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 |

- 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

- 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,

- The affine Lie algebra as an extended loop algebra
- The affine Weyl group action on 𝔥
_{ℂ}^{*} - The Weyl character formula for a Kac-Moody Lie algebra
- The Weyl character formula for an affine Lie algebra

- The double affine linear groups, affine linear groups and Heisenberg groups
- The affine orthogonal group and isometries
- Heisenberg groups and Weyl algebras
- Diff(S
^{1}) and Virasoro algebras - Weight lattices for GL
_{n}, SL_{n}and PGL_{n} - Affine root systems of classical type
- Examples of affine braid groups
- The affine braid group of type C
- Root system type C
_{2} - Serre's SL
_{2}(ℤ) picture - The group SL
_{2}and the Lie algebra sl_{2} - The group SU
_{2}and the Lie algebra su_{2} - The group PGL
_{2} - The group SL
_{3}and the Lie algebra sl_{3} - The group Sp
_{4}and the Lie algebra sp_{4} - Chevalley groups of type G
_{2} - The general linear group GL
_{n}(F) - The symplectic group Sp
_{2n}(F) - The group Sp(n)=U
_{n}(H)

- Parkinson-Ram-Schwer Section 1: Cartan, Bruhat, Iwahori and Iwasawa decompositions
- Parkinson-Ram-Schwer Section 2: Borcherds-Kac-Moody Lie algebras
- Parkinson-Ram-Schwer Section 3: Steinberg-Chevalley groups
- Parkinson-Ram-Schwer Section 4: Labeling points of the flag variety
- Parkinson-Ram-Schwer Section 5:Loop Lie algebras
- Parkinson-Ram-Schwer Section 6: Loop groups and the affine flag variety
- Parkinson-Ram-Schwer Section 7: The folding algorithm
- Parkinson-Ram-Schwer Section 8: Example of the folding algorithm
- Examples of reductive group data
- Flags and Grassmannians
- Buildings, The Fano plane as a flag variety (pdf file), Incidences and projective geometries (pdf file)
- The Borel-Weil-Bott theorem
- Mirković-Vilonen cycles
- Geometric lifting

- 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 A
_{1}and A_{2} - Schubert products G
_{2} - 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: N/T presentation, reflection presentation, Coxeter presentation, Bruhat decomposition indexing
- Braid groups: Coxeterish presentation, π
_{1}presentation, - Finite Hecke algebras: Finite Hecke algebras: Weylish presentation, Coxeterish presentation, C(B\G/B) presentation,
- Affine Weyl groups: Semidirect product presentation, Coxeterish presentation, reflectionish presentation, Iwahori decomposition indexing
- Affine Braid groups: Bernsteinish presentation, Coxeterish presentation, π
_{1}presentation, - Affine Hecke algebras: Bernstein presentation, Coxeterish presentation, Affine Weylish presentation, C(I\G/I) presentation,

- 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
- 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 A
_{1} - Matt Davis PhD Thesis type A
_{1}xA_{1} - Matt Davis PhD Thesis type A
_{2} - Matt Davis PhD Thesis type C
_{2} - Matt Davis PhD Thesis type G
_{2} - Matt Davis PhD Thesis Geometric indexing type C
_{2} - Matt Davis PhD Thesis Geometric indexing type G
_{2}

- Symmetric functions
- Interpolating symmetric functions: Elementary, homogeneous and power sum symmetric functions
- Hall-Littlewood polynomials at roots of unity
- The rings Z[P] and Z[P]^W
- Some q-Weyl dimension formula examples
- MV (Mirković-Vilonen) polytopes
- Examples

- Lecture 1. Reflection groups and groups of Lie type
- Lecture 2. Representations of the Symmetric group
- Lecture 3. The Weyl character formula
- Lecture 4.Crystals from paths and MV polytopes
- Lecture 5. Crystals from paths and preprojective algebras

- 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 quantum group U
_{q}sl_{2} - Lie bialgebras (pdf file )
- Enveloping algebras (pdf file )
- coPoisson Hopf algebras (pdf file )
- Lie algebra cohomology (pdf file )
- Lie bialgebra structures for Lie algebras with triangular decomposition (pdf file )
- Drinfeld-Jimbo quantum groups (pdf file )
- Drinfeld-Jimbo quantum groups
- Definition of the integral form of Drinfeld-Jimbo quantum groups
- Proof of the quantum Steinberg identity
- The quantum groups of type GL
_{2}, SL_{2}and PGL_{2} - The quantum double (pdf file)
- sl2 (pdf file)
- Quantization (pdf file)
- two dimensional (pdf file)
- Hopf algebras
- Quasitriangular Hopf algebras
- The quantum double
- RTT algebras
- Spectral subalgebras, more spectral subalgebras
- Transfer Matrices

- Cartan matrices, Dynkin diagrams and Quiver data (pdf file)
- Tensor and shuffle algebras
- The braid group action
- PBW bases and Canonical bases
- PBW bases for sl
_{3} - The shuffle realization of U-
- Lyndon words
- Dual PBW bases
- Quiver Hecke algebras
- Quiver Hecke modules
- Quiver Hecke algebra representations and characters
- Seminaire Lotharingien Combinatoire I
- Seminaire Lotharingien Combinatoire II
- Seminaire Lotharingien Combinatoire III
- Quiver varieties
- Preprojective Algebras
- Hall Algebras
- Universal Verma modules
- Translation
- The category O
- Projectives in the Category O
- Multisegments

- The degenerate affine braid algebra
- The degenerate affine BMW algebra
- A basis of the degenerate affine BMW algebra
- The center of the degenerate affine BMW algebra
- Group algebra of the affine braid group
- Affine BMW algebra
- A basis of the affine BMW algebra
- The center of the affine BMW algebra
- Central element recursions
- Classical type Lie algebras
- Actions of tantalizers
- Central elements in classical type quantum groups
- Subalgebras of the partition algebra
- Schur-Weyl dualities
- The tower A
_{k}(r,p,n) - Murphy elements in A
_{k}(p,r,n) - The symmetric group and Brauer algebras
- Affine Type C Temperley-Lieb algebras
- Representation theory Lecture 1: Basic representation of finite dimensional algebras via Templerley-Lieb algebras

- Spaces, What is a Space?
- simplicial complexes, CW complexes, simplicial objects
- Spectra
- Varieties and Schemes, C
^{r}-manifolds, smooth manifolds, complex manifolds - Affine varieties, affine schemes (i.e. spec)
- Sheaves and Ringed spaces, Stalks, O-modules and D-modules
- D-modules and the characteristic variety
- Constructible functions
- Maximal ideals, prime ideals, and primary ideals
- Homotopy groups
- Fibrations, covers, central extensions, fibre bundles, principal bundles, vector bundles
- Bundles, bundles and homotopy groups
- Covers: subgroups of π
_{1}and ramification - Derivations, Tangent spaces and differential forms
- grad, curl, div (and, to come, Stokes theorem)
- Connections
- The Gauss-Manin connection
- Koszul and DeRahm complexes
- spheres
- tori
- Riemann surfaces
- Moduli spaces
- Complex tori and Abelian varieties
- The Klein bottle
- L functions and zeta functions
- Deligne Diff Eq Section 1-3
- Deligne Weil I: Section 2, Section 3, Section 4, Section 6

- Introduction to categories
- Morphisms and products
- Universal objects: examples (free groups, tensor algebras, symmetric algebras, exterior algebras, polynomial rings, quotients, completions, products, disjoint unions, tensor products, inverse limits, direct limits, fields of fractions, localisation...)
- Functors: examples (tensor product and Tor, Hom and Ext, localisation=fractions, Spec, associated graded)
- Generators and relations: presentations
- Homology, group cohomology, Tor
- Homology of complexes, totalization and spectral sequences
- cones, cylinders, the connecting homomorphism, long exact sequences, derived functors and axioms for cohomology theories
- homotopy groups and homotopy long exact sequences
- Singular homology and cohomology, Simiplicial homology and cohomology, cellular homology and cohomology, homology and cohomology for simiplicial sets
- Ext and Tor, Lie algebra homology and cohomology, Group homology and cohomology
- Sheaf and Cech cohomology

- Algebras
- Schur's lemma and Double centralisers
- The Regular representation
- Complete reducibility
- Tits deformation theorem
- Clifford theory
- Radicals
- Representations - Examples
- Generalised Matrix algebras
- Almost semisimple algebras
- Lecture Notes in Representation theory 1993 (notes taken by Rob Leduc and Tom Halverson and Mark Mckinzie) Outline, Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11, Lecture 12, Lecture 13, Lecture 14, Lecture 15, Lecture 16, Lecture 17, Lecture 18, Lecture 19, Lecture 20, Lecture 21, Lecture 22,
- Operator algebras
- Functions, measures, distributions
- Admissible Representations
- Group algebras
- General Hecke algebras
- Fourier analysis on compact Lie groups
- Haar measure
- Monomial groups
- The Heisenberg group
- Kirillov's classification
- Abelian varieties (pdf fileA) (pdf fileB)

- The symmetric group G
_{1,1,n} - The hyperoctahedral group G
_{2,1,n} - The cyclic group G
_{r,1,1}of order r - The dihedral group G
_{r,r,2} - Representations of the dihedral Hecke algebra
- The groups G
_{r,p,n} - The groups G
_{r,p,n}number two - The affine symmetric group G
_{∞,1,n} - The Iwahori Hecke algebra of type A
- Iwahori-Hecke algebras of type B
- The Weyl group of type D
- The Hecke algebra of type G
_{r,p,n} - The groups G
_{H}(H,k,n) - Representations of the groups GH(H,k,n)
- The groups G
_{H,1,n}

**Numbers**- Lecture 1: Numbers
- The integers ℤ, The Rationals ℚ, The Real numbers ℝ, The complex numbers ℂ, The quaternions $\mathbb{H}$
,

- 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 ℝ
^{n}, complex space C^{n}, quaternionic space H^n - The 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)); x
^{n}- 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)

**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
- Derivatives: Examples
- Derivatives: Exercises
- More derivatives: Exercises
- Indefinite integrals - Exercises
- Exponential, trig and hyperbolic functions: Exercises
- additional trig identities: Exercises
- Trigonometry: Exercises
- exponential functions and motion applications: Exercises
- 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, 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

- 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

- 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

- 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,∞) - R
^{n}and T^{n} - M
_{nxm}(R) and GL_{n}(R) - S
^{n-1}and B_{n} - P
^{n}and G_{n,p}

**Groups**

- Groups, homomorphisms
- Quotients and Normal subgroups
- Group actions
- Group actions (again but with proofs? 28 May 2010)
- p-groups and Sylow theorems
- Nilpotent and solvable Groups
- Exercises
- Examples of finite groups
- Cyclic groups
- Dihedral groups
- Symmetric groups
- Alternating groups
- List of Groups of low order
- Cyclic group of order 2
- The Klein four group
- Nonabelian group of order 6
- Group action of dihedral group of order 6
- Dihedral group of order 8
- Group action of dihedral group of order 8
- Quaternion group, order 8
- Tetrahedral group
- Group action of the tetrahedral group
- Octahedral group
- Group action of the octahedral group

- Examples of infinite groups

**Rings**

- ????
- Ideals incomplete
- Exercises
- Examples of rings
- Z, Z
_{(p)}, Z_{p} - F[x] and A[x]
- F[x
_{1},...,x_{n}] and A[x_{1},...,x_{n}] - F[[x
_{1},...,x_{n}] and A[[x_{1},...,x_{n}]]

- Z, Z

**Fields**

**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.

- Limits-Examples (calculations Ramulus)
**Sequences**- Sequences-Examples
- 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

**Applications of limits**- Derivatives and limits
- 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

- 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

- 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

- ?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

- 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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

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

- 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

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

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

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

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

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

- 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

- Maps
- Loops and suspension
- BG
- The Borel construction

- Affine Root systems
- Weyl character formula

- twisted K-theory
- The Steinberg tensor product theorem
- The BGG resolution
- Modules of sl
_{2}hat

- P
^{1} - P
^{n} - Grassmannians

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.

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

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

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

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

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 z
^{n}, 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

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

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.

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

- 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.