Arun Ram: Notes

Notes

Arun Ram

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

This collection of notes is slowly, but continually, being added to and polished. I hope that these resources will be useful to the community. Most of these pages are in MathML. The newest of them use HTML5. Firefox has supported MathML for a long time, and other browsers are quickly implementing rendering for it, as it is included in the W3C HTML5 recommendation. It may be necessary to get the STIX fonts, see https://developer.mozilla.org/en/Mozilla_MathML_Project/Fonts. A recent version of Firefox (3 and later) with the STIX fonts should render these notes correctly.

Notes-pre2005	Notes2005	Notes2007
References	Math Grammar and How to do Proofs	SVG Pic Library
Glossary	Exercises	SVG primer
Template, Proofstyleidea	Notes with collapsible lists	ScriptFont

Linear algebra

Matrices M_n(F), in other words End(V)
Invertibility, The general linear group, More general linear group, Invertible matrices GL_n(F), in other words Aut(V)
Permutations, Determinants
Gaussian elimination, row reduction, echelon form, and Smith normal form
Linear dependence bases and dimension from PartV.pdf
matrices of bilinear forms
Gram-Schmidt: normal form for matrices of symmetric bilinear forms
Frobenius normal form for matrices of skew symmetric bilinear forms
Eigenvalues and eigenvectors, and characteristic and minimal polynomials
Jordan normal form: normal form for matrices under conjugation
Determinants and traces
cofactors and inverse matrices
The Cayley-Hamilton theorem
polar decomposition
Bruhat decomposition
Cartan decomposition
Iwasawa decomposition
Tensor, symmetric, and exterior algebras
Bilinear forms
Bilinear, sequilinear and quadratic forms
Absolute value
- The triangle and Cauchy Schwartz inequalities
- Absolute value - Exercises
C[t]-modules, invariant subspaces, minimal polynomials, Cayley-Hamilton theorem, Jordan normal form
The spectral theorem, polar decomposition
Euclidean isometries and the affine orthogonal group
Lecture: Reflection groups and Braid groups
Lecture: Matrix groups and Lie groups
Linear structures
- Fields and Vector spaces
- Rings and Modules, more modules
- Groups and Group actions
- Matrices (see link above)
- Bases of vector spaces
- Banach and Hilbert spaces

Groups of Lie type

General Lie group and Lie algebra stuff

Lecture: Reflection groups and Braid groups
Lecture: Matrix groups and Lie groups
Weights, coweights, Weyl groups
Indexing representations of G and T, Weyl's theorem
Chevalley-Steinberg-Tits groups
Lie groups
Lie groups and Lie algebras
Lie algebra of a Lie group and the exponential map
The adjoint representation
Abelian Lie groups
Compact groups
Linear algebraic groups
Lie algebras
Virasoro algebra, More Virasoro algebra
The Casimir
The Harish-Chandra homomorphism
More classical type Lie algebras
Central elements in classical type quantum groups
Weights, roots and Weyls integral formula
The Weyl character formula
Ram -- 1992 Rota class lectures parts 1 and 2: Representation Theory of Semisimple Lie algebras,
Ram -- 1992 Rota class lectures part 3: Hecke algebras,

Examples of Lie algebras

Examples of algebraic groups

Groups of Lie type

Reflection groups

Definitions
Examples and the Shephard-Todd classification
Presentations
Reflection representations and lattices
The ring of generalized theta functions
Polynomial invariants and degrees
The Chevalley, Shephard-Todd theorem, CSTsomepdfnotes
Affine Root systems
Schubert Calculus
- The rings H_T(pt), K_T(pt), E_T(pt), Omega_T(pt)
- Borel picture
- Bott towers
- Bott Tower picture
- BGG-Demazure operators and the Pieri-Chevalley formula
- The top class
- Schubert products A₁ and A₂
- Schubert products G₂
- Old Ganter-Ram Section 1: Flag and Schubert varieties
- Old Ganter-Ram Section 7: Schubert classes and products in rank 2
- Old Ganter-Ram Section 8: Calculus of BGG-Demazure operators

Weyl groups, Braid groups and Hecke algebras of Weyl groups

Weyl groups: N/T presentation, reflection presentation, Coxeter presentation, Bruhat decomposition indexing
Braid groups: Coxeterish presentation, π₁ presentation,
Finite Hecke algebras: Finite Hecke algebras: Weylish presentation, Coxeterish presentation, C(B\G/B) presentation,

affflags1.14.07.pdf

Affine Weyl groups: Semidirect product presentation, Coxeterish presentation, reflectionish presentation, Iwahori decomposition indexing
Affine Braid groups: Bernsteinish presentation, Coxeterish presentation, π₁ presentation,
Affine Hecke algebras: Bernstein presentation, Coxeterish presentation, Affine Weylish presentation, C(I\G/I) presentation,

Affine Weyl groups, their Hecke algebras and their representations

The centre of the affine Hecke algebra is K_G(pt)
Weyl character formula, Whittaker functions, Casselman-Shalika and Hecke algebras
Type GL_n affine Hecke algebra examples:

The affine symmetric group
The affine Hecke algebra of Type GL_n
The degenerate affine Hecke algebra of Type GL_n
The affine braid group of type GL_n and its action by R-matrices

Double affine Hecke algebras
- Double affine Weyl groups
- Double affine braid groups
- Double affine Hecke algebras of type C_n
- Double affine braid group and double affine Hecke algebra of type (Cn^∨,Cn)
- Intertwiners
- Koornwinder polynomials
Kazhdan-Lusztig basis elements and Kazhdan-Lusztig polynomials
OLD Kazhdan-Lusztig basis elements and Kazhdan-Lusztig polynomials
0-Hecke algebras
0-Hecke algebra examples
Representations of affine Hecke algebras

Intertwiners and calibration graphs
Constructions of calibrated representations
Deducing the H_L representation theory from the H_P representation theory
Matt Davis PhD Thesis type A₁
Matt Davis PhD Thesis type A₁xA₁
Matt Davis PhD Thesis type A₂
Matt Davis PhD Thesis type C₂
Matt Davis PhD Thesis type G₂
Matt Davis PhD Thesis Geometric indexing type C₂
Matt Davis PhD Thesis Geometric indexing type G₂

Symmetric functions and crystals

The Brazil lectures

The CBMS Lectures

Lecture 1. The Weyl Character formula: Presenting the affine Hecke algebra: Iwahori and Bernstein presentations and the path model CBMS Lecture 1
Lecture 2. The Weyl Character formula: The Weyl character formula from the affine Hecke algebra point of view CBMS Lecture 2
Lecture 3. Representations of symmetric groups: Quiver Hecke algebras: The Brundan-Kleshchev presentation of the affine Hecke algebra
Lecture 4. Representations of symmetric groups: Representations of Quiver Hecke algebras: Bead diagrams as generalizations of standard Young tableaux
Lecture 5. Quantum groups and representations: Shuffle algebras and characters of quiver Hecke algebra representations: PBW bases and canonical bases
Lecture 6. Quantum groups and representations: Verma modules and crystals, i.e. Crystals coming from quantum groups and quiver Hecke algebras
Lecture 6. Verma modules and crystals: Crystals coming from quantum groups and quiver Hecke algebras
Lecture 7. Cohomology of flag varieties: Affine Hecke algebra to cohomology computations, i.e. Schubert calculus by BGG and Demazure operators
Lecture 8. Cohomology of flag varieties: Moment graphs for cohomology computations: i.e. the GKM technique for Schubert calculus
Lecture 9. Loop groups and affine flag varieties: Definition of MV cycles
Lecture 10. Loop groups and affine flag varieties: Crystals and Schur functions from MV-cycles: i.e. MV polytopes and their crystal structure

The groups G_r,1,n, their Hecke algebras and their representations

Number systems and polynomials

Numbers
- Lecture 1: Numbers
- The integers ℤ, The Rationals ℚ, The Real numbers ℝ, The complex numbers ℂ, The quaternions $ℍ$ ,
- The clock Z/nZ, the cyclic group μ_n of nth roots of unity and cyclotomic polynomials, and the Finite fields 𝔽_p,
- Euclidean space ℝⁿ, complex space Cⁿ, quaternionic space H^n
- T he rings Z/pZ, Z[1/p], Z_(p), the p-adic integers Z_p, and the p-adic numbers Q_p
- Polynomials F[x], F[[x]], F(x) and F((x)); xⁿ
  - The ring Z, Euclidean algorithm, arithmetic modulo m, gcd, and partial fractions for Z
  - The rings F[x], Euclidean algorithm, gcd and partial fractions
- Matrices M_n(F), GL_n(F), Smith normal form, Gram-Schmidt, and Jordan normal form (M_n(R) and GL_n(R), in other words, End(V) and Aut(V))
- Q[x]/(f)
- Octonions
- Adeles
- Z-hat
- W(Z)
  - Numbers - Exercises
  - Z, Q, R, C - Exercises
Polynomials
- Quadratic and cubic formulas, solvability by radicals, the fundamental theorem of algebra,
- Derivations of C[x], C(x), C[[x]], C((x)), and Taylor series, and partial fractions,
- The chain rule and implicit differentiation
- Indefinite integrals = antiderivatives: substitution, partial fractions, trig substitions, hyperbolic substitutions
- Integration by parts
- The binomial theorem and the exponential function; e^x and a^x
- Trig and hyperbolic functions
- Inverse expressions: √x, x^1/n, log(x), sin^-1 x, etc
- Binomial series
- The beta and gamma functions
- The Genesis Lecture
Graphing
- The Pythagorean theorem, Lines, parabolas, circles hyperbolas, shifting and scaling
- Angles and trig functions by geometry
- Basic graphs, Shifting, scaling, flipping and asymptotes
- Continuity, differentiability, increasing, decreasing, concavity, maxima and minima, critical points
- Tangent and normal lines
- Graphing rational functions, Graphing examples
- Basic graphing-Exercises
- Graphing polynomials - Exercises
- Graphing rational functions - Exercises
- Other graphing-Exercises

Sets and functions

Sets and Functions
- Sets, functions, cardinality and composition of functions, proofs
- Relations, Ordered sets
  - Sets: Examples
  - Functions: Examples
  - Sets and functions exercisesproofs
  - Sets: Exercises
  - Functions: Exercises
  - Cardinality: Exercises
  - Ordered sets: Exercises
  - Orders: Exercises
  - Orders on Z,Q, R,C: Exercises
  - Inequalities: Exercises

Linear structures

Algebraic structures

Operations
- Operations - Exercises
Monoids, groups, rings and fields, Fields and ordered fields, Ordered fields
- Groups and monoids - Exercises
Fields and integral domains (text)
Fields and integral domains (proofs)
Fields of fractions, localization, more localization, and valuations
Euclidean domains, Principal ideal domains and Unique factorization domains (text)
Euclidean domains, Principal ideal domains and Unique factorization domains (proofs)
Divisors, Fractional ideals
Polynomial rings (text)
Polynomial rings (proofs)
Finitely generated modules over a PID, finitely gen PID mods partB
Finiteness conditions: Noetherian rings, Artinian rings, and composition series

Topological structures

Filters, nets, sequences and convergence
Topological spaces and continuous functions, Interiors and closures
limits and continuous functions, Examples in R and C: Limits, and Sequences, and Series
Uniform spaces, metric spaces and completion, completion in the h-adic topology
Generating filters, topologies and uniformities
Uniform spaces, Metric spaces
Hausdorff and separable spaces
Compact spaces and proper mappings
Connected, Irreducible and Noetherian topological spaces
Measurable spaces, measurable sets and measurable functions
Measures and Integration
Lebesgue convergence theorems
Hölder, Minkowski, Cauchy-Schwarz and triangle inequalities
Function spaces
The Radon-Nikodym and Reisz representation theorems
Distributions: The Riesz representation theorem
Exercises
- Topology: Exercises
- Metric and Hilbert Space problems 2014
- Integration Exercises: Exercises BR Chapt 11
- Integration: Exercises R Chapt 1
- Integration: Exercises R Chapt 2
- Integration: Exercises R Chapt 3
- Integration: Exercises R Chapt 4
- Integration: Exercises R Chapt 5
- Integration: Exercises R Chapt 6
- Measure and Integration: More Exercises
Examples of topological spaces
- R and R/Z
- [0,∞] and [-∞,∞]
- (R^x)⁰=(0,∞)
- Rⁿ and Tⁿ
- M_nxm(R) and GL_n(R)
- S^n-1 and B_n
- Pⁿ and G_n,p

Groups

Rings

????
Ideals incomplete
Exercises
- Commutative algebra homework
Examples of rings
- Z, Z_(p), Z_p
- F[x] and A[x]
- F[x₁,...,x_n] and A[x₁,...,x_n]
- F[[x₁,...,x_n] and A[[x₁,...,x_n]]

Fields

The extension F(α)
The Galois correspondence
Extensions of Q
Galois cohomology and cyclic and abelian extensions
Splitting fields
Separability
Exercises
- Groups: Exercises
Examples of field stuff
- Cyclic groups

Limits

Limits
- Limits-Examples (calculations Ramulus)
  - Limits - Exercises
  - Existence of limits: Exercises
  - Limits as x goes to 0: Exercises
  - Limits as x goes to a: Exercises
  - Limits as x to infinity: Exercises
  - Limits with exp and log: Exercises
  - Limits with trig functions: Exercises
  - Limits with inverse trig functions: Exercises
  - L'Hopitals rules: Exercises
  - Where is a function continuous - Exercises
  - Continuous functions - Exercises
  - etc.
Sequences
- Sequences-Examples
  - Sequences - Exercises
  - Sequences and order (boundedness, increasing, monotone) - Exercises
  - Convergence theorems for sequences - Exercises
  - Picard and Newton iteration - Exercises
- Sequences of functions: pointwise and uniform convergence
Series
- Convergence tests (comparison test, root test, ratio test, integral test, alternating series)
- Taylor series and radius of convergence
- Fourier series
- Special sequences and series
  - Geometric series
  - The Riemann zeta function
- Series-examples
  - Series: Exercises
  - Sequences and series - Exercises
  - Fourier series - Exercises
Applications of limits
- Derivatives and limits
  - Derivatives (using the limit definition)
  - derivatives and limits
  - Taylor's theorem and the Mean Value theorem
  - Derivatives - Examples
    - Differentiability - Exercises
    - Taylor approximations - Exercises
    - Mean Value theorem - Exercises
    - Rolle's theorem and the MVT - Exercises
    - increasing, decreasing and concavity - Exercises
    - L'Hopital's rule
    - Rates of change - Exercises
- Integrals and limits
  - The Riemann integral
  - The fundamental theorem of calculus
  - Improper integrals
  - Trapezoidal and Simpsons integrals
  - Integrals-examples
    - fundamental theorem of calculus-Exercises
    - Definite integrals and areas - Exercises
    - Computing volumes by integration - Exercises
    - Improper integrals - Exercises
    - Trapezoidal and Simpson's integrals - Exercises
    - Surface areas and centers of mass - Exercises
    - Additional differentiation and integration - Exercises

Graphing and geometry

Distances, angles and the number π
plotting, shifting, scaling, asymptotics
lines, circles, parabolas, ellipses, hyperbolas
rational functions
exponential and trigonometric functions
parametric curves
polar coordinates, cylindrical and spherical coordinates
vectors, dot products, cross products
lines and planes, planes through the origin = subspaces
conic sections and surfaces
level curves

continuity and graphing
derivatives and graphing
- slopes, tangents and normals, tangent planes,
- maximums, minimums, concavity, inflection,
- related rates
- optimization
integrals and graphing
- fundamental theorem of calculus
- areas,
- solids of revolution
- arc length

Differential geometry

partial derivatives
chain rules for partial derviatives
scalar functions and vector functions
limits, continuity, differentiability in the multivariable case
Taylor polynomials (multivariable)
tangent planes and normals for surfaces, parametric surfaces
max-min, concavity multivariable: classification of stationary points
Lagrange multipliers
vector fields
gradient and directional dervative
curvature, torsion, tangent and normal vectors
grad, div, curl

Integration - multivariable

?partial integration

double integrals
triple integrals
Jacobian
change of variable for multiple integrals
surface integrals
path integrals, line integrals, parametrisation of surfaces
Green's theorem, divergence theorem, Stokes theorem
conservative fields
arc length
surface area
other Applications: velocity and acceleration in space
Other applications: ?Laplacian, scalar potentials

Solving equations: linear, polynomial, differential

linear equations, solution space
Newton's method
polynomials in one variable

Weyl algebra
separable, integration factor (algebra)
homogeneous constant coefficient linear ODEs
inhomogeneous constant coeff linear ODEs

slope fields (vector fields)
equilibrium points (sink in vector field)

population models, radioactive decay, interest, cooling,
logistic populations models with and without harvesting
mixing problems
circuits
springs (including air resistance)

Linear algebra

Matrices -- ring structure
Row reduction -- generators and relations for GL_n
determinants
vector spaces, linear transformations, subspaces
free modules, bases, spanning, linear combinations and independence, dimension
matrix representation of linear transformations and change of basis
kernels, images, quotients, nullity, rank
eigenvectors, eigenvalues, normal forms (wrt conjugacy)
Cayley-Hamilton theorem
dual vector spaces, inner products, Gram-Schmidt, least squares
Spectral theorem

Real Analysis

Numbers, N, Z and Q
proof techniques: contradiction, induction
Real numbers R
supremum, infimum
irrationals, approximation by rationals
absolute value
inequalities
open sets
Heine-Borel theorem and applications
sequences
limits of sequences, limits
monotone convergence and applications
sandwich rule with applications
subsequences, Cauchy convergence
bounded sequences, upper and lower limits
functions, injective, surjective, bijective
limits of functions, epsilon delta
algebra of limits, standard limits
continuity
one sided limits
uniform continuity
intermediate value theorem
differentiability
Rolles Theoren, Mean value theorem,
newton's method
Riemann sums and integrals
integrality and L^1
trapezoidal rule, simplon's method, error estimates
fundamental theorem of calculus
mean value theorem for the integral
integrals depending on a prameter
improper integrals
comparison test
recurrence relations
Gamma function
Series, partial sums
convergence of series, telecoping
harmonic p-series
integral test, ratio test, root test, comaprison test
absolute and conditional convergence, text for conditional convergence
Taylor polynomials, approximation
power series
radius of convergence
fourier series and convergence

Group theory and Linear algebra (2nd year)

Z/nZ
fields and vector space definitions
invariant subspaces, minimal polynomials
Cayley-Hamilton, Jordan canonical form
complex inner products
orthogonal complemements, adjoints,
spectral tehorem
hermitian and normal matrices
groups, subgroups, isomprhism
permutations, symmetric group
cyclic group
group actions: stabilizer and orbits, cosets
group action exmplles: conjugacy, left multiplication
application: cryptography
quotients, normal subgroups, isomorphism theorem
application: finite groups of isometries

Dynamical systems and Chaos

definitions: continuous and discret dynamical systems
state space, phase space, Poincare sections, equilibria and fixed points
unimodal maps, p-cycles
linear stability
period doubling
renomrlaization transformation
transition to chaos
statstical anaylsis
probability density functions
Models: discrete, population, Markov chains
higher dimensional maps
Henon maps
attractors, strange attractors
fractals, fractal dimension, Cantor sets, Koch curve,
complex interation, Mandelbrot and Julia sets
fixed points
Examples of cintinuous dynamical systems
homogeneous/inhomogeneous linear systems
systems of oscillators
equilibria, stability and phase plnaes for diff eqs
nonlinear systems, linealization, limit cycles
3D problems
periodic trajectories and Poincare sections
numerical methods as dynamicla systems

Discrete maths

Scheduling problems: directed graphs, scheduling algorithms, precednce relations, graph colorings
Voting: fairness criteria, preferential voting, ranking procedures, Arrow's theorem
Fair division and apportionment: schemes, inidivisble objects and paradoxes
Bargaining and arbitration: Nash's bargaining game, application ot Fair division, Shapley value

Operations Research

Linear programming: standard and cononical form
geometry of linear programming, graphical solution in 2dim
optimal solutions, feasible regions
Simplex algorithm
solution of problems in standard and nonstandard form
revised simplex method
duality theory

Probability and Statistics (2nd year)

events, sample space, random variables,
Set theory: unions, intersection, complements, DeMorgans laws
Conditional probability and independence
Examples: Monty Hall, prisoner's dilemma
Law of total probabilty, Bayes Theorem
Distributions: Bernoulli, binomial, geometric, negative binomial, hypergeometric
Panjer's recursion
random smapling, cumulative distirbution functions, conditional distirbutions
continuous distributions: Probability Density Functions, Quantiles
poisson, Gamma, Uniform, normal
Fubini theorems
variance, stanadrd diveiation, expectations, revlative fequencies,
bivariate randomvariable, discrete and continuous
bivariat normel dist
bivariant expectation
Makov and Chebyshev inequaties
weak law large numbers
variance: ANOVA
moments
covariance
generating functions
approximation of ditributions
Markov chains, equilibrium
point estimation and sufficiency
estimation of means and variances
order statistics and confidence intervals
Regression, hypothesis testing
Asymtptic distributions of MLEs
Bayesian methods
hypothesis testing rank tests, chi square test, contingency tables

Complex analysis

convergence, seuences and series, Laurent series
holomorphic, meromorphic functions, CR equations
harmonic functions, conformal mappings
contour integrals, Cauchy integral theorem, examples
singularities, residue theorem
Gamma function, Riemann zeta function
Dolbeault cohomology?

Metric and Hilbert spaces

Metric and normed spaces, Hilbert spaces
balls, limits, topology, compactness, completeness
contraction mapping theorem
orthonormal systems
bounded linear operators and bounded linear functionals

Algebra

Z/nZ, C[x], Z, C, C[[x]]
rings, quotients
Unique factorization
Euclidean domains
fields, extensions,
finite fields
ruler and compass constructions

Geometry

differential geometry of curves and surfaces
tnagent spaces, differential forms
Frenet formula
curvature, minimal surfaces
classification of surfaces, Gauss-Bonet theorem
plane curves: conics, cubics, complex curves

PDEs

separation of variables
eigenfunction expanstions
Greens' functions
similarity solution
method of images
methamtical modelling

Graph theory (Aleks Owczarek, SanMing Zhou)

graph category: connectedness, bipartite, paths, cycles, trees,
weighted graphs, distance, Steiner trees,
matchings, flows, Eulerian circuits
Applications

Discrete Math (3rd year)

permutations, combinations
Ramsey theorey
tiling and combinatorial logic,
recursive structures, Fibonnaci, Penrose tilings
finite groups and permutations
beginnings of combinatorial group theory

Decision making: really an OR course = Game theory

techniques in OR

Stochastic models (Kostya)

Linear models (Ray Watson)

Probability and Satistical inference

Modern applied statistics

Reflection groups

The tetrahedral group
The octahedral group
The icosahedral group
The rank 2 reflection groups
The primitive irreducible reflection groups

Hodge theory

Elliptic curves
Abelian varieties
Hodge structures
The Tate Hodge structure
polarization
The period domain
variation of Hodge structure
The period mapping
Mixed Hodge structure

Homological algebra

Spectral sequences
The Hodge-to-de Rahm spectral sequence
The Gauss-Manin connection
The Kodaira-Spencer map
Spectra
Eilenberg-Maclane spaces
Bundles
BU and BU x Z
K-theory
Bott periodicity

Homotopy theory

Maps
Loops and suspension
BG
The Borel construction

Symmetric functions

Affine Root systems
Weyl character formula

Representations of Chevalley groups and Lie algebras

twisted K-theory
The Steinberg tensor product theorem
The BGG resolution
Modules of sl₂hat

Examples of spaces

P¹
Pⁿ
Grassmannians

MASTERS-Core 1

This course will cover basic category theory, basic topology, and basic definitions of manifolds, varieties, schemes, bundles and sheaves. A primary goal of the course will be to provide a tour of the fundamental examples in pure mathematics -- examples of number systems, algebraic stuctures, spaces, and bundles.

basic category theory: objects, morphisms, functors, adjoint functors, push forwards, pullbacks, initial and final objects, limits and colimits
basic topology: Fundamental group, homotopy groups, CW complexes
basic definitions of Spaces: charts, manifolds=varieties=schemes, sheaves=bundles
Examples of categories: Sets, abelian groups, groups, rings, fields, algebras, topological spaces, the category of categories, the category of functors, sheaves, vector bundles, varieties, manifolds, groupoids, stacks, Lie algebras, Lie groups
Examples of number systems: real numbers, finite fields, local fields (power series and p-adic numbers), hamiltonians
Examples of algebraic stuctures: polynomials, matrix algebras, symmetric algebras, exterior algebras, general linear groups, orthogonal groups, Weyl algebras, braid groups.
Examples of spaces: projective space, Grassmannians, flag varieties, Riemann surfaces, Elliptic curves, spheres, tori, lens spaces, the upper half plane
Examples of bundles: covering spaces and deck transformations, monodromy, fiber bundles, vector bundles, connections, principal bundles, classifying spaces=nerves of categories,

Prior knowledge: Undergraduate linear algebra (matrix algebra, vector spaces, bases, linear transformations), algebra (groups, rings fields), and topology (continuous functions, connected and compact)

Suggested texts:

Gelfand-Manin, Methods of homological algebra,
Hatcher, Algebraic topology
Geometry ????
Examples of spaces ????

Thank you for your comments on the pure masters program. I have spent many hours processing your comments and making a draft design of a masters program in pure maths. I have tried my best to be faithful to the content and intention of the comments that I received. Having completed the draft, I have a few opinions:

Point of view: As I progressed through this exercise, it became more and more clear to me that pure mathematics is huge and our goal should be to give a tour of it. In analogy, we must plan 2 days in Sydney, 2 days in Melbourne, 2 days in Adelaide, a day in the SA wine regions, 1 day in Perth, 1 day to the Kalgoorlie gold mines, 1 day in Darwin, 2 days in Kakadu, 1 day at Uluru, 2 days at the Barrier reef diving, 2 days on the Gold Coast, etc. The alternative: to give a semester long course on Kakadu National park, will be fantastically interesting but leave the tourist with a very limited picture of Australia.
Skills: If we expect our students to be able to use mathematics we will have to teach them about and give them some basic experience with tools that have wide use in pure mathematics research: homological algebra, categories and functors, generators and relations, LOTS of examples--particularly of spaces and actions. We should try to put focus on these things in every course. Of course these tools change with time: homological algebra was probably not one of these fundamental tools in the 1960's, but not including it now would be like expecting our future students to be successful in in life without any knowledge or concept of computers and the internet.
Marketability: Looking back now at what has come out of my efforts, I think the proposal is pretty attractive to the customer. If I were a beginning masters student, I would be quite attracted by a program that would give me this much overview of the huge world of pure mathematics and the possibility to get some understanding of so many sexy modern words.
Suggested Texts: I tried but failed -- we're going to have to write our own text(s).
The seminar course: I went back and forth over this because of some strange feeling that it wasn't politically correct and that I would be the recipient of admonition if I included it. Finally I decided that I must face the fact that I think this is as important, and probably more important, than all the other courses combined. Professional mathematicians can't do business without attending and participating in seminars, conferences and workshops and I can't see how I can expect my students to be competitive without analogous experience. It certainly takes plenty of work to run this, keeping track of the students the schedule, the topics and the speakers and tutoring the students as they prepare for their talks. I see no good reason someone couldn't receive a teaching credit for running this.
Deadline 30 Oct and reality: Comments, suggestions and changes are welcome, but they need to be very constructive and very specific. For example, please do change the 100 word description, not by suggesting it be changed but by drafting a new 100 word decription; please do change the list of courses and topics, not by saying that it should be changed but by writing a new list of courses, 100 word descriptions, list of topics and prerequities.

MASTERS-Core 2

This course will cover the basics of actions (orbits and modules), universal objects, and homological algebra. There will be a focus on basic definitions and examples. The course will begin with basic linear algebra from the point of view of modules: the Jordan normal form will be presented in the context of a classification of finitely generated modules over C[x] and the classification of finitely generated abelian groups in the context of a classification of Z-modules. Another point of view on these same results is as a classification of orbits in matrix groups. After a review of unversal properties and examples (particularly the exterior algebra), the second part of the course will be an introduction to homological algebra, with a discussion of differential forms, the Koszul and deRahm complexes, and derived functors.

vector spaces and modules: homomorphisms, free modules, composition series, radicals, semisimplicity
C[x]-modules: eigenvectors=1dimensional submodules, Jordan canonical form, finitely generated modules over a PID
Examples of universal objects: quotients, tensor products, symmetric algebras, exterior algebras, polynomial rings, symmetric functions, free associative algebras, localization, completion, free groups and presentations by generators and relations
Orbits and Conjugacy classes: Sylow subgroups, maximal compact subgroups, tori, Borel subgroups, spectral theorems, Bruhat decomposition
Homological algebra: exact sequences, complexes--Koszul and deRahm complexes, cohomology, derived functors--ext and tor,
Examples of bundles: tangent spaces, differential forms, and cohomology
Examples of cohomology theories: DeRahm cohomology, singular homology, sheaf cohomology, etale cohomology, group cohomology, Lie algebra cohomology, K-theory, Hochschild cohomology

MASTERS-Algebra

This course will be a tour of algebraic techniques in modern pure mathematics research. By its nature, the research is motivated by examples, and the first half of this course will have a focus on examples. The first part of the course will focus on the motivation and algebraic construction of the types of spaces used in the study of mirror symmetry, in the Langlands program, and mathematical physics. The second part of the course will be focused on developing the algebraic tools used for the study of these spaces: the derived category, spectral sequences, pushforward and pullback functors.

More examples of algebras: Polynomials, tensor algebras, symmetric algebras, exterior algebras, Weyl algebras, Lie algebras and enveloping algebras, group algebras, Temperley-Lieb algebras, centralizer algebras
More examples of groups: Lie groups and reductive groups -- general linear groups, orthogonal groups, symplectic groups, reflection groups, braid groups, SL_2(R), SL_2(Z), congruence subgroups
Examples of orbit spaces: projective space, Grassmannians, flag varieties, the upper half plane, elliptic curves, abelian varieties
Spec, Hilbert nullstellensatz, basic invariant theory, definition of varieties and schemes
Examples of categories of "spaces": varieties, manifolds, schemes, groupoids, stacks, Lie algebras, Lie groups
Additional category theory: abelian categories, simple objects, kernels, cokernels, projectives, injectives, exact sequences, derived functors, spectral sequences, the derived category
Examples of cohomology theories: group cohomology, Lie algebra cohomology, K-theory, Hochschild cohomology
Chern character, characteristic classes, push forwards, pullbacks, Grothendieck-Riemann-Roch, localization and the index theorem

MASTERS-Geometry

This course will be a tour of the four main types of geometry: differential geometry, complex geometry, algebraic geometry and hyperbolic geometry. For each of these subfields, there will be a tour of the main structures, and the main examples. For differential geometry this will include connections and curvature, for complex geometry there will be treatment of Riemann surfaces and conformal mappings, in the algebraic geometry postion there will be discussion of divisors, line bundles and sheaf cohomology, and the section on hyperbolic geometry will provide an introduction to geometrization and rigidity.

differential geometry: connections, curvature, symmetric spaces, Gauss-Manin, the fundamental group, monodromy, and the Riemann-Hilbert correspondence
complex geometry: Riemann surfaces, conformal mappings, classification of Riemann surfaces
algebraic geometry: sheaf cohomology, Picard groups, pencils, ample line bundles, divisors, blowups, Riemann-Roch, schemes
hyperbolic geometry: groups of isometries, rigidity, geometrization

MASTERS-Topology

This course will be an introduction to modern topology. The course will begin with review of "basic topology" including, in particular, the concepts and use of Euler characteristic and Poincare duality. The remainder of the course will have two primary foci: homotopy theory and low dimensional topology. In the homotopy theory part of the course there will be a treatment of basic structures and basic examples: suspensions, loop spaces, classifying spaces, etc, The low dimensional topology part of the course will describe the Thurston classification, discuss surgery methods and study knot complements and the basics of combinatorial group theory and geometric topology.

"Basic" topology: Euler characteristic, Euler classes, Poincare duality
Homotopy theory
- homotopy groups, CW complexes, classifying spaces, path spaces, the bar construction, Eilenberg-Maclane spaces
- basic constructions: suspensions, loop spaces, spectra, homotopy fibres, homotopy fixed points, Postnikov towers
- homotopical algebra, stable homotopy theory, rational homotopy theory
- homology theories and homotopical algebra
Low dimensional topology
- homology, surgery, lens spaces, knot complements, manifold topology
- Introduction to combinatorial group theory
- Introduction to geometric topology: geometrization and the Thurston classification

MASTERS-Analysis

This course will cover basic tools from analysis. It is loosely divided into subtopics: integration, functional analysis, complex analysis, and harmonic analysis. It will begin with an brief incursion into the world of the logarithm, the Gamma function and Riemann zeta function, using these as the first examples of the ideas of analytic continuation and monodromy. These functions provide motivating examples and tools for many parts of mathematics. The section on harmonic analysis will treat basic Hilbert spaces, proceed to Fourier analysis (the harmonic analysis of R and R/Z), and give an overview of noncommutative harmonic analysis from the point of view of the Laplacian and geometric analysis on symmetric spaces. The section on functional analysis will give an introduction to Banach spaces and the world of bounded operators and convexity. The complex analysis section will study the basics of Riemann surfaces, conformal mapping and harmonic maps. The section on measure theory is the basis of the theory of integration.

The functions zⁿ, the Riemann zeta function, the Gamma function, analytic continuation, and monodromy
Hilbert spaces, Fourier analysis, noncommutative harmonic analysis, the Laplacian, symmetric spaces
Banach spaces, Bounded operators, Topological vector spaces and convexity, the Hahn-Banach theorem and its geometric version, the Stone-Weierstrass theorem
Holomorphic functions, Cauchy-Riemann equations, harmonic functions, Riemann surfaces, conformal mapping, SL_2(R), SL_2(Z), Elliptic curves and the Weierstrass P-function
Measure theory: measures, distributions, the Riesz representation theorem, Lebesgue and Haar measure

MASTERS-Logic

This will be a basic tour of the field logic providing an introduction to P vs NP and each of the main subfields of logic: set theory, model theory, recursion theory, and proof theory. The section on set theory will include discussion of the Zemelo-Fraenkel axioms, the axiom of choice and cardinals, the section on model theory will include treatment of universal algebra and the completeness theorem, the section on recursion will treat Turing computability, decidability versus undecidability and Godel's incompleteness theorem and the section on proof theory will discuss and compare logics.

Completeness theorem for first order logic (start of model theory)
Decidability-undecidability (recursive vs. nonrecursive, computable versus noncomputable)
Godel incompletness theorem, example: undecidability of the halting problem
P vs NP and RSA encryption as an example, example of an NP complete problem
Important examples of decidable theories: Algebraically closed fields, Real closed fields, O-minimality
countable versus uncountable: uncountability of R, countability of Q

MASTERS-Pure maths seminar

Pure mathematics research does not take place in textbooks but in seminars and this is an important part pure mathematics education. Students should attend a variety of seminars, including a research seminar with lectures from outside visitors and a working seminar for students. The research seminar will normally meet once per week, and the student seminar will normally meet once per week. Usually the research seminar will be 1 hour and the student seminar will be 2 hours so that together the usual meeting time for the seminars will total 3 hours per week. Students will be expected to attend both seminars and give a lecture presentation in the working seminar at least once during the semester. Each semester the focus of the student seminar will be chosen based upon current interests of the students and the mentoring faculty.

MASTERS-Other Analysis

Martingale convergence
Measure disintegration
Convergence
Ergodic
Spectral theorem for infinite dimensional operators
Sobolev
Amenability (for groups)

MASTERS-Other algebra

modular forms, class field theory basics, Eisenstein series, L-functions
Dynkin diagrams
reflection groups
combinatorial group theory: decision problems, cayley complex, metrics on finitely generated groups, isoperimetric inequality, hyperbolic groups, mapping class groups

(1) Homogeneous representations of Khovanov-Lauda algebras (with A. Kleshchev), preprint 2008, arXiv0809.0557.

Linear algebra

Groups of Lie type

General Lie group and Lie algebra stuff

Examples of Lie algebras

Examples of algebraic groups

Groups of Lie type

Reflection groups

Weyl groups, Braid groups and Hecke algebras of Weyl groups

Affine Weyl groups, their Hecke algebras and their representations

Symmetric functions and crystals

The Brazil lectures

The CBMS Lectures

Quantum groups

Tantalizers

Geometry

Categories

Representation Theory-- The category of modules

The groups Gr,1,n, their Hecke algebras and their representations

Number systems and polynomials

Sets and functions

Limits

Graphing and geometry

Differential geometry

Integration - multivariable

Solving equations: linear, polynomial, differential

Linear algebra

Real Analysis

Group theory and Linear algebra (2nd year)

Dynamical systems and Chaos

Discrete maths

Operations Research

Probability and Statistics (2nd year)

Complex analysis

Metric and Hilbert spaces

Algebra

Geometry

PDEs

Graph theory (Aleks Owczarek, SanMing Zhou)

Discrete Math (3rd year)

Decision making: really an OR course = Game theory

techniques in OR

Stochastic models (Kostya)

Linear models (Ray Watson)

Probability and Satistical inference

Modern applied statistics

Reflection groups

Hodge theory

Homological algebra

Homotopy theory

Symmetric functions

Representations of Chevalley groups and Lie algebras

Examples of spaces

MASTERS-Core 1

MASTERS-Core 2

MASTERS-Algebra

MASTERS-Geometry

MASTERS-Topology

MASTERS-Analysis

MASTERS-Logic

MASTERS-Pure maths seminar

MASTERS-Other Analysis

MASTERS-Other algebra

The groups G_r,1,n, their Hecke algebras and their representations