss-Konservatorium, Munich) and Arun Ram (University of Melbourne) Wyselaskie Auditorium, Parkville, Melbourne, Australia, 20 July 2017.
   Abstract: Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
   
   
8. Level 0 representations and Macdonald polynomials, Mathematical Physics Seminar, University of Melbourne, 16 May 2017.
   Abstract: I will describe the passage from extremal weight modules, to global Weyl modules and then to local Weyl modules (for affine algebras and currents). Then I will explain how characters of these modules are (a specialisation) of Macdonald polynomials. If time permits I will outline why these level 0 modules are fundamental to some conformal field theory constructions (Fock spaces, Khnizhnik-Zamolodchikov, R-matrices, and energy functions).
   
   
9. R-matrices, elliptic stable envelopes, Schubert calculus, Working seminar, University of Melbourne 10 April 2017.
   Abstract: This talk provided some summary of the recent paper by G. Felder, R. Rimanyi and A. Varchenko entitled Elliptic dynamical quantum groups and equivariant elliptic cohomology
   
   
10. Niveaux de la forêt cristal, Colloquium, University of Reims, 21 March 2017.
    Abstract: La forêt de cristal affine est mystérieuse, traversée par de nombreux chemins d'un homme petit (Littelmann). Les chemins montent aux niveaux positifs et descendent aux niveaux négatifs. À chaque endroit de la forêt on découvre les traces des groupes des chats de mauvaise humeur (Kac-Moody groups), mais les groupes des chats de mauvaise humeur sont toujours cachés dans l'ombre. Caché ... et voilà! (Kashiwara) "tien le saké!" (Tanisaki) ... Les japonais habitent aussi les tunnels au niveau zéro: cas d'eaux (Kato) japonaises, les tunnels sont d'ici au semi-infini.
    
    
11. A Fock space plan, Invited speaker, Workshop on Asymptotic Representation Theory, part of the trimester on Combinatorics and Interactions, Institut Henri Poincaré, Paris 23 February 2017.
    Abstract: This talk has two goals: (a) to explain the construction of 'abstract Fock space' following the recent paper of Lanini-Ram-Sobaje arXiv1612.03120, (b) to outline the relationship of Fock space to symmetrisation/antisymmetrization, Boson-Fermion correspondence, Brownian motion to Whittaker process, and geometric Langlands.
    
    
12. Really cool numbers and Research methodology through Berlioz, Special lecture, University of Melbourne 18 January 2017.
    Abstract: This was a lecture for High School students (Year 9/10) attending the "Melbourne Summer School 2017", a program run by the MGSE (Melbourne Graduate School of Education) specifically for the Master of Teaching (Secondary) Internship program (this is different to the "VCE Summer School (VCESS)). Many thanks to interns Andres Alzate and Andrew Jacobs who organised and invited me to give this lecture. The goal was to provide the students a "University experience" in a lecture theatre and discussion of what a professional mathematician does.
    
    

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Talks of Arun Ram in 2016

1. What is fusion of representations, Working seminar, University of Melbourne 4 October 2016.
   Abstract: This will be a survey of Kazhdan-Lusztig's paper "Tensor Structures on Affine Lie algebras I", J. Amer. Math. Soc. 1993, where the notion of fusion of representations (as used in representations of affine Lie algebras and conformal field theory) is put on rigorous mathematical footing in 3 different points of view.
2. Vertex algebras, Associative algebras and Poisson algebras, Working seminar, University of Melbourne 24 August 2016.
   Abstract: This talk is a summary of the definitions of Vertex algebras, associative filtered algebras and Poisson algebras, and the Zhu functors between them. The content was mostly taken from Arakawa's paper "Rationality of W-algebras: principal nilpotent cases" arXiv:1211.7124.
3. Alcove walks and the Peterson isomorphism, Banff conference on Whittaker Functions: Number Theory, Geometry and Physics (16w5039), BIRS Banff Canada Banff conference on Whittaker Functions: Number Theory, Geometry and Physics (16w5039), BIRS Banff Canada 24-29 July 2016.
   Abstract: I will describe a combinatorial (alcove walk) labelling of the points of the moduli space of curves (of genus 0) in the flag variety. The idea is that this geometric labelling can explain the sometimes magical "quantum to affine" phenomena relating quantum cohomology of the flag variety to the cohomology of the affine Grassmannian. This is joint work with Liz Milicevic.
4. A combinatorial gadget for decompositions numbers for quantum groups at roots of unity , invited speaker at the conference Representation Theory and Physics, University of Leeds, 18-22 July 2016.
   Abstract: This is joint work with Martina Lanini and Paul Sobaje in which we produce a generalization of the q-Fock space (as used, for example, by Ariki, Lascoux-Leclerc-Thibon, Hayashi, Misra-Miwa, Kashiwara-Miwa-Stern) to all Lie types. This gadget captures the decomposition numbers of standard modules for representations of quantum groups at roots of unity in the same way that the usual q-Fock space does for type A. In classical type, via Schur-Weyl duality, it will also see the decomposition numbers of affine BMW algebras in the same way that the usual q-Fock space does for the affine Hecke algebras of type A.
5. The door is ajar, Concluding speaker, Strengthening Engagement and Achievement in Mathematics and Science (SEAMS) Senior Residential Camp, 29 June - 1 July 2016, Melbourne University.
   Abstract: The door is ajar for the next generation of Mathematicians and Scientists.
6. Combinatorics of affine Springer fibres, Algebra seminar, Sydney University 17 June 2016.
   Abstract: This talk will be a survey of the relation between affine Springer fibres and representations of the double affine Hecke algebra. I will likely focus on a favourite example of the elliptic homogeneous case where I can draw a nice picture illustrating how the affine Springer fibre is decomposed into cells indexed by connected components of complement of a hyperplane arrangement called the Shi arrangement (the same one that appears in the K-theory version of the Chevalley-Shephard-Todd theorem for reflection groups). These regions then correspond to a Macdonald polynomial basis of the corresponding representation of the double affine Hecke algebra.
7. Combinatorics of representations of affine Lie algebras, Algebra seminar, Sydney University 16 June 2016.
   Abstract: This will be a survey of my current understanding of the combinatorial representation theory of affine Lie algebras. For category O at negative level, Verma modules have finite composition series with decomposition numbers determined by Kazhdan-Lusztig polynomials. The structure of affine Weyl group orbits controls the pretty patterns. For category O at positive level, Verma modules have infinite compositions with decomposition numbers given by inverse Kazhdan-Lusztig polynomials, and at critical level, the patterns correspond to the periodic Kazhdan-Lusztig polynomials. I’ll also discuss parabolic category O. Finite dimensional modules (which are level 0) are indexed by Drinfeld polynomials and then there are various collections of smooth representations where our combinatorial understanding has increased greatly in recent years.
8. Combinatorics of the loop Grassmannian, Algebra seminar, Sydney University 14 June 2016.
   Abstract: I will explain what the loop Grassmannian and the affine flag variety are and how to label their points. This labelling is a refinement of the labelling of crystal bases by Littelmann paths. I’ll show the picture which summarises the connection to the affine Hecke algebra and the spherical affine Hecke algebra. I’ll give a summary of the relationship between Mirkovic-Vilonen cycles and the crystal bases and explain how this is reflected in the path model indexing.
9. The BRST Complex and quantised Hamiltonian reduction, Working seminar, University of Melbourne, 16 May 2016.
   Abstract: This will be a description of the Kac-Roan-Wakimoto construction of the BRST complex. This process, often called quantised Hamiltonian reduction produces a chain complex generated by charged free fermions, neutral free fermions, currents. The cohomology of this complex produces vertex algebras called W-algebras (whose representations correspond to minimal models from conformal field theory).
10. Does BRST reduction produce branes?, MUMS seminar, University of Melbourne, 13 May 2016.
    Abstract: This talk will be a story about a recent experience in my role as a professional mathematician. The experience drew me into a new field with new collaborators and new project to work on. I will describe how this came about, what the process was, and how one learns and expands into new subjects and directions, in this case, String Theory.
11. Parking functions, the Shi arrangement and Macdonald polynomials, Philadelphia Area Combinatorics and Algebraic Geometry seminar, Drexel University 3 May 2016.
    Abstract: A parking function is a sequence (b₁, ..., b_n) of positive integers which, when rearranged in increasing order (a₁ ≤ a₂ ≤ ... ≤ a_n), is such that a_i≤ i. I will first convert parking functions to elements of the affine Weyl group which correspond to regions of the Shi hyperplane arrangement and bases of a module for the rational Cherednik algebra (or double affine Hecke algebra). As explained, for example, in papers of Varagnolo-Vasserot and Oblomkov-Yun, this module can be realized as the cohomology (or K-theory) of an affine Springer fiber. These bases are closely connected to Macdonald polynomials. Goresky-Kotwitz-Macpherson explain how to chop up the affine Springer fiber into tractable pieces indexed by the (generalised) parking functions (paving by Hessenbergs). I'll start by drawing the pictures and then explain how to read the connections off the picture.
12. The Geometric Peterson isomorphism, University of Ottawa, Workshop on Equivariant generalized Schubert calculus and its applications, 30 April 2016.
    Abstract: I will describe a combinatorial (alcove walk) labelling of the points of the moduli space of curves (of genus 0) in the flag variety. The idea is that this geometric labelling can explain the sometimes magical "quantum to affine" phenomena relating quantum cohomology of the flag variety to the cohomology of the affine Grassmannian. This is joint work with Liz Milicevic.
13. A combinatorial gadget for decompositions numbers for quantum groups at roots of unity, CUNY Representation Theory seminar, CUNY Graduate Center, New York, 29 April 2016.
    Abstract: This is joint work with Martina Lanini and Paul Sobaje in which we produce a generalization of the q-Fock space (as used, for example, by Ariki, Lascoux-Leclerc-Thibon, Hayashi, Misra-Miwa, Kashiwara-Miwa-Stern) to all Lie types. This gadget captures the decomposition numbers of standard modules for representations of quantum groups at roots of unity in the same way that the usual q-Fock space does for type A. In classcal type, via Schur-Weyl duality, it will also see the decomposition numbers of affine BMW algebras in the same way that the usual q-Fock space does for affine Hecke algebras of type A.
14. Double affine Hecke algebras, Working seminar, University of Melbourne 8 April 2016.
    Abstract: This will an introductory talk about the double affine Hecke algebra and the algebraic and geometric construction of its "polynomial representation".
15. Hitchin Fibers, Higgs bundles and Springer fibers, Working seminar, University of Melbourne 8 April 2016.
    Abstract: This will be a summary of the section 6 of the paper of Oblomkov-Yun arxiv:1407.5685 which explains the relationship between affine Springer fibers and homogeneous Hitchin fibres. This relationship is a key part of the setup of Ngo in his proof of Langlands “Fundamental Lemma”.
16. Affine Springer fibers, Summer working seminar, University of Melbourne 12 January 2016.
    Abstract: In a paper of Goresky-Kottwitz-Macpherson they explain how to view many affine Springer fibers as towers with a Hessenberg foundation. I will hopefully explain how they use the Moy-Prasad filtration to do this.
17. Cohomology of flag varieties, Summer working seminar, University of Melbourne 12 January 2016.
    Abstract: In a recent paper of Oblomkov-Yun they view the cohomology of the affine flag variety as a module for a Hecke algebra. I will survey their paper.
18. Curves in flag varieties, Summer working seminar, University of Melbourne 5 January 2016.
    Abstract: In his 1997 unpublished notes setting up his description of the quantum cohomology of flag varieties, D. Peterson described a beautiful correspondence between curves in flag varieties and points of the affine flag variety. I will try to give an interpretation (and some examples) of this correspondence in terms of alcove walks.

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Talks of Arun Ram in 2015

1. Picturing representation rings, Talk at the conference Geometric and categorical representation theory, Mooloolaba Queensland, 17 December 2015.
   Abstract: The goal is to provide pictures, yes PICTURES, for navigating the Grothendieck ring of category O for the affine Lie algebras. The representations are divided into levels (by the action of the central element) and blocks (by the dot action of the affine Weyl group) and are connected together by Kazhdan-Lusztig polynomials for the affine Weyl group (via the affine Hecke algebra). The negative level representations with integral highest weight correspond (by a Khnizhnik-Zamolodchikov functor) to representations of quantum groups at a root of unity, and this part of the Grothendieck ring appears, in type A, in a different incarnation, as a Fock space of partitions (boxes in a corner). We explain how the Fock space generalises to all affine Kac-Moody Lie algebras. This is joint work with Martina Lanini and Paul Sobaje.
2. Parking functions and the Shi arrangement, Summer working seminar, University of Melbourne 7 December 2015.
   Abstract: A parking function is a sequence (b₁, ..., b_n) of positive integers which, when rearranged in increasing order (a₁ ≤ a₂ ≤ ... ≤ a_n), is such that a_i≤ i. Persi has raised the question of random walks on parking functions, and for computing statistics for various possible distributions on parking functions. In order to provide some tools for solving this, I want to convert parking functions to elements of the affine Weyl group which correspond to regions of the Shi hyperplane arrangement and bases of a module for the rational Cherednik algebra (or double affine Hecke algebra). As explained, for example, by a recent paper of Oblomkov-Yun, this module can be realized as the cohomology of an affine Springer fiber. I'll start by drawing all the pictures and then explaining how to read all the connections off the picture.
3. Picturing the flag variety, Talk at the Victorian Algebra Conference 2015, University of Western Sydney, Paramatta, 30 November 2015.
   Abstract: What is the flag variety? A lattice? A projective space? The Veblen-Young theorem? Cosets? Subspaces? A building? A simplicial complex? A generalised generalised quadrangle? An incidence geometry? A Chevalley group? Borel subalgebras? Borel subgroups? The LUP factorisation? The Bruhat decomposition? The Schubert calculus? A polarization? A twisted Chevalley group? An isotropic subsapce? An ovoid? A finite simple group? Perhaps the right answer is the title of Wassily Kandinsky’s book, POINT AND LINE TO PLANE.
4. Some projects to work on: Fusion, Elliptic, glass beads TQFTs, moduli spaces, Hecke algebras, etc, Summer working seminar, University of Melbourne 29 October 2015.
   Abstract: This goal here was to produce a list of possible projects for Vacation scholars, Masters students, PhD students, for postdocs, for research careers and for grant proposals. A problem was only considered suitable for the list if it could be naturally shaped for any one of these purposes.
5. Unitary, Orthogonal and Symplectic groups, Summer working seminar, University of Melbourne 27 October 2015.
   Abstract: This talk was an effort to unify approaches to these groups: (1) exponentiation from the Lie algebra, (2) as twisted Chevalley groups of invariants of general linear groups with respect to an involution, (3) as ordinary Chevalley groups coming from a root system, (4) as given by generators and relations as found in Section 10 of Bruhat-Tits paper "Groupes reductifs sur un corp local I".
6. The glass bead game, Public Lecture as part of the BrisScience series at The EDGE, Brisbane, 7 July 2015.
   Abstract: This talk will take a virtual tour of the toy store with our friends Maria Callas, Alexander Grothendieck and Hermann Hesse. There are pleasant games with glass beads, athletic games skiing the moguls, and violent games where everything gets smashed. There are crystals and hurricanes and, of course, a few polynomials. The point of the talk is to tell some stories related to current research in symmetry.
7. Views from Castalia, Guest lecture at the AMSI Winter School, University of Queensland, 7 July 2015.
   Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
8. Theta functions as matrix coefficients for Heisenberg representations, Number theory seminar, University of Melbourne, 16 March 2015.
   Abstract: This talk is the third of a series of working seminar talks with the aim of reading the first few sections of Mumford's Tata Lectures on Theta III.
9. Heisenberg groups, abelian varieties and theta functions, Number theory seminar, University of Melbourne 9 March 2015.
   Abstract: This talk is the second of a series of working seminar talks with the aim of reading the first few sections of Mumford's Tata Lectures on Theta III.
10. Two boundary Hecke and Temperley-Lieb algebras, Pure Math Seminar, University of Melbourne 6 March 2015.
    Abstract: This talk will be a survey of recent work with Z. Daugherty. I will introduce the two pole braid group, the two pole Hecke algebra, and the two pole Temperley-Lieb algebra. I will explain the connection to the affine Hecke algebra of type C and the indexing of irreducible calibrated representations. Then I will explain how to get actions on tensor space from R-matrices and give descriptions of the irreducible representations which appear this way. As a consequence one obtains some understanding of the answers to some questions and conjectures of de Gier and Nichols. A preprint with the first part of this story is now available at http://researchers.ms.unimelb.edu.au/~aram@unimelb/preprints.html
11. The glass bead game, Talks for the Melbourne Universty Mathematics Students Society MUMS, University of Melbourne, 6 March 2015.
    Abstract: This talk will take a virtual tour of the toy store with our friends Maria Callas, Alexander Grothendieck and Hermann Hesse. There are pleasant games with glass beads, athletic games skiing the moguls, and violent games where everything gets smashed. There are crystals and hurricanes and, of course, a few polynomials. The point of the talk is to tell some stories related to the topic of my research.
12. Heisenberg groups and representations, Number theory seminar, University of Melbourne 2 March 2015.
    Abstract: This talk is the first of a series of working seminar talks with the aim of reading the first few sections of Mumford's Tata Lectures on Theta III.

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Talks of Arun Ram in 2014

1. The Peterson isomorphism between quantum cohomology and affine Schubert calculus for flag varieties, Moduli spaces seminar, University of Melbourne 28 November 2014.
   Abstract: Lam and Shimozono, in the paper arXiv:0705.1386, say "We show that the quantum cohomology of a flag variety is a quotient of the homology of the loop Grassmannian after localization and describe the map explicitly on the level of Schubert classes. As a consequence, all three-point genus zero Gromov-Witten invariants of G/P are identified with homology Schubert structure constants of the loop Grassmannian, establishing the equivalence of the quantum and homology affine Schubert calculi. This is an unpublished result stated by Dale Peterson in 1997." Allen Knutson talks about the Peterson isomorphism at https://plus.google.com/+AllenKnutson/posts/Dz6eTTtG3ML When I looked at Peterson notes (which I have had typed and put on the resources link from my web page), I was amazed to see an explicit bijection between the moduli spaces, essentially the same as that used in the form of "quasimap spaces" by Braverman and coauthors. This renders the "mysterious Peterson isomorphism" completely unmysterious. I will see what I can do in the talk to present this bijection between the moduli spaces.
2. Heisenberg groups and Theta functions, Three talks in the Number Theory seminar, University of Melbourne 7, 19 and 24 November 2014.
   Abstract: These talks were a survey of Chapter 1 of Igusa's book on Theta function. We introduced the Heisenberg group, the Schrodinger representation, discussed the symplectic group Sp_2n(R), its maximal compact subgroup, the Siegel upper half space, the integral symplectic group, the normalizer of the Heisenberg group in Aut(L²(R)), its maximal compact subgroup, the favourite discrete subgroup arising as normalizer of a lattice, and the expression of periodic functions under a lattice in terms of theta functions of characteristic m,m^* and modulus τ.
3. Two boundary Hecke algebras and Schur-Weyl duality, Talk at the Representation Theory conference/retreat at the ANU coastal campus at Kioloa, organised by Scott Morrison and Tony Licata, 12 November 2014.
   Abstract: This was a summary of the forthcoming paper with Zajj Daugherty.
4. Affine and degenerate affine BMW algebras, Talk at the Workshop on Diagram Algebras, University of Stuttgart, Germany, 8-12 September 2014.
   Abstract: This talk will be a survey of recent and forthcoming papers with Zajj Daugherty and Rahbar Virk. In this work the affine and degenerate affine BMW algebras are completely parallel: with parallel presentations, parallel structure and parallel representation theory. These algebras act on tensor space and are in Schur-Weyl duality with orthogonal and symplectic groups and quantum groups. The centers of these algebras are described by symmetric functions with a cancellation property as in the description of the cohomology and K-theory of symplectic and orthogonal Grassmannians. The representation theory of these algebras can be described by multisegments and the decomposition numbers are given by Kazhdan-Lusztig polynomials.
5. The glass bead game, Public lecture at ICMS as part of the workshop Algebraic Lie theory and representation theory, International Centre for Mathematical Sciences, Edinburgh, Scotland, 3 September 2014.
   Abstract: This talk will take a virtual tour of the toy store with our friends Maria Callas, Alexander Grothendieck and Hermann Hesse. There are pleasant games with glass beads, athletic games skiing the moguls, and violent games where everything gets smashed. There are crystals and hurricanes and, of course, a few polynomials. The point of the talk is to tell some stories related to the topic of the concurrent ICMS research workshop on Algebraic Lie theory.
6. Generalized Fock space for decomposition numbers for quantum groups at roots of unity, Invited talk at the ICM 2014 Satellite Conference on Representation Theory and Related Topics, 6-9 August 2014, EXCO, Deagu, South Korea
   Abstract: In this work we provide a generalization of the Type A Fock space. This provides a combinatorially constructed "vector space" with a bar involution such that the bar-invariant basis captures the decomposition numbers of Weyl modules for quantum group (of a finite dimensional complex semisimple Lie algebra) at a root of unity. This is joint work with Martina Lanini and Paul Sobaje.
7. Double affine Artin groups and the modular group (or, more to the point, Did you know there are bugs on your donut?), Algebra seminar at the University of North Texas, Denton, June 16, 2014.
   Abstract: The double affine Artin group is used to construct the double affine Hecke algebra which is used to construct Macdonald polynomials. In this work we analyse the topological construction of the double affine Artin group to uncover the underlying elliptic curves and the action of the modular group. This is joint work with Z. Daugherty and S. Griffeth. (Note: Double affine Hecke algebras are undegenerated symplectic reflection algebras and Macdonald polynomials are generalized spherical and Whittaker functions for p-adic groups).
8. Lyndon words and the loop Grassmannian, Workshop on Categorification and geometric representation theory, at the Thematic semester: New Directions in Lie Theory June 9-13, 2014.
   Abstract: In the representation theory of KLR algebras, the cuspidal modules are indexed by the good Lyndon word giving the maximal weight, and the simple modules are heads of the standard modules (shuffle products of cuspidal modules). This is a categorification of the relation between the PBW basis and the canonical basis of the quantum group. In this work we analyse how a similar phenomenon appears inside the loop Grassmannian. The key which makes it work is that the generators of the group are exponentials of the generators of the Lie algebra, and the combinatorics of the Lie algebra (which produces the Lyndon words) is reflected in the Steinberg-Tits relations for the Chevalley group. This enables one to use Lyndon words to control the combinatorics of expressions in the Chevalley group. This is joint work with A. Ghitza and S. Kannan.
9. Are 3-pole braids elliptic?, Colloquium, Dartmouth College, 26 April 2013.
   Abstract: The goal of this work, partly joint with Zajj Daugherty and Stephen Griffeth, is to understand the source and implications of the action of the modular group on the double affine braid group. The example of the double affine braid group of type C_n has a pleasant presentation as braids with n strands and three poles, and the modular group moves the poles around. To gain understanding of our objects it is enlightening to draw lots of fun pictures of braids. PRETEND THERE IS A PICTURE OF A 3-POLE BRAID HERE.
10. Are 3-pole braids elliptic?, Workshop on Combinatorial Representation Theory, at the Thematic semester: New Directions in Lie Theory April 21-25, 2014.
    Abstract: The goal of this work, partly joint with Zajj Daugherty and Stephen Griffeth, is to understand the source and implications of the action of the modular group on the double affine braid group. The example of the double affine braid group of type C_n has a pleasant presentation as braids with n strands and three poles, and the modular group moves the poles around. To gain understanding of our objects it is enlightening to draw lots of fun pictures of braids. PRETEND THERE IS A PICTURE OF A 3-POLE BRAID HERE. The modular group action on the double affine braid group should be of great importance in the study of Macdonald polynomials but has not yet become a familiar object in the Macdonald polynomial theory. In his PhD thesis, van der Lek gave three presentations of the double affine braid group, and these were followed by the presentations of Cherednik, Ion-Sahi and Haiman. In this work, we compare these presentations, analyze their relationship to the family of abelian varieties that appear in the character theory of affine Lie algebras, and do some explicit examples.
11. Artin groups and affine Artin groups related to Coxeter groups, Talk, Talk preparation, Deformation retract, Notes from an old seminar on 14.10.2013, Coxeter groups seminar, University of Melbourne, 11 April 2014.
    Abstract: This was a survey of the results of the first three chapters of van der Lek's PhD thesis, focusing on the construction of the configuration spaces which have as fundamental group the Artin group and the affine Artin group corresponding to a Coxeter group W.
12. Introduction to Elliptic functions, Talk 1, Talk 2, Talk 3, Number theory seminar, University of Melbourne, 7,14 and 21 March 2014.
    Abstract: This was a leisurely 3-talk introduction to elliptic functions and modular forms: the analogy to trigonometry, theta functions, the Weierstrass P-function, Eisenstein series, elliptic curves, the moduli space of elliptic curves, abelian varieties, and the moduli space of elliptic curves. The punchline was that elliptic functions are global sections of line bundles on elliptic curves and modular forms are global sections of line bundles on the moduli space of elliptic curves.
13. Proof Machine, MUMS Seminar, University of Melbourne, 7 March 2014.
    Abstract: There is one tool that has saved my mathematical confidence (and career) more times than any other, my Proof Machine. In this talk I will discuss the theory behind how (and why) it works, and power it up and show how it works on a few examples. With a little practice wielding the hammer anyone can use the Proof Machine to build rock solid proofs, both within mathematics and also outside mathematics.
14. Proof Machine, Department Seminar, University of South Australia, 14 February 2014.
    Abstract: There is one tool that has saved my mathematical confidence (and career) more times than any other, my Proof Machine. In this talk I will discuss the theory behind how (and why) it works, and power it up and show how it works on a few examples. With a little practice wielding the hammer anyone can use the Proof Machine to build rock solid proofs, both within mathematics and also outside mathematics.

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Talks of Arun Ram in 2013

1. Fock space and representations of quantum groups, Taipei Conference on Representation Theory IV, Insitute of Mathematics, Academica Sinica, 20-23 December 2013.
   Abstract: The q-deformed Fock space has several incarnations: Hayashi's formulation via spin and oscillator representations, or via semiinfinite wedges, or by a quotient of a polynomial representation as in Leclerc-Thibon. Misra and Miwa made it a representation of quantum affine sl(l) and Kashiwara-Miwa-Stern provided an intimate relationship with the affine Hecke algebra of type A. Ariki used it in his study of the representations of cyclotomic Hecke algebras at roots of unity, and it has played an important role in much further work, notably the works of Uglov and Leclerc-Thibon. All this is mostly for type A. In this work, joint with Martina Lanini and Paul Sobaje, we use affine Hecke algebras to generalise this construction to all Lie types, in such a way that we produce a good model for the (graded) Grothendieck group of finite dimensional (and category O) representations of quantum groups at roots of unity. Our construction makes clear the relationship to the periodic module introduced in Lusztig's work on Jantzen's generic decomposition patterns, the inverse Kazhdan-Lusztig polynomials for affine Weyl groups, the Steinberg tensor product theorem, and the Kazhdan-Lusztig conjectures for affine Lie algebras at negative integer level (as proved in the work of Kashiwara-Tanisaki). The result is a combinatorial model for working with the decomposition numbers for quantum groups at roots of unity in general type, completely analogous to the Fock space construction ubiquitous to the type A case.
2. What is a Schubert polynomial?, Pure Math Seminar, University of Queensland, 26 November 2013.
   Abstract: A Schubert polynomial is what comes out of the black box called "cohomology" when you put a Schubert variety in. As technology gets more advanced generalised cohomologies (say T-equivariant elliptic cohomology) produce new Schubert polynomials, ... or do they? It seems that the connecting cable for the actual Schubert varieties doesn't quite fit the socket in the new cohomologies and the old workaround of replacing the input by a Bott-Samelson variety no longer produces the right answer. But the new cohomologies definitely have some killer apps so it's hard to resist them.
3. Views from Castalia, Insitute for Basic Sciences Symposium Representation Theory and Algebraic Structures, Seoul Korea, 7 August 2013.
   Abstract: The title of this talk comes from the novel "The Glass Bead Game" by Hermann Hesse. The setting of the story is a serene location in the mountains called Castalia. There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
4. Views from Castalia, Algebra seminar, University of Sydney, 12 July, 2013.
   Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
5. Alcove walks and Mirkovic-Vilonen cycles, Number theory seminar, University of Melbourne, 10 July 2013.
   Abstract: In this talk I gave some background on the relation between characters of irreducible representations of the Langlands dual group and Mirkovic-Vilonen cycles in the loop Grassmanian. The method of Gaussent-Littelmann provides a construction of the MV-cycles using the combinatorics of alcove walks. I illustrated this method with the computation of the 27 MV-cycles corresponding to the irreducible representation of PGL_3(ℂ) with highest weight 2α₁+2α₂.
6. Combinatorics, representations, homgeneous spaces and elliptic cohomology, Plenary lecture at PRIMA 2013 Pacific Rim Mathematical Association Congress, 24-28 June 2013, Xuhui (downtown) campus of Shanghai Jiao Tong University.
   Abstract: This talk will give an elementary survey of recent developments in combinatorial representation theory, where input from stable homotopy theory and algebraic topology enters. The theory of p-compact groups founded by Dwyer and Wilkerson gives rise to homogeneous spaces that generalize the combinatorics of Lie theory, and the framework of generalized cohomology theories (elliptic cohomology and cobordism) provides new insight into classical intersection theory computations for Grassmannians and flag varieties. In both cases, the study of possible generalizations of the Weyl character formula, and their relation to Atiyah-Bott localization formulas provides useful motivation and insight. The combinatorial aspect provides powerful elementary models for all of these objects.
7. Combinatorics, Representations and elliptic cohomology, Colloquium at Stanford University, Stanford, California USA, 9 May 2013.
   Abstract: This talk will summarize generalizations of the Weyl character formula to spherical functions and Whittaker functions (for groups over local fields), to Macdonald polynomials (generalizations of Askey-Wilson polynomials) and to affine Lie algebras (using theta functions on (very special) abelian varieties). This leads to the combinatorial study of the equivariant elliptic cohomology (and equivariant cobordism) of flag varieties.
8. A probabilistic interpretation of the Macdonald polynomials, talk in the RTGC seminar at University of California, Berkeley, 8 May 2013.
   Abstract: P. Hanlon studied a random walk on partitions that has the Jack polynomials as eigenvectors. This random walk arises from a Markov chain on permutations by "lumping". In this work (joint with Persi Diaconis) we generalise this process to a much more vigorous walk which has eigenvectors, the Macdonald polynomials.
9. Combinatorial representation theory and elliptic cohomology, Colloquium at University of Oregon, Eugene, Oregon USA, 6 May 2013.
   Abstract: On passing through a pedestrian underpass near the train station in Bonn one might notice graffiti declaring passion between CRT and EC and wonder and speculate about what kind of relationship this is. Actually this graffiti was put there late one night by a mathematician visiting the Hausdorff Institute of Mathematics. Indeed this is a soap opera of the best sort with Combinatorial Representation Theory and Elliptic Cohomology being an uncomfortable pair from different echelons of society that, as a rule, never mix. Of course the relationship is manipulated by family (generalised cohomology theories), some friends (the Schubert classes) that were once true have now become elusive, and there are other, somewhat uncomfortable, acquaintances (abelian varieties and theta functions) who cannot seem to keep their nose out of what we think is not their business. And who is it that is spying and scheming for their own purposes? (shh...the affine Lie algebras).
10. Bringing together Demazure, Macdonald and Whittaker, talk at the research seminar of the ICERM special program in Spring 2013, Brown University, Providence, 30 April 2013.
    Abstract: I will draw my favourite "Langlands" picture and explain how this specializes to the Weyl character formula and how it generalizes to Macdonald polynomials. Using the affine Hecke algebra to unify the perspective clarifies the role of Demazure operators and provides a definition of Macdonald Whittaker functions and motivates some tantalizing positivity conjectures on (q,t)-weight multiplicities.
11. Generalized Schubert calculus, talk at special session on "???" at the Southwestern section American Mathematical Society Meeting, Boulder, Colorado USA 14-15 April 2013,
    Abstract: We show how to study the generalized cohomology (cobordism) of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the Bott-Samelson classes. We will illustrate the results by showing some rank 2 computations which generalize various earlier computations of Griffeth-Ram (in equivariant K-theory) and of Calmes-Petrov-Zainoulline (in non-equivariant cobordism).
12. Elliptic and Cobordism Schubert calculus, talk at Whittaker functions, Schubert calculus and crystals, a conference at the ICERM special program in Spring 2013, ICERM, 4-8 March, 2013, Brown University, Providence, 4 March 2013.
    Abstract: In this work we explain combinatorial methods for studying the equivariant generalised cohomology of flag varieties via the Borel model and the moment graph model. We will discuss BGG-Demazure operators and explain why Bott-Samelson classes cannot be the same as Schubert classes. As examples, we compute some Schubert products in equivariant cobordism. This is a report on joint work with Nora Ganter. In the actual talk there was not time for doing examples, instead I announced a conjectural definition of Schubert classes and gave a conjectural formula for the class of a divisor Schubert class.
13. Schubert calculus, Tutorial on Schubert calculus for the ICERM special program in Spring 2013, ICERM, Brown University, Providence, 1 March 2013.
    Abstract: This talk was an introduction to the Schubert calculus using the example of type C₂. We reviewed the Borel model and the moment graph model in the context of equivariant cohomology. We defined BGG operators and Bott-Samelson classes, and defined Schubert classes in terms of the moment graph picture using the support condition and the degree condition. We computed these in the cases of type C₂.
14. Alcove path models and Macdonald polynomials, Tutorial on Macdonald polynomials for the ICERM special program in Spring 2013, ICERM, Brown University, Providence, 25 February 2013.
    Abstract: This talk was a description of the double affine Hecke algebra, Macdonald polynomials and the path model formulas for Macdonald polynomials from the paper of Ram-Yip, A combinatorial formulas for Macdonald polynomials.
15. p-adic integrals to sums over crystals, Tutorial on Crystals for the ICERM special program in Spring 2013, ICERM, Brown University, Providence, 18 February 2013.
    Abstract: This talk was a description of the main results, and the extended example in the paper of Parkinson-Ram-Schwer Combinatorics in affine flag varieties. I described, in detail, the indexing of points in the affine flag variety by (Littelmann) paths with additional labels. The focus was on the application of these tools to the p-adic integrals that arise in the local representation theory of the Langlands program.

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Talks of Arun Ram in 2012

1. Modular forms, Hecke algebras and G₂ flags, Number Theory seminar, University of Melbourne, 18 December 2012.
   Abstract: In this talk, we reviewed the relationship between modular forms and spaces of intertwining operators between induced representations and the associated Hecke algebra actions. The second part of the talk was focused on providing representatives for the (G(Q), K) double cosets in G(A), where A is the field of finite adeles, in terms of partial flag varieties. We used G₂ as the primary example, decribing null flags in the 7-dimensional representation. Everything followed Gross and the PhD thesis of Pollack.
2. The Octonions and G₂, Number Theory seminar, University of Melbourne, 7 November 2012.
   Abstract: This talk was an attempt to begin to understand the construction of the integral form of G₂ for which G₂(ℝ) is compact. This group is fundamental to the study of the algebraic modular forms for the G₂, as defined by Gross. Preliminary work on decomposing the space of algebraic modular forms as a Hecke module for this case has i been done by Pollack and Lansky-Pollack. The group is constructed as the automorphism group of a maximal order in the octonions.
3. Clifford Theory and Hecke algebras, Groups and Combinatorics Seminar, University of Western Australia, 23 October 2012.
   Abstract: The usual Clifford theory describes the irreducible representations of group G in terms of those of a normal subgroup. Generalizing, Clifford theory constructs the irreducible representations of semidirect product rings and invariant rings. In this work with Z. Daugherty we use Clifford theory to index the irreducible representations of two pole Hecke algebras and relate this indexing to a labeling coming from statistical mechanics (following work of de Gier and Nichols) and to a geometric labeling (coming from K-theory of Steinberg varieties following Kazhdan-Lusztig). Despite the maths-physics and geometric motivations for the project, in the talk I shall assume only that the audience is familiar with the notions of groups, rings, and modules.
4. Soergel bimodules and Rouquier complexes, Algebra-Geometry-Topology seminar, University of Melbourne, 19 October 2012.
   Abstract: Soergel bimodules are the primary tools in the recently announced proof by Elias-Williamson of the positivity of the coefficients of Kazhdan-Lusztig polynomials for Coxeter groups. I will define these and explore some examples. Time permitting, I will explain how these are used in Schubert calculus.
5. Flags, Cohomology and Positivity, MUMS (Melbourne University Mathematics Society) lecture, University of Melbourne 03 August, 2012.
   Abstract: Flags are fundamental in linear algebra. I'll explain how they relate to lower triangular, upper triangular decompositions of matrices, and provide a method for representing collections of flags in equivariant cohomology. Then we'll do a few little computations and explain a couple little questions for which we'd like to understand the answers better.
6. Combinatorial Representation Theory: Minicourse of five lectures: Representation Theory, Reflection groups and Groups of Lie type, Representations of the Symmetric group from KLR algebras, The Weyl character formula and Geometric Langlands, Crystals from paths and MV polytopes, and Crystals from KLR and preprojective algebras, XXII Brazilian Algebra meeting, 15-20 July 2012.
   Abstract: This would be an introduction to modern methods in combinatorial representation theory. The primary tool is the affine Hecke algebra, which comes in many forms: as the group algebra of the symmetric group, as a quiver Hecke algebra, as a convolution algebra of functions on a p-adic group. From this we can derive the structure and combinatorics of the cohomology of flag varieties, projective spaces and Grassmannians, and the detailed "crystal" structure of the representations of complex simple Lie algebras. I will give an introductory tour to these techniques in the interplay between combinatorics, representation theory and geometry.
   
7. FI-modules: a new approach to stability for S_n representations, Seminar, University of Melbourne, 22 June 2012.
   Abstract: This talk will be a summary of a recent paper of Church, Ellenberg and Farb, arXiv:1204.4533 FI-modules: a new approach to stability for S_n representations.
8. Schubert calculus for the affine Grassmannian, Seminar, University of Melbourne, 11 May 2012.
   Abstract: This talk will be a summary of a recent paper of Lam and Shimozono. arXiv:1105.2170, k-Double Schur functions and equivariant (co)homology of the affine Grassmannian.
9. Views from Castalia, Colloquium, University of Southern California, 27 April 2012.
   Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
10. Elliptic Schubert Calculus, Colloquium, University of California, Los Angeles, 26 April 2012.
    Abstract: Traditional Schubert calculus is the combinatorial study of intersections of Schubert varieties inside the flag variety. This is usually done by computing cup products in cohomology or equivariant cohomology. This talk is a summary of our study of how to extend the story to other equivariant cohomology theories: K-theory, elliptic cohomology and cobordism. The talk is intended to be a general audience survey: to give a feel for what flag varieties "look like", what cohomology theories do for us, and the combinatorial structure (reflection group symmetry) that makes the game go. The subject is fascinating in the confluence of different parts of mathematics: Lie groups, loop groups, symmetries of regular polytopes, and algebraic topology. This talk is based on ongoing joint work with Nora Ganter.
11. Generalized equivariant cohomology of flag varieties, Geometry/Physics seminar, Northwestern University, 24 April 2012.
    Abstract: I will review some of the Kac-Peterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, K-theory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter.
12. Views of Castalia, Geometry/Physics seminar, Northwestern University, 24 April 2012.
    Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). At three particular moments of each day (dawn, noon, dusk) the shadow of the muses on the valley below is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
13. Views from Castalia, Special Geometry seminar, University of Texas at Austin, 23 April 2012.
    Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
14. Generalized equivariant cohomology of flag varieties, Infinite dimensional algebra seminar, MIT, 20 April 2012.
    Abstract: I will review some of the Kac-Peterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, K-theory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter.
15. Schubert Calculus I, II and III, Seminar, University of Melbourne, 9, 11 and 13 April 2012.
    Abstract: Schubert calculus I -- Examples of the Borel picture for generalised cohomologies of flag varieties, Schubert calculus II -- Generalised cohomology of the Bott-Samelson map, Schubert calculus III -- BGG/Demazure operators for generalized cohomology theories: These talks are intended as a (hopefully) leisurely tour of the generalised Schubert calculus that we are setting up in our join work with Nora Ganter. In each case I will try to indicate how to work with the appropriate rings and maps, both in terms of generators and relations, and via moment graphs. Traditional Schubert calculus is the combinatorial study of intersections of Schubert varieties inside the flag variety. This is usually done by computing cup products in cohomology or equivariant cohomology. This talk is a summary of our study of how to extend the story to other equivariant cohomology theories: K-theory, elliptic cohomology and cobordism.
16. Combinatorics and growth in Chevalley groups and their representations, Connecting finite and infinite mathematics through symmetry, 1-3 February 2012, University of Wollongong, 2 February 2012.
    Abstract: I will endeavour to illustrate the ‘path model’, a powerful tool for labeling points in Chevalley groups in a way which is consistent with all finite dimensional representations (‘finite’ quotients of the group) at once. The main tools are to have a set of generators and relations which interacts in a controllable way with the reflection group geometry of the group (coming from automorphisms) and a powerful combinatorics for manipulating computations using these generators and relations.
17. Views from 20 years trekking on the LS path, CMI-IMSc Mathematics Colloquium, in celebration of the 80 birthday of CS Seshadri, 23-27 January 2012, Chennai Mathematical Institute, Chennai, 23 January 2012.
    Abstract: My travels on the LS (Lakshmibai-Seshadri) path have brought many personal realizations and its vistas have allowed me to look out over many beautiful structures. As best I can, I shall give a brief summary (1 hour instead of 20 years) including: Kashiwara crystals, combinatorial aspects of Schubert calculus, affine Hecke algebras and structure of Chevalley groups.

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Talks of Arun Ram in 2011

1. RTT algebras and Algebraic Bethe ansatz, working seminar at University of Melbourne, 6 December 2011.
   Abstract: This is a review of RTT algebras (as found in Drinfeld's ICM paper in section 10), spectral subalgebras (following Drinfeld's ICM paper section 11), lattice models (following deGier's thesis chapter 5), the source of the Yangian (following Drinfeld and Chari-Pressley) and a formulation of the algebraic Bethe ansatz (following Takhtajan and Faddeev, Kulish and Reshitikhin, Kirillov and Reshetikhin, and Nazarov and Tarasov). If time permits I will formulate the L-matrices in the Drinfeld double context (following Reshetikhin-Takhtajan-Faddeev).
2. Schubert Calculus, working seminar at University of Melbourne, 28 October 2011.
   Abstract: In connection with my work with Nora on the Elliptic Schubert Calculus I have found that I need to read two recent papers on Complex Cobordism and Schubert calculus: Calmes-Petrov-Zainoulline arXiv:0905.1341 and Kritichecko-Krishna arXiv:1104.1089 Viewing this talk as a working seminar, I will try to present some survey of the contents of these papers from my perspective.
3. An introduction to the Knizhnik-Zamolodchikov equations, working seminar at University of Melbourne, 19 September 2011.
   Abstract: The Knizhnik-Zamolodchikov connection provides a remarkable passage between representation theory, differential equations and conformal field theory. I will try to give a brief introduction to this fascinating correspondence.
4. Elliptic Schubert calculus, invited talk at Perspectives in Algebraic Lie Theory, at the Isaac Newton Institute, Cambridge, 12-16 September 2011.
   Abstract: Bernstein-Gelfand-Gelfand and Demazure operators (also called divided difference operators) are the foundation of the Schubert calculus, the study of the cohomology and K-theory of the flag variety in terms of the natural basis coming from the Schubert varieties. In this work, joint with Nora Ganter, we define elliptic (double) Schubert polynomials by using rings of theta functions. The elliptic Schubert polynomials are combinatorial realisations of the classes of Schubert varieties in equivariant elliptic cohomology. We set up the BGG calculus and the corresponding moment graph perspective for elliptic cohomology of flag varieties and make the connection to the representation theory of affine Lie algebras (following Kac-Peterson, Grojnowski, Ando).
5. Elliptic Schubert calculus, invited talk at Algebraic Cycles and the Geometry of Group Orbits, A Conference on the occasion of the 60th birthday of Peter O'Sullivan, at the Australian National University, Canberra, 2-4 September 2011.
   Abstract: Bernstein-Gelfand-Gelfand and Demazure operators (also called divided difference operators) are the foundation of the Schubert calculus, the study of the cohomology and K-theory of the flag variety in terms of the natural basis coming from the Schubert varieties. In this work, joint with Nora Ganter, we produce an analogous study of the elliptic cohomology of the flag varieties, by using rings of theta functions and the appropriate elliptic cohomology versions of the Thom isomorphism and localisation at T-fixed points. Additionally, we make the connection to the representation theory of affine Lie algebras (following Kac-Peterson, Grojnowski, Ando).
6. Cohomology of Grassmannians and isotropic Grassmanians, working seminar talk at University of Melbourne, 2 June 2011.
   Abstract: The classical example of the Schubert calculus is the case of the Grassmannian where the Schubert classes can be represented as Schur functions, and the problem is then solved with representation theory. The case of the isotropic Grassmanians (orthogonal and symplectic groups) was treated by Pragacz, making contact with the Schur Q-functions that arise in the projective representation theory of the symmetric group (and Nora's recent work). I recently understood how this generalises to p-compact groups, what the generalizations of the Schur Q-functions are and note, following Morris, an interesting connection to the evaluation of characters of the symmetric groups at r-regular conjugacy classes (the type of phenomena that Jamie is noting ought to happen in the p-compact setting). A future goal here is to get the whole picture clear also in K-theory and T-equivariant K-theory.
7. Affine Weyl group, Heisenberg groups and classifying complex reflection groups, working seminar talk at University of Melbourne, 26 May 2011.
   Abstract: I will try to outline the generalised Cartan matrix approach to the affine Weyl group, as found in Kac's book on Infinite dimensional Lie algebras, and explain its relation to the Heisenberg group, and an attempt to extend it so that it might have a chance of providing a new way to approach the classification of complex reflection groups. This is some meld of current work with Nora on elliptic stuff, and what Don Taylor and I came up with during his visit in April.
8. Heisenberg groups, abelian varieties and theta functions, working seminar talk at University of Melbourne, 12 May 2011.
9. Rank 1 reductive Lie groups, in perspective, working seminar talk at University of Melbourne, 5 May 2011.
   Abstract: I will discuss the Lie groups SU(2), Spin(3), Sp(1), SL(2), PGL(2) their relationships, and their adjoint representations in the context of the other reductive Lie groups of Lie types A,B, C and D. I'll derive the adjoint representation in terms of Hamiltonians=Quaternions, Pauli matrices and Chevalley generators.
10. The Borel-Weil-Bott Theorem, working seminar talk at University of Melbourne, 29 April 2011.
    Abstract: The lecture will essentially be a version of a lecture I gave at the WinterSchool on The interaction of Geometry and Combinatorics in Representation Theory at the Hausdorff Institute in Bonn in January. Understanding this is, hopefully, a path to a full understanding of the Weyl character formula, cohomology and K-theory for compact and p-compact groups, the elliptic cohomology of flag varieties, theta functions, Looijenga line bundles, and modular forms from abelian varieties.
11. Polytopes, shuffles, quivers and flags, seminar talk at University of Melbourne, 8 March 2011.
    Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of Braverman-Gaitsgory and Gaussent-Littelmann and Kashiwara-Saito and Kamnitzer-Baumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of Leclerc-Geiss-Schroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties.
12. A probabilistic interpretation of Macdonald polynomials , invited speaker at the Combinatorial Representation Theory day at the Leibniz Universität Hannover, 18 February, 2011.
    Abstract: P. Hanlon studied a random walk on partitions that has the Jack polynomials as eigenvectors. This random walk arises from a Markov chain on permutations by "lumping". In this work we generalise this process to a much more vigorous walk which has eigenvectors, the Macdonald polynomials.
13. Polytopes, shuffles, quivers and flags, invited speaker at the La troisième du séminaire de combinatoire énumérative et analytique at the Institut Henri Poincaré, Paris, 3 February, 2011.
    Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of Braverman-Gaitsgory and Gaussent-Littelmann and Kashiwara-Saito and Kamnitzer-Baumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of Leclerc-Geiss-Schroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties. Following my recent joint work Ghitza and S. Kannan, I will explain the purely combinatorial approach for seeing the moment polytopes and the shuffle elements.
14. Combinatorics of the flag variety: Minicourse of three lectures: Chevalley groups and Hecke algebras, Cohomology, and The Borel-Weil-Bott theorem, Hausdorff Insitute of Mathematics, special Trimester on "The interaction of geometry and combinatorics in Representation Theory", Winterschool, 10-14 January 2011.
    Abstract: This was a review of the cohomology and K-theory of G/B, following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506.
    

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Talks of Arun Ram in 2010

1. On the cohomology of G/B, Working seminar, University of Melbourne, December 15, 2010.
   Abstract: This was a review of the cohomology and K-theory of G/B, following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506.
2. Elliptic cohomology and Weyl character formulas, Invited speaker at the IGA/AMSI Workshop "Dirac Operators in Geometry, Topology, Representation Theory, and Physics" at University of Adelaide October 18-22, 2010.
   Abstract: In this work, joint with Nora Ganter, we establish an elliptic cohomology version of the Atiyah-Segal-Lefschetz fixed point formula and apply it to the flag variety of a compact Lie group. We make contact with the work of Looijenga on Root systems and Elliptic Curves and the work of Kac and Peterson on Affine Lie algebras and Modular Forms and obtain Weyl characters for the loop group as push forwards in elliptic cohomology.
3. The Glass Bead Game, Colloquium, University of Queensland, Brisbane, 11 October 2010.
   Abstract: This title is taken from the novel of Hermann Hesse. In joint work with A. Kleshchev, we were amused to discover a glass bead game for constructing representations of quiver Hecke algebras (algebras recently defined by Khovanov-Lauda and Rouquier whose representation theory categorifies quantum groups of Kac-Moody Lie algebras). In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store. I will explain how this game works, and some of the fascinating numerology that appears in the scoring of the plays.
4. What are KLRBMW algebras?,Invited speaker at the International Conference on the Non-Commutative Rings and Combinatorial Representation Theory" at Pondicherry University September 2-3, 2010.
   Abstract: In 2008, Khovanov-Lauda and Rouquier defined a family of diagram algebras whose representations have the property that their
   characters are elements of the quantum group. The characters of the simple modules of the KLR algebras are the canonical basis elements of the quantum group. In the type A case, the KLR algebras are a graded version of the affine Hecke algebra. This talk will be a survey,
   with the question of the definition of the KLR Birman-Wenzl-Murakami and Brauer algebras, as the motivation.
5. Towards elliptic Chevalley groups and flag varieties,Seminar at Chennai Mathematical Institute, India, September 1, 2010.
   Abstract: Assuming that the double affine Hecke algebra is a shadow of a double loop group or an "elliptic" Chevalley group we learn about the structure of the elliptic flag variety. Though we are not yet ready to make a proper definition we can see many properties of the object which should be the analogue of the flag variety for the elliptic case. This talk is based on joint work with Martha Yip on the combinatorics of the double affine Hecke algebra and with Nora Ganter on obtaining Weyl character formulas from elliptic cohomology using an elliptic cohomology analogue of the Atiyah-Segal Lefschetz fixed point formula and localization.
6. Musings towards elliptic buildings,Invited speaker at the ICM Satellite conference "Buildings, Finite Geometries and Groups" at the Indian Statistical Institute, Bangalore, India, during August 29 - 31, 2010.
   Abstract: Assuming that the double affine Hecke algebra is a shadow of a double loop group or an "elliptic" Chevalley group we learn about the structure of the elliptic building. Though we are not yet ready to make a Tits style definition we can see many properties of the object which should be the analogue of the Tits building for the double loop and the elliptic cases. This talk is based on joint work with Martha Yip on the combinatorics of the double affine Hecke algebra and with Nora Ganter on obtaining Weyl character formulas from elliptic cohomology.
7. Symmetry and identities, Melbourne University Mathematics Society (MUMS) seminar, University of Melbourne, 20 August 2010.
   Abstract: I will explain some relationship between classical partition identities of Euler, Gauss and Jacobi are related to the symmetries of rigid polyhedra and the way that they fill up space.
8. On affine BMW algebras,Invited speaker at the International Conference on Representation Theory, Xian China August 9 -August 14, 2010.
   
9. The Glass Bead Game, Colloquium, University of Adelaide, 25 June 2010.
   Abstract: This title is taken from the novel of Hermann Hesse. In joint work with A. Kleshchev, we were amused to discover a glass bead game for constructing representations of quiver Hecke algebras (algebras recently defined by Khovanov-Lauda and Rouquier whose representation theory categorifies quantum groups of Kac-Moody Lie algebras). In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store. I will explain how this game works, and some of the fascinating numerology that appears in the scoring of the plays.
10. Affine BMW algebras, Pure Mathematics Seminar, University of Adelaide, 25 June 2010.
    Abstract: I will describe a family of algebras of tangles (which give rise to link invariants following the methods of Reshetikhin-Turaev and Jones) and describe some aspects of their structure and their representation theory. The main goal will be to explain how to use universal Verma modules for the symplectic group to compute the representation theory of affine BMW (Birman-Murakami-Wenzl) algebras.
11. Combinatorics and Spherical functions, invited talk at the BIRS Workshop 10w5096, Whittaker Functions, Crystal Bases, and Quantum Groups, Banff Canada June 6-11, 2010.
    
12. What is a line bundle?, Informal Seminar, University of Melbourne, 4 June 2010.
    
13. Three examples when the space of global sections of line bundles is interesting (Borel-Weil-Bott, modular forms, and toric varieties from polytopes), Informal Seminar, University of Melbourne, 2 June 2010.
    
14. Lecture 1: Quantum groups and Lyndon words, Lecture 2: Graded quiver algerbas and their representations, Lecture 3: Indexings of canonical bases: Lyndon words, MV polytopes and the path model, Invited speaker at the 64th Séminaire Lotharingien de Combinatoire, Program, Institut Camille Jordan - Bâtiment Braconnier, Lyon, Sunday, March 28th, 2010 (evening) to Wednesday March 31st, 2010.
    

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Talks of Arun Ram in 2009

1. The Glass Bead Game, Short talk to the Vacation Scholars, University of Melbourne, 18 December 2009.
   This was a short presentation of the bead game in recent work with A. Kleshchev on homogeneous representations of Quiver Hecke algebras.
2. Moment maps on flag varieties and piecewise linear functions, Seminar/Introduction, University of Melbourne, 26 November 2009.
   We discuss the theorem of Borel-Bott-Weil and the Weyl character formula via localization.
3. Introduction to equivariant cohomology, Seminar/Introduction, University of Melbourne, 19 November 2009.
   We will introduce foundational material on Brion-Vergne lattice point counting via the Jeffrey-Kirwan localization formula.
4. Introduction to categories, Seminar/Introduction, University of Melbourne, 12 November 2009.
   We introduce the notions of chain complexes, categories, natural transformations, totalization, homotopy and derived categories.
5. Universal Verma modules and Translation, at the 53rd Annual meeting of the Australian Mathematical Society, Special session in Algebra and Number Theory, University of South Australia, Adelaide 28 Spet.- 1 October, 2009.
   Abstract: We will introduce a framework for studying the combinatorics of translation functors in a "universally integral" framework and explain a unified perspective on Gabber-Joseph's approach to the Kazhdan-Lusztig conjectures, Kleshchev and Brundan's approach to modular branching rules, and the Misra-Miwa Fock space. This talk is based on joint work with Peter Tingley.
6. Universal Verma modules and Translation, at the International workshop on combinatorial and geometric approach to representation theory, Seoul National University, Korea, 21-24 September, 2009. YouTube
   Abstract: We will introduce a framework for studying the combinatorics of translation functors in a "universally integral" framework and explain a unified perspective on Gabber-Joseph's approach to the Kazhdan-Lusztig conjectures, Kleshchev and Brundan's approach to modular branching rules, and the Misra-Miwa Fock space. This talk is based on joint work with Peter Tingley.
7. Why I care about p-compact groups, Reading seminar, University of Melbourne, 21 August 2009.
   Abstract: A survey of symmetric functions, Schur functions, Weyl characters, the Borel-Weil-Bott theorem, the cohomology and K-theory of flag varieties, the classification of p-compact groups and the Clark-Ewing formula.
8. Poles, strings, braids and lattices, Colloquium, La Trobe University, 1 May 2009.
   Abstract: The double affine braid group has important applications to Macdonald polynomials, group representations, mathematical physics and combinatorics. The classical type double affine braid groups have nice pictorial presentations which exhibit the tantalizing symmetries at play. In this talk I'll draw some of these pictures and explain their role in topology, harmonic analysis, combinatorics and the study of symmetry.
9. Lyndon Bases, "blackboard seminar", University of Melbourne, 31 March 2009.
   Abstract: I will define Lyndon words and good Lyndon words and explain how we associate standard and simple quiver Hecke algebra modules to these words. I will not assume any memory of last week's talk.
10. Quiver Hecke alagebras, "blackboard seminar", University of Melbourne, 24 March 2009.
    Abstract: Quiver Hecke algebras were recently defined by Khovanov-Lauda and, independently, by Rouquier. The importance of these algebras is that the category of graded modules for the quiver Hecke algebras is a categorfication of the Drinfeld-Jimbo quantum group. I will give a survey of this exciting new subject, perhaps highlighting some of our recent results joint with Kleshchev.
11. A path model formula for Macdonald polynomials,Séminaire sur les Algèbres Enveloppantes et Théorie des Représentations, Paris Jussieu, 6 March 2009.
    Abstract: The path model of Littelmann provides a combinatorial formula for Weyl characters. In this talk we shall explain the generalization of the Littelmann formula to Macdonald polynomials.
12. A path model formula for Macdonald polynomials, Seminar Algebra and Topologie, University of Basel, 20 February 2009.
    Abstract: The path model of Littelmann provides a combinatorial formula for Weyl characters. In this talk we shall explain the generalization of the Littelmann formula to Macdonald polynomials.
13. Two boundary Hecke algebras and tantalizer algebras, Algebra seminar, Maxwell Institute for the mathematical sciences, University of Edinburgh, 17 February 2009.
    Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will be a medley of vignettes around double affine type C braid groups and quotient algebras.
14. Two boundary Hecke algebras and tantalizer algebras, Algebra seminar at Cambridge University, 28 January 2009.
    Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will be a medley of vignettes around double affine type C braid groups and quotient algebras.
15. Symmetry, Polynomials and quantisation Lecture1 Lecture2 Lecture3 Lecture4, Minicourse of four lectures at the program Algebraic Lie Theory, Isaac Newton Institute, 12-23 January 2009.
    Abstract: These talks will provide a pictorial approach to Weyl groups, braid groups and their Hecke algebras. With the pictures in hand, we can use them to study orthogonal polynomials, representations of braid groups, solutions of difference and differential equations, integrable systems and the quantisations that produce them.

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Talks of Arun Ram in 2008

1. Beads on runners, Invited talk at the special session Group actions and Representation Theory at the 7th Australia-New Zealand Mathematics Convention, Christchurch, New Zealand, 8-12 December 2008.
   Abstract: Khovanov-Lauda algebras are a family of algebras whose representation theory provides a categorification of quantum groups. In this work we classify and construct homogeneous representations of these algebras. The construction generalises the construction of irreducible representations of the symmetric groups and the notions of partitions, skew shapes, and abaci.
2. The mysteries of symmetry, Colloquium, Australian National University, 20 November 2008.
   Abstract: In recent joint work with Martha Yip we gave a combinatorial formula for Macdonald polynomials. The formula is a weighted sum of paths and the construction of the paths is completely elementary. The mystery is that these paths are describing subtle information about fancier objects: loop groups, integrable hierarchies of differential equations, representation theory and cohomology theories. I will try to formulate some of my speculations about how these objects are related. The underlying symmetry is certainly touching many parts of modern mathematics and it is all the more amazing that the elementary combinatorics of paths has something deep to say about it all.
3. Beads on runners, Colloquium, Monash University, 6 November 2008.
   Abstract: We think of beads on runners like an abacus, or like one of those games for toddlers where the children slide the beads on the runners (these games are sometimes found in waiting rooms of the offices of pediatricians). In joint work with A. Kleshchev we have shown this is a perfect model for representations of Khovanov-Lauda algebras, the recently discovered algebras whose representations categorify quantum groups. I shall explain the bead and runner model and how to have your toddlers compute representations of Khovanov-Lauda algebras while waiting for the doctor at the medical centre. The model generalizes partitions and their classical connection to the symmetric group. At the end of the talk I will explain how these algebras are related to Lie algebras and quantum groups and why they are considered a great new advance in the art of "categorification".
4. Short lecture at the University of Melbourne/BHP Billiton School Mathematics competition, 11 October 2008.
   Abstract: This was a 10 minute talk to school students -- maths competition winners. I told them that I went into mathematics for the lifestyle and pointed out the existence of a coffee shop/restaurant on the lakefront in Lugano on Lago Como in Swizerland. Then we looked at the wonderful Bratelli diagram on Tom Halverson's web page, and finally I told them that Persi Diaconis has a knack for finding uses of pure maths in other arenas and will be visiting Melbourne in 2010.
5. A combinatorial formula for Macdonald polynomials, Victorian Algebra Conference, RMIT Melbourne, 2-3 October 2008.
6. Generalising Pascal's triangle, Melbourne University Mathematics and Statistics Society (MUMS), lunchtime seminar, 12 September 2008.
7. Two boundary Hecke algebras and tantalizer algebras, Invited speaker at the International conference on Combinatorics and Representation Theory, Graduate School of Mathematics, Nagoya University, 1-5 September 2008.
   Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. One interesting quotient is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will survey this family of algebras.
8. Introduction to Buildings, Algebra-Geometry-Topology Discussion session, University of Melbourne, 21 August 2008.
   Abstract: This is a brief introduction to buildings in hopes of strengthening the analogy to the curve complex and Fenchel-Nielsen coordinates.
9. M_g,n and Fock space, Algebra-Geometry-Topology Processing seminar, University of Melbourne, 18 August 2008.
   Abstract: In his January lectures at MSRI, Okounkov outlined how to use partition combinatorics and Fock space to give formulas for the coefficients of the M_{g,n} volume polynomial. I shall try to explain this combinatorics and summarize the results of Okounkov-Pandharipande.
10. M_g,n, counting and recursions (pdf file), Algebra-Geometry-Topology Processing seminar, University of Melbourne, 4 August 2008.
    Abstract: I will attempt an elementarification/bigpicturification of the topic of Paul Norbury's talk last week.
11. Introduction to the Path Model, Seminar/Introduction, University of Melbourne, 2 July 2008.
    About the path model: I mean the model of P. Littelmann which generalises Dyck paths to give combinatorial models for representations of compact Lie groups. I am interested in using it to determine the polytope whose integer points describe the places where Littlewood-Richardson coefficients (also called Clebsh-Gordon or tensor product coefficients) are nonzero.
12. Type C Hecke and Temperley-Lieb algebras, Seminar/Introduction, University of Melbourne, 27 June 2008.
13. A combinatorial formula for Macdonald polynomials, Stanford Combinatorics and Geometry seminar, Stanford University, April 30, 2008.
    Abstract: We will explain a common generalization of Littelmann's formula for Weyl characters and Schwer's formula for spherical functions for a p-adic group. These formulas hold for arbitrary Lie type.
14. Symmetric functions d'après Macdonald, Keynote presentation at the 4th annual Graduate Student Combinatorics Conference, UC Davis, April 12-13, 2008.
    Abstract: This talk will be a road map to Macdonald's classic book on Symmetric functions, highlighting the combinatorics, representation theory and geometry coded by symmetric function identities.
15. Path Models (pdf notes), invited talk at Topics in Combinatorial Representation Theory, MSRI, Berkeley, March 17-21, 2008.
    Abstract: Recent years have seen big developments in the theory and applications of path models. The new applications are in understanding the combinatorics of the affine Hecke algebra, spherical functions, and the geometry of points in affine flag varieties. This talk will survey some of these recent results.
16. Tantalizer algebras, Colloquium University of Utah, March 6, 2008.
    Abstract: Abstract: Tantalizer is short for tensor power centralizer. These algebras often come as algebras of diagrams or of tangles, and so working with them requires drawing lots of pictures. Their structure and representation theory contains and immense amount of information about the representation theory of groups and quantum groups of types GL, SO, Sp, and they can be used to construct corresponding link polynomials and 3-manifold invariants. This talk will be a survey of some recent developments in tantalizer algebras.
17. Path Models, Representation Theory seminar, University of Utah, March 7, 2008.
    Abstract: This talk will be a survey of applications of path models: The Weyl character formula, Schubert calculus, spherical functions, normal forms in Chevalley groups, and indexing of points in affine flag varieties and Mirkovic-Vilonen cycles.
18. Tantalizers, invited talk at University of California Lie Theory Workshop, a conference in honor of Georgia Benkart, University of California, San Diego, February 16-18, 2008.
    Abstract: A tensor power centralizer algebra (tantalizer) is the algebra of commuting operators for a Lie group or quantum group action on tensor space. The favourite examples are the group algebra of the symmetric group and the Brauer algebra. This talk will survey some recent work on tantalizers: giving definitions and recent results for affine and graded BMW algebras and some two boundary tantalizers.
19. Minicourse: Combinatorics of Lie Type, three lectures at the Introductory Workshop on Combinatorial Representation Theory at MSRI, January 22-25, 2008.

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Talks of Arun Ram in 2007

1. Combinatorial Representation Theory 2008-2018, Colloquium, University of Minnesota, November 15, 2007.
   Abstract: The 1997 survey article of Barcelo-Ram entitled Combinatorial Representation Theory “defined” the field and set out its structure. In 2007 this field is thriving and vibrant. In Spring 2008 there will be a full semester program at MSRI entitled Combinatorial Representation Theory. Where is the field now? What has happened in the interim 1997-2007? More importantly, what will happen in Combinatorial Representation Theory in 2008-2018?
2. Generalizing partitions and standard tableaux, combinatorics seminar, University of Minnesota, November 15, 2007.
   Abstract: The irreducible representations of the symmetric group are indexed by partitions and bases of these representations are indexed by standard tableaux. The representation theory of the affine Hecke algebras provides a generalization of partitions and standard tableaux. I will explain these combinatorial indexings and how they arise.
3. Two row partitions and the Temperley-Lieb algebra, Combinatorics seminar, University of Wisconsin, Madison, October 15, 2007.
   Abstract: Following a good idea of V. Rittenberg, two boundary diagram algebras are getting more and more attention, with two boundary Temperley-Lieb algebras being a fundamental example. This talk will begin to answer the question: Which Type C affine Hecke algebra representations are two boundary Temperley-Lieb representations and what is a good combinatorial set for indexing these representations?
4. Boundary diagram algebras, Representation theory seminar, University of Wisconsin, Madison, October 12, 2007.
   This talk was a repeat of a talk given at University of Koln on 19 November 2005.
5. Centers of tantalizers, Representation theory seminar, University of Wisconsin, Madison, September 14, 2007.
   Abstract: Many diagram algebras arise as tantalizers. The Schur-Weyl duality makes it possible to steal most of the center of the tantalizer from the corresponding dual object in the duality. I will outline this process and explain how combinatorial results pop out of the picture. This talk is based on joint work with Zajj Daugherty and Rahbar Virk.
6. Today I feel like a mathematician - personality, music and geometry, The 21st Behrend Memorial Lecture, a public lecture at the University of Melbourne, August 21, 2007.
   Abstract: What does it feel like to be a mathematician? Who are the people who discovered and proved the Weil conjectures (one of the great human achievements of the 20th century)? Are they artists, musicians, or scientists? So, what does it feel like to be a mathematician, really?
7. Combinatorics in affine flag varieties, 6 July 2007; invited talk at GL07, Geometry and Lie Theory, a conference in honor of Gus Lehrer's 60th birthday, University of Sydney, July 2-6 and July 9-13, 2007.
   Abstract: This talk is about the combinatorics of indexing points in affine flag varieties. It is possible to make choices so that the points are indexed by a refinement of Littelmann's path model in such a way that the Schubert cell and the Mirkovic-Vilonen slice are easily read off the "path" indexing of the point. From this, the relations for the affine Hecke algebra can be derived, both in the Iwahori-Matsumoto and in the Bernstein generators. If time permits I will discuss the action of the "root operators" on points, and/or the relation to the Kamnitzer and Baumann-Gaussent indexings of Mirkovic-Vilonen cycles.
8. What is a Weyl group?, Summer Representation Theory seminar, University of Wisconsin-Madison, 14 June 2007.
9. Level l Fock spaces and the polynomial representation of Cherednik's double affine Hecke algebra,American Institute of Mathematics workshop: Arithmetic harmonic analysis on character and quiver varieties, American Institute of Mathematics, Palo Alto, June 4-8, 2007.
10. Introduction to Buildings and Combinatorial Representation Theory (pdf file), American Institute of Mathematics workshop on Buildings and Combinatorial Representation Theory, Palo Alto, March 26, 2007.
11. Introduction to moment maps on flag varieties, Lie Theory seminar, University of Wisconsin, Madison, 21 March, 2007.

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Talks of Arun Ram in 2006

1. Plenary lecture at the 2006 Fall American Mathematical Society Southeastern Section Meeting, University of Arkansas, Fayetteville, Arkansas, 3-4 November 2006.
2. Lecture at the workshop Modern Math: An Introduction to 2007-08 Programs at MSRI, at the Society for the Advancement of Chicanos and Native Americans in Science , National Conference, Tampa, Florida, October 24-25, 2006.
3. Combinatorial Hopf algebras: An outsider's survey, Minicourse at the conference Hopf algebras, Combinatorics and Quantum field Theory, Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany, 25-28 September 2006.
4. Path models and Chevalley groups, Oberwolfach meeting on Finite groups and representation theory, March 25-31, 2006.
   
5. Representations and translation, Special lecture in Quantum groups course, KdV Institut, Amsterdam, March 22, 2006.
6. Row reduction and loop groups, Lie Theory seminar, University of Wisconsin, Madison, February 28, 2006.
7. The Schur Hopf algebra, Combinatorics seminar, University of Wisconsin, Madison, February 27, 2006.
8. Alcove walks and reductive groups over local fields, Lie group and Representation Theory seminar, University of Maryland, February 3, 2006.
   In this talk I presented the precise combinatorial construction of the generalized MV cycles by labeled alcove walks.
9. Examples of groups: Lecture 1 - Reflection groups and braid groups, Lecture 2 - Matrix groups and Lie groups, Special minicourse, University of Rome "La Sapienza", January 23-24, 2006.
   These lectures were for advanced undergraduates in mathematics in Italy: introductory lectures on braid groups, reflection groups, matrix groups and Lie groups.
10. Hecke algebras and spherical functions, Harmonic analysis seminar, University of Rome "La Sapienza", January 18, 2006.
    

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Talks of Arun Ram in 2005

1. Boundary diagram algebras (pdf file), Seminar on Transformation groups & mathematical physics, a joint seminar of the Universities of Koln, Hamburg, Bochum, Bremen and Darmstadt, University of Koln, November 19, 2005.
   This talk is about diagram algebras which come from the two-boundary braid group (braids with two poles). This is a generalization of recent work (from statistical mechanics) on two-boundary Temperley-Lieb algebras. The generalized setting naturally includes two boundary Hecke algebras and two-boundary BMW algebras. These algebras are like affine Hecke algebras (of type A) and affine BMW algebras except with two poles.
2. p-compact groups, Oberseminar Geometrie, University of Fribourg, Switzerland, October 26, 2005
   This talk was the first time I realised that applying Milnor's construction of the classifying space to the p-compact groups of Clark-Ewing gives the space where path models (such as the Littelmann path models) live.
3. Random walks, spherical functions and representations, Colloquium, University of Fribourg, Switzerland, October 25, 2005.
4. p-compact groups, Algebra seminar, University of Rome "La Sapienza", October 20, 2005.
   This talk was my first attempt to learn something about p-compact groups.
5. Diagram algebras as tantalizers, Colloquium, University of Rome "Tor Vergata", October 19, 2005.
   This talk was the first time I realized and defined the graded version of the group algebra of the affine braid group which has, as quotients, the graded BMW algebras (also called cyclotomic Nazarov-Wenzl algebras), and the graded Hecke algebras.
6. Representations of affine Hecke algebras, Algebra seminar, University of Rome "La Sapienza", October 13, 2005.
   This talk was a survey on the representations of affine Hecke algebras.
7. Alcove walks and Iwahori cosets Algebra seminar, University of Rome "La Sapienza", October 6, 2005.
   This talk was where I first worked out the generalization of the MV cycles to G/I, in the example SL_2. In other words, coset representatives for the cosets in U^+vI\cap IwI, coset representatives for the cosets in U^+vI\cap IwI, where I is an Iwahori, and v and w are elements of the affine Weyl group.
8. Picturing Hecke algebras and loop groups, Algebra seminar, University of Rome "La Sapienza", September 29, 2005.
   This talk was an attempt to explain the alcove walk method of looking at affine Hecke algebras and loop groups.
9. Random walks, spherical functions and representations, Colloquium, University of Stuttgart, July 4, 2005.
   
10. q-crystals , Invited speaker in the special session in honor of Adriano Garsia at the conference Formal Power Series and Algebraic Combinatorics 2005, June 24-25, 2005.
11. Commuting elements in diagram algebras, Algebra seminar, University of Wuppertal, June 7, 2005.
12. Verma crystals, Algebra seminar, University of Lyon 1, May 27, 2005.
13. Random walks, spherical functions and representations, Colloquium, University of Lyon 1, May 26, 2005.
14. Walks, crystals and polytopes, Algebra seminar, University of Caen, May 24, 2005.
15. Random walks, spherical functions and representations, Colloquium, University of Freiburg, May 13, 2005.
16. Murphy elements in diagram algebras, Plenary speaker at the conference Cellular and diagram algebras and their applications in mathematics and physics, University of Leicester, England, April 3-10, 2005.
17. Combinatorial Representation Theory, Colloquium, Max-Planck-Institut fur Mathematik, March 24, 2005.
    
18. Combiantorial Representation theory II-Crystals, Talk 2 of a lecture series at University of Zaragoza, Spain, February 24, 2005.
19. Combinatorial Representation theory I-Towers and Centralizers, Talk 1 of a lecture series at University of Zaragoza, Spain, February 22, 2005.
20. Representations of Reflection groups, Seminar, Bernoulli Centre, EPFL, Lausanne, January 19, 2005.
    

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Talks of Arun Ram in 2004

(12) Algebra seminar, University of Virginia, November 19. 2004

(11) Colloquium, University of Virginia, November 18, 2004.

(10) Colloquium, Michigan Sate University, November 4, 2004.

(9) Distinguished lecture series "The Shoemaker Lectures", University of Toledo, November 1-3, 2004.

(8) Invited speaker and the AMS conference special session on Algebraic Representations and Deformations, Evanston Illinois, October 22-23, 2004.

(7) Algebra seminar, University of Wuppertal, Germany, July 9, 2004.

(6) Plenary speaker at the 11th conference of the International Linear Algebra Society, University of Coimbra, Portugal, July 19-22, 2004.

(5) Lecture series (joint with P. Diaconis) on Representation Theory and its Applications University of Coimbra, Portugal, July 15-16, 2004.

(4) Algebra seminar, University of Illinois, Urbana-Champaign, April 24, 2004.

(3) Algebra seminar, Stanford University, March 29, 2004.

(2) Invited speaker at the Groups and Representations Conference dedicated to the 60th birthday of Gary Seitz March 24 - 27, 2004, University of Oregon, Eugene, Oregon.

(1) Plenary speaker at the International Colloquium on Algebraic Groups and Homogenous Spaces, Tata Institute for Fund. Research, Bombay, India, January 6-14, 2004.

Talks of Arun Ram in 2003

(5) Plenary speaker at the KIAS International Conference on Lie algebra and related topics, Korea Institute for Advanced Study, Seoul Korea, October 20-24, 2003.

(4) Lecture series for the Summer school on the theory of group representations, University of Lisbon, Portugal, July 21-25, 2003.

(3) Plenary speaker at the Nineteenth British Combinatorial Conference, University of Wales, Bangor, Wales, June 29-July 4, 2003.

(2) Invited speaker at the BIRS Workshop Recent Advances in Algebraic and Enumerative Combinatorics, Banff International Research Station, May 3-8, 2003.

(1) Invited speaker at the conference Representations of finite groups, Oberwolfach, Germany, March 23-29, 2003.

Talks of Arun Ram in 2002

1. Plenary speaker for the 46th Annual conference of the Australian Mathematical Society, Newcastle, Australia, September 30-October 3, 2002.
2. Plenary speaker for the Quantum groups day at the XXIV International Colloquium on Group Theortical Methods in Physics, Paris, France, July 15-20, 2002.
3. Geometry seminar at University of Wuppertal, Wuppertal, Germany, July 2, 2002.
4. Invited speaker at the conference on Computational Lie theory, at the Centre Recherches Mathematiques of the University of Montreal, May 27-June 7, 2002.
5. Colloquium, University of Massachusetts, Amherst, March 28, 2002.
6. Plenary speaker, two one hour lectures, at the Mid-Atlantic Algebra Conference, Wake Forest University, March 16-17, 2002.

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Talks of Arun Ram in 2001

1. February 1, 2001, Symmetric functions seminar, Isaac Newton Institute, Cambridge, England.
2. February 16, 2001, Colloquium, University of Southampton, England.
3. February 21, 2001, Algebra seminar, Cambridge University, England.
4. February 26, 2001, Speaker, Isaac Newton Insitute Colloquium for General Scientific Audience.
5. March 8, 2001, Colloquium, City University, London, England.
6. March 21, 2001, Colloquium, University of Birmingham, England.
7. March 25-31, 2001, Invited speaker at the conference, ``Representations of Finite groups'' Oberwolfach, Germany.
8. April 30, 2001, Algebra seminar, University of Copenhagen, Denmark.
9. May 2, 2001, Algebra semianr, University of Aarhus, Denmark.
10. May 11, 2001, Colloquium, University of Warwick, England.
11. May 21, 2001, Algebra seminar, University of Leicester.
12. May 23, 2001, Algebra seminar, University of Glasgow, Scotland.
13. May 31, 2001, Seminaire Chevalley, Institut Henri Poincare, Paris, France.
14. May 31, 2001, Colloquium, Ecole Normale Superieur, Paris, France.
15. June 2-5, 2001, Invited plenary speaker at ``The Heritage of Schur's 1901 dissertation: a conference in honor of J.A. Green''.
16. June 12, 2001, Algebra seminar, University of Oxford.
17. June 12, 2001, Representation theory seminar, University of Oxford.
18. June 18, 2001, Speaker, Symmetric functions seminar, Isaac Newton Institute, Cambridge, England.
19. October 6-7, 2001, Invited speaker at the conference ``Midwest Lie algebras and Related Topics'' conference, DePaul University.
20. October 22-26, 2001, Invited speaker at the conference ``Combinatorial and Geometric Representation Theory'', Seoul, Korea.
21. December 7-16, 2001, Invited speaker at the conference ``Algebra and Geometry'', University of Hyderabad, India.
22. December 16-22, 2001, Invited speaker at the special year on ``Recent results and conjectures on Hilbert functions'', IIT Bombay, India.

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Talks of Arun Ram in 2000

1. December 1, 2000, Colloquium, University of Wisconsin--Milwaukee.
2. November 22-23, 2000, Series of two talks, algebra seminar, Instituto de Matematica, UNAM, Morelia,
3. November 21, 2000, Colloquium, Instituto de Matematica, UNAM, Morelia, Mexico.
4. October 16-20, 2000, Invited speaker at ``Algebres de Hecke affines et groupes reductifs'', Luminy-Marseille, France.
5. Algebra seminar, University of Sydney, Sydney Australia, August 11, 2000.
6. Colloquium, Macquarie University, Sydney Australia, August 7, 2000.
7. Algebra seminar, Mathmatisches Institut B, Universität Stuttgart, June 8, 2000.
8. Algebra seminar, University of Strasbourg, France, June 7, 2000.
9. Basel-Freiburg-Strasbourg joint Algebraic Groups Seminar, May 30, 2000.
10. Combinatorics seminar, University of Michigan, Ann Arbor, March 31, 2000.
11. Algebra seminar, University of Oregon, Eugene, March 7, 2000.
12. Colloquium, University of Oregon, Eugene, March 6, 2000.

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Talks of Arun Ram in 1999

1. Invited speaker at the KIAS Lie Theory Conference at the Korea Institute for Advanced Study (KIAS), Seoul, Korea, October 5-8, 1999.
2. Invited speaker at the conference Quantum groups and knot theory, at L'Institut de Recherche Mathématique Avancée, Strasbourg, France, September 27-29, 1999.
3. Invited lecturer at Seoul National University Math Camp, Chunan, Korea, June 21, 1999.
4. Invited lecture series (Minicourse on Hecke algebra representations) at the Korea Institute for Advanced Study (KIAS), Seoul, Korea, June 15-17, 1999.
5. Tableaux, hyperplanes and representations, plenary talk at the 11th Conference on Formal Power Series and Algebraic Combinatorics, Barcelona, June 7-11, 1999.
6. Tableaux, hyperplanes and representations, Colloquium, Center for Communications Research, Princeton, May 18, 1999.
7. Affine braids, quantum groups, and Jantzen filtrations, Invited speaker at the special session on Representations of Lie algebras at the American Math. Society meeting, Buffalo, NY, April 24-25, 1999.
8. Lie Theory seminar, MIT, April 21, 1999.
9. Combinatorics seminar, MIT, April 19, 1999.
10. Algebra and Geometry seminar, Stanford University, April 8, 1999.
11. Lie group seminar, Rutgers University, March 5, 1999.
12. Tableaux, hyperplanes and constructing representations, a lecture series at the Institute of Advanced Study, February 1999.

Talks of Arun Ram in 1998

(10) Combinatorics, representations and vector bundles on flag varieties, Colloquium, Princeton University, December 1998.

(9) Young tableaux, root systems and affine Hecke algebras and A Pieri-Chevalley formula for the K-theory of the flag variety, two one hour invited talks at the workshop on The Interaction of Combinatorics and Representation Theory, Research Institute for Mathematical Sciences, Kyoto University, October 1998.

(8) The fine structure of representations of affine Hecke algebras, 1 hour invited talk at the ICM 1998 Satellite conference on Representations of finite groups and combinatorics, Magdeburg, Germany, August 1998.

(7) Connecting to affine Hecke algebras, 1 hour invited talk at the conference Representations of affine and quantum affine algebras and their applications, Raleigh, North Carolina, May 1998.

(6) Young tableaux in the space age, Haverford-Bryn Mawr Mathematics Colloquium, Haverford College, April 1998.

(5) A group tour of combinatorial representation theory, Colloquium, George Washington University, April 1998.

(4) Young tableaux in the space age and Why are centralizers so stringy?, invited talks at the GWU Combinatorics Day, George Washington University, April 1998.

(3) Modular representation theory for affine Hecke algebras, invited talk in the special session Representation theory of Lie algebras, algebraic groups, and quantum groups, Amer. Math. Soc. conference, Kansas State University, March 1998.

(2) Combinatorics and representations of affine Hecke algebras, Colloquium, Columbia University, February 1998.

(1) Combinatorics and representations of affine Hecke algebras, Colloquium, University of Virginia, January 1998.

Talks of Arun Ram in 1997

(1) Combinatorial representation theory, 1 hour invited talk in the special session Representations of algebras and groups, Amer. Math. Soc--Mexican Math. Soc. Joint meeting, Oaxaca, Mexico, December 1997.

(2) Combinatorics and representations of affine Hecke algebras, Colloquium, Temple Univ., November 1997.

(3) Combinatorics and representations of affine Hecke algebras, Colloquium, Univ. of Illinois--Chicago, November 1997.

(4) Combinatorics and representations of affine Hecke algebras, Colloquium, Univ. of Wisconsin--Madison, October 1997.

(5) Geometry and the combinatorics of affine Hecke algebras, Alperin/Glauberman seminar, Univ. of Chicago, October 1997.

(6) A global formula for the quantum Weyl group, in the special session on Enveloping algebras and Quantum groups, Amer. Math. Soc. conference, Milwaukee, Wisconsin, October 1997.

(7) Representations of affine Hecke algebras, Colloquium, Tata Institute of Fundamental Research, Mumbai, India, August 1997.

(8) Combinatorics and representation theory: Recent results for affine Hecke algebras, plenary address at the l Fifth Australasian Math. Convention, Auckland, New Zealand, July 1997.

(9) Representations of affine Hecke algebras, Colloquium, University of Newcastle, Australia, July 1997.

(10) Combinatorial representation theory, invited speaker in the special session on Combinatorics and Algebra, Mathematical Association of America conference, Logan, Utah, April 1997.

(11) Seminormal representations, affine Hecke algebras, Jucys-Murphy elements and the Shi arrangement, invited speaker at the Workshop in Representation theory and symmetric functions, MSRI, Berkeley, April 1997.

(12) Another look at Hoefsmit's representations, Colloquium, Univ. of British Columbia, Vancouver, February 1997.

(13) Iwahori-Hecke algebras of type A, bitraces and symmetric functions, invited talk at the special session in algebraic combinatorics, AMS-MAA joint meetings, San Diego, January 1997.

Talks of Arun Ram in 1996

(3) Hyperplanes, Hecke algebras and combinatorics, Colloquium, North Carolina State University, November 1996.

(2) A construction in combinatorial representation theory, Colloquium, University of Georgia, Athens, October 1996.

(1) Quantum groups, series of 5 one hour lectures, Workshop on Algebra, Geometry and Topology, Australian National University, Canberra, January 1996.

Talks of Arun Ram in 1995

(7) Quantum groups, R-matrices, and centralizer algebras, one hour invited talk, Australian Lie Groups Conference '95, University of Adelaide, November 1995.

(6) A construction in combinatorial representation theory, Joint colloquium of University of Sydney and University of New South Wales, November 1995.

(5) On a (traditionally complicated, but actually trivial) piece of combinatorial representation theory, Colloquium, University of Melbourne, September 1995.

(4) An elementary approach to the representations of Iwahori-Hecke algebras, Colloquium, University of Wisconsin--Milwaukee, April 1995.

(3) Invited speaker in special session on Lie theory, Amer. Math. Society meeting in Chicago, Illinois, March 1995.

(2) Representations and characters of Iwahori-Hecke algebras, Colloquium, Michigan State University, January 1995.

(1) Representations and characters of Iwahori-Hecke algebras, Colloquium, Notre Dame University, January 1995.

Talks of Arun Ram in 1994

(3) Representations and characters of Iwahori-Hecke algebras Colloquium, Binghamton University, December 1994.

(2) Representations of Iwahori-Hecke algebras and centralizer algebras, Invited speaker, one-hour talk, Midwest group theory conference, Madison, Wisconsin, October 1994.

(1) Centralizer algebras+Path algebras+Quasitriangular Hopf algebras=Irreducible representations, Invited speaker, one-hour talk, Third conference in algebraic combinatorics at Ann Arbor, Michigan, June 1994.

Talks of Arun Ram in 1993

(3) Orthogonality of characters, regular representations and Weyl group symmetric functions, Invited talk in special session on algebraic combinatorics, SIAM annual meeting, Philadelphia, July 1993.

(2) On the second orthogonality relation for characters of the Brauer algebras, Invited speaker at E'cole internationale de combinatoire de Bordeaux, June 1993.

(1) Braids, quantum groups, and centralizer algebras, Colloquium, University of Wisconsin-Parkside, February 1993.

Talks of Arun Ram in 1992

(2) Weyl group symmetric functions and the representation theory of Lie algebras, 4th conference Formal Power series and Algebraic combinatorics, Univ. de Quebec a Montreal, June 1992.

(1) Character classes in Hecke algebras, Colloquium, Virginia Polytechnic Institute, May 1992.

Talks of Arun Ram in 1991

(2) Quantum groups and the characters of the Hecke algebras, Colloquim, Trinity College, Dublin, Ireland, April 1991.

(1) Characters of Hecke algebras, Colloquim, Universitat Basel, Switzerland, May 1991.

Talks of Arun Ram in 1990

(2) A Frobenius formula for the characters of the Hecke algebras, Jones/Grunbaum seminar, Univ. of Calif., Berkeley, October 1990.

(1) Frobenius formulas for the characters of the Hecke algebras and Brauer's centralizer algebras, Colloquium, Univ. of Wisconsin, Madison, September 1990.

Talks of Arun Ram in 1989

(1) Matrix units for the Brauer algebras, Colloquium, New Mexico State Univ., April 1989.