2007-2008 Graduate Course Descriptions

CORE COURSES

1. Lebesgue integration, measure theory, convergence theorems, the Riesz representation theorem, Fubini’s theorem, complex measures.
2. L^p-spaces, density of continuous functions, Hilbert space, weak and strong topologies, integral operators.
3. Inequalities.
4. Bounded linear operators and functionals. Hahn-Banach theorem, open-mapping theorem, closed graph theorem, uniform boundedness principle.
5. Schwartz space, introduction to distributions, Fourier transforms on the circle and the line (Schwartz space and L₂ ).
6. Spectral theorem for bounded normal operators.

1. Review of elementary properties of holomorphic functions. Cauchy's integral formula, Taylor and Laurent series, residue calculus.
2. Harmonic functions. Poisson's integral formula and Dirichlet's problem.
3. Conformal mapping, Riemann mapping theorem.
4. Elliptic functions and Riemann surfaces.
5. Analytic continuation, monodromy theorem, little Picard theorem.

Some topics to be covered:

1. Nonlinear first-order PDE. Method of characteristics.
2. The Fourier Transform. Distributions.
3. Sobolev spaces on Rⁿ. Sobolev spaces on bounded domains. Weak solutions.
4. Second order elliptic partial differential operators. The Laplace operator. Harmonic functions. Maximum principle. The Dirichlet and Neumann problems. The Lax-Milgram Lemma. Existence, uniqueness and eigenvalues. Green's functions. Single layer and double layer potentials.
5. Hyperbolic partial differential equations. The wave equation. The Cauchy problem. Energy methods. Fundamental solutions. Domain of influence. Propagation of singularities.

The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.

Reference:
Lawrence Evans: Partial Differential Equations


MAT 1100YY
ALGEBRA
P. Selick

1. Linear Algebra. Students will be expected to have a good grounding in linear algebra, vector spaces, dual spaces, direct sum, linear transformations and matrices, determinants, eigenvectors, minimal polynomials, Jordan canonical form, Cayley-Hamilton theorem, symmetric, alternating and Hermitian forms, polar decomposition.
2. Group Theory. Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
3. Ring Theory. Rings, ideals, rings of fractions and localization, factorization theory, Noetherian rings, Hilbert basis theorem, invariant theory,Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties.
4. Modules. Modules and algebras over a ring, tensor products, modules over a principal ideal domain, applications to linear algebra, structure of semisimple algebras, application to representation theory of finite groups.
5. Fields. Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.

Main text: Bredon's "Topology and Geometry".

Optimistic plan:

8 weeks of local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.

5 weeks of differential forms: exterior algebra, forms, pullbacks, d, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.

5 weeks of fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.

8 weeks of homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.

The class will be hard and challenging and will include a substantial component of self-study. To take it you must feel at home with point-set topology, multivariable calculus and basic group theory.

2007-2008 TOPICS COURSES, including CROSS-LISTED COURSES

An introduction to the theory of several complex variables.

Prerequisites: a undergraduate course in one complex variable and real analysis.


MAT 1016HF
INTRODUCTION TO OPERATOR ALGEBRAS
M.-D. Choi

This course is part of the Fields Institute Fall 2007 thematic program on Operators Algebras.

There will be self-contained lectures covering basic aspects of operator algebras. No previous knowledge of operator algebras is required. Students taking this course for academic credit will be required to write a written report on a topic of related interest.


MAT 1035HF
STRUCTURE OF C*-ALGEBRAS
G.A. Elliott

This course is part of the Fields Institute Fall 2007 thematic program on Operators Algebras.

Recent investigations have revealed that, for the most common class of C*-algebras, the amenable ones, there may exist a complete classification scheme, in some ways as simple as the periodic table of the elements. Since it also appears feasible to construct very special algebras---the structure of which is quite transparent---exhausting the range of the invariant arising in the classification, completing the classification of amenable C*-algebras will likely answer all (or most) questions about the structure of these algebras as well.

References:
Huaxin Lin: An Introduction to the Classification of Amenable C*-Algebras, World Scientific Press, Singapore, 2001.
Mikael Rordam, Flemming Larsen, and Niels Jakob Laustsen: An Introduction to K-Theory for C*-Algebras, London Mathematical Society Student Texts 49, Cambridge University Press, 2000.
Mikael Rordam: Classification of Nuclear, Simple C*-Algebras, Encyclopaedia of Mathematics, Springer-Verlag, 2002.


MAT 1045HS
ERGODIC THEORY
K. Khanin

The course is an introduction to some of the basic notions and methods of the ergodic theory. It covers such notions as minimality, topological transitivity, ergodicity, unique ergodicity, weak mixing and mixing. These notions will be explained by examining simple concrete examples of dynamical systems such as translations and automorphisms of tori, expanding maps of the interval, Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, and the Birkhoff ergodic theorem will be presented. We also plan to outline the thermodynamic formalism, entropy theory, and the theory of Lyapunov exponents.

Prerequisites: Knowledge of real analysis, basic topology and measure theory.

Intent: This course is the second part of a two semester core course that prepares students for research in dynamics and ergodic theory. This year is a trial of the concept. Charles Pugh will give the first component of the course (Introduction to Dynamics, MAT 1844HF) in the fall semester. If you want to study dynamics and/or ergodic theory you should take the course.

Textbooks:
I. Kornfeld, S. Fomin, Ya. Sinai, Ergodic Theory, Springer, 1982.
Ya. Sinai, Topics in Ergodic Theory, Princeton University Press, 1993.


MAT 1062HS
COMPUTATIONAL METHODS FOR PDE
M. Pugh

We'll study numerical methods for solving partial differential equations that commonly arise in physics and engineering. We will pay special attention to how numerical methods should be designed in a way that respects the mathematical structure of the equation.

Outline:

Parabolic PDE

explicit and implicit discretizations in 1-d

consistency, stability, and convergence in 1-d

boundary conditions in 1-d

multi-dimensional problems

Elliptic PDE

solution of sparse linear systems

variational formulations and finite element methods

Hyperbolic PDE

CFL stabilty condition

nonlinear conservation laws, shock capturing

Special topics

pseudospectral methods

Prerequisites: You should be familiar with the material that would be taught in a serious undergraduate PDE course. Sample programs will be provided in matlab. If you know matlab, great! If you don't, you're expected to be sufficiently comfortable with computers that you can learn matlab on the fly. Which isn't actually hard at all, unless you hate computers.


MAT 1063HF
MICROLOCAL ANALYSIS AND APPLICATIONS I
V. Ivrii

This course will be more introduction and theory than applications:

Content:

Theory of Distributions.

Classes D, E, S and their dual D', E', S'. Basic operations, Fourier transform.

Sobolev spaces H^s on R^d.

Paley-Wiener theorem.

Calculus of Pseudodifferential Operators:

Symbols, Quantization, Calculus.

Oscillatory Front Sets, coherent states, microlocalization.

Inverse of elliptic operator, resolvent.

Functional calculus.

Analysis of Pseudodifferential Operators.

L²-estimates

Garding inequalities.

Pseudodifferential Operators and Boundary Value Problems.

Classical pseudodifferential operators.

Parametrix construction for elliptic boundary value problem.

Other types of operators appearing in parametrix construction.

Dirichlet-to-Neumann operator.

Non-elliptic boundary value problems.

Applications to Hyperbolic Systems.

Proof of well-posedness of the Cauchy problem for strictly hyperbolic systems.

Recommended prerequisites:
Real Analysis (graduate or undergraduate);
Complex Analysis (graduate or undergraduate, or even non-specialist);
Ordinary Differential Equations (graduate or undergraduate);
Partial Differential Equations (graduate or undergraduate).


MAT 1075HS
MICROLOCAL ANALYSIS AND APPLICATIONS II
V. Ivrii

This follow-up course of MAT 1063HF will cover theory and applications.

Content:

Fourier Integral Operators:

Oscillatory solutions, relation to classical dynamics.

Elements of classical dynamics.

Lagrangian distributions, phase functions and amplitudes, Lagrangian manifolds.

Canonical graphs and Fourier Integral Operators.

Metaplectic operators.

Oscillatory solutions near and beyond caustics.

Maslov Canonical Operator.

Propagation of singularities.

Fourier Integral Operators Approach.

Heisenberg approach.

Coherent States approach.

Energy estimates approach.

Propagation near boundary (survey)

Applications to Spectral Asymptotics.

Tauberian Approach.

Poisson Relations, Spectrum and Length Spectrum.

Sharp spectral asymptotics.

Recommended prerequisites:
MAT 1063HF Microlocal Analysis and Applications I (as of Fall 2007); while not an official pre-requisite one needs to know the material;
Real Analysis (graduate or undergraduate);
Complex Analysis (graduate or undergraduate, or even non-specialist);
Analysis on Manifolds (elements);
Ordinary Differential Equations (graduate or undergraduate);
Partial Differential Equations (graduate or undergraduate).


MAT 1101HF
GROUP THEORY TODAY
B. Szegedy

Group theory is useful in almost every area of mathematics but it is interesting (and very aesthetic) in its own right. The goal of the course is to give an overview about modern group theory. We will touch the following topics (and more): finite groups, pro-finite groups, arithmetic groups, Kazhdan's property, amenable groups, geometric group theory. To make the course as self-contained as possible I will start with a short introduction to group theory.

Prerequisites are Linear Algebra and some open-mindedness towards abstract stuff.


MAT 1120HF
LIE ALGEBRAS
J. Repka

An introduction to the theory of Lie algebras, with particular emphasis on semisimple Lie algebras, their classification and representation theory. Topics include parabolic subalgebras, the universal enveloping algebra, and the Weyl Character Formula.

Recommended preparation: MAT1100Y or equivalent.

Useful References:
1. Humphreys: Introduction to Lie Algebras and Representation Theory.
2. Your favourite text on Linear Algebra.


MAT 1121HS
LIE GROUPS AND ALGEBRAIC GROUPS
F. Murnaghan

Introduction to Lie groups, compact Lie groups, semisimple and reductive Lie groups, classical groups, Iwasawa decomposition, Bruhat decomposition, parabolic subgroups, Cartan subgroups.

If time permits, we will include a general discussion of reductive linear algebraic groups.

Prerequisites: core algebra, analysis and topology courses. Some knowledge of Lie algebras will be assumed (for example, material from MAT 1120HF).


MAT 1155HF
COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY
K. Kaveh

This course will provide a basic introduction to algebraic geometry and the underlying commutative algebra. Topics will include:

Affine varieties: Nullstellensatz, Zariski topology, ideal theory in Noetherian rings, Krull dimension, examples

projective varieties: graded rings and ideals, P^n, examples, cones, quadrics, Grassmannians

morphisms: regular functions, localization, local rings, rational functions, morphisms, products, examples

rational maps: dominant and birational morphisms, birational equivalence, blowing up, examples

nonsingular varieties: singular and nonsingular points, regular local rings, completions, analytic isomorphism, curve singularities,

nonsingular curves: function fields, valuations, discrete valuation rings, fundamental theorem on nonsingular curves and function fields, examples.

intersection theory: dimensions, Hilbert polynomials, intersection multiplicities, Bezout's theorem

If time permits:

Riemann-Roch (for curves) and

Introduction to toric varieties.

Basic reference:
R. Hartshorne, Algebraic Geometry, Chapter I.

Additional references:
I. Shafarevich, Basic algebraic geometry I.
J. Harris, Algebraic geometry, a first course.
W. Fulton, Algebraic curves: an introduction to algebraic geometry.
D. Mumford, The red book of varieties and schemes. LN 1358
_________, Algebraic geometry I: complex projective varieties.

Prerequisite: One year of algebra at a graduate level, e.g. MAT 1100Y.


MAT 1190HS
INTRODUCTION TO SCHEMES
S. Kudla

This course will provide a basic introduction to the theory of schemes with an emphasis on the arithmetic examples.

Topics will include:

Basic definitions: Spec, affine schemes, schemes, morphisms, fiber products and base change, reduced and nonreduced schemes, the functor of points, Proj, separated, proper and projective morphisms, tangent spaces, regular schemes, flat, étale and smooth morphisms, coherent sheaves and their cohomology, Kaehler differentials, blowing up, examples, curves, surfaces and arithmetic surfaces, groups schemes.

References:
Q. Liu, Algebraic geometry and arithmetic curves.
D. Eisenbud and J. Harris, The geometry of schemes.

Additional references:
R. Hartshorne, Algebraic geometry, Chapters II and III.
J. Silverman, Advanced topics in the arithmetic of elliptic curves.
D. Mumford, The red book of varieties and schemes. LN 1358.

Prerequisites: One year of algebra at a graduate level, e.g. MAT 1100Y. Some familiarity with algebraic number theory, basic algebraic geometry (e.g., the first chapter of Hartshorne) and sheaves will be assumed.


MAT 1199HF
SPECTRAL METHODS OF AUTOMORPHIC FORMS
V. Blomer

Automorphic Forms are functions that are invariant under a discrete subgroup of SL₂(R), so they are a natural generalization of periodic functions (invariant under a discrete subgroup of R). The rich theory of automorphic forms combines number theory, non-abelian harmonic analysis, representation theory and Riemannian geometry. We cover

classical automorphic forms and Hecke theory

invariant integral operators and spectral theory

Selberg's trace formula and Kuznetsov sum formula

Representation theory of GL₂(R)/SL₂(Z)

Applications to L-functions, hyperbolic lattice point problems, etc.

This course prepares one to read original papers in the field.

Literature:
H. Iwaniec, Spectral Methods of Automorphic Forms, 2002.
D. Bump, Automorphic Forms and Representations, 1997.
Y. Motohashi, Spectral Theory of the Riemann Zeta-Functions, 1997.
P. Sarnak, Some Applications of Modular Forms, 1990.

Prerequisites: Analysis, Algebra, Number Theory


MAT 1202HS
ANALYTIC NUMBER THEORY
J. Friedlander

This course is part of the Fields Institute Winter 2007 thematic program on New Trends in Harmonic Analysis.

Prerequisites: A basic course in complex variables and one in abstract algebra.

Fourth year undergraduate students welcome.


MAT 1202HF
INTRODUCTION TO THE THEORY OF L-FUNCTIONS
H. Kim

In this course we want to give an overview of the variety of L-functions, their importance in number theory and automorphic forms. An L-function is a type of generating function formed out of local data associated with either an arithmetic-geometric object (such as elliptic curves and Shimura varieties) or with an automorphic form. After introducing the classic examples such as the Riemann zeta function and Dirichlet L-functions, we study their zero free region which implies the prime number theorem. Next, we study modular forms and their L-functions, mainly their meromorphic continuation and functional equations.

Prerequisite: Complex analysis.

References:
Serre, A Course in Arithmetic
Titchmarsh, The Theory of the Riemann Zeta Function
Jorn Steuding, Introduction to the Theory of L-functions


MAT 1302HS (APM 461H1S/CSC 2413HS)
COMBINATORIAL METHODS
S. Tanny

We will cover a selection of topics in enumerative combinatorics. Some of these include: a focus on direct counting methods and their application in various contexts; general approaches for solving linear, constant coefficient recursions, more advanced methods in recursions; an analysis of some unusual recursions, including the Josephus recursion and self-referencing recursions; binomial coefficients and their identities; and some special combinatorial numbers and their identities (Fibonacci, Stirling, and Eulerian).

Prerequisite:
Linear algebra.

Recommended preparation:
an introductory combinatorics course, such as MAT 344H.


MAT 1312HS
COMPLEX HYPERBOLIC GEOMETRY
J. Bland

This course will be an introduction to Kobayashi hyperbolicity. It will serve as a preparation for a special semester at the Fields Institute in the fall of 2008 on Arithmetic Geometry, Hyperbolic Geometry.

Topics: Complex spaces, Schwarz Lemma and its generalization, Kobayashi metric, Kobayashi hyperbolicity, Brody's theorem, complex line bundles, complex vector bundles, metrics, connections, curvature.

Prerequisites: Manifolds, differential forms, metrics, complex analysis.

References:
S. Kobayashi, Hyperbolic manifolds and holomorphic mappings.
S. Lang, Introduction to complex hyperbolic spaces.


MAT 1313HF
COMPARISON GEOMETRY
V. Kapovitch

The course will cover the following topics: Applications of Toponogov angle comparison, Volume comparison, Laplace comparison and eigenvalue estimates under various Ricci curvature bounds. Critical point theory for distance functions and its applications. Grove-Shiohama sphere theorem, finiteness theorems under various geometric bounds, Gromov Betti number estimates.

Prerequisite: Differential geometry, e.g. MAT 464H/1342H.


MAT 1314HS
INTRODUCTION TO NONCOMMUTATIVE GEOMETRY
R. Ponge

This course aims to give an introduction to some basics concepts of Connes' noncommutative geometry with an emphasis on some recent applications in differential geometry, local index theory and mathematical physics. Topics to be covered are:

Spectral triples as the noncommutative substitutes for manifolds. Example of the Dirac spectral triple and how noncommutative geometry enables to define the area of a 4-dimensional Riemannian manifold.

Cyclic cohomology as the noncommutative substitute for differentiable forms and the natural receptacle for the noncommutative Chern Character. Examples of the Cyclic cohomology of the algebra of smooth functions on a manifold and of the noncommutative torus.

Local index formula and higher index formula in noncommutative geometry. Example of the Atiyah-Singer index formula for Dirac operators. Higher index formula of Connes-Moscovici for Dirac operators on Galois coverings. Applications to the Novikov conjecture and the quantum Hall effect.

Smooth manifolds, Sard's theorem and transversality. Immersion and embedding theorems. Mapping degree and Pontryagin's construction. Intersection theory. Euler characteristic, Hopf theorem, Morse theory. Classification of compact surfaces. Applications of smooth topology to algebra and calculus.

References:
J. Milnor: Topology from the Differential Viewpoint.
J. Milnor: Morse Theory.


MAT 1341HS
DIFFERENTIAL TOPOLOGY II: Algebraic Topology from a Differnetial Viewpoint
A. Khovanskii

Differential forms and De Rham cohomology, its isomorphism to simplicial and singular cohomology. Mayer-Vietoris sequence. Poincare duality. Vector bundles. Tom class. Spectral sequences, Lefschetz fix points theorem. Characteristic classes. Applications to algebra and geometry.

References:
R. Bott and L.W. Tu: Differential Forms in Algebraic Topology.
J. Milnor and J. Stasheff: Characteristic Classes.


MAT 1345HF
HOMOLOGICAL ALGEBRA
S. Arkhipov

1. Homology and cohomology of finite groups. Examples. 1-, 2- and 3-cocycles, group extensions etc. The standard complexes. Cohomology of Z and Z/n.
2. Language of categories. Categories of modules over an algebra. Bimodules and functors. Adjoint functors. Representable functors. Kernel, image and cokernel of a map.Short and long exact sequences. 5-lemma.
3. Projective and injective objects. Resolutions. Homotopy equivalence. Derived functors.
4. Homology and cohomology of groups, associative algebras, Lie algebras as classical derived functors.
5. Sheaves on topological spaces. Global sections. Stalks. Presheqaves and sheaves, sheafication. Godement resoution of a sheaf. Cohomology with coefficients in a sheaf. Cech and De Rham cohomology of a topological space.
6. Inverse and direct images of sheaves. Derived functors. Cohomology with compact support. Verdier duality.

The numbers below approximately correspond to the week numbers:

1)    Preliminaries/reminder:
      Symplectic manifolds, Hamiltonian fields, Darboux theorem, Lagrangian
      manifolds and foliations, integrable systems.

2)    Symplectic properties of billiards, geodesics on an ellipsoid.

3-4)  Symplectic fixed points theorems:
      Poincare-Birkhoff theorem, Arnold's conjecture, Conley-Zehnder theorem.

5-6)  Morse theory: Morse inequalities, Lusternik-Schnirelmann category,
      applications to geodesics, other ramifications (Morse-Witten complex,
      Morse-Novikov theory); the end of proof for Conley-Zehnder.

7)    Glimpse of generating functions for symplectomorphisms, non-squeezing
      results, symplectic capacities, Floer homology.

8)    Hofer metric, geometry of and geodesics on symplectomorphism groups.

9-10) Contact structures, Legendrian knots, their invariants and
      Bennequin inequality; glimpse of contact homology of Legendrian knots.

11)   The Lie-Poisson bracket, compatible brackets, the shift argument
      method,  integrability.

12)   Toda lattices and the KdV equation.

References:
S. Tabachnikov, "Introduction to symplectic topology" Lecture notes, (PennState U.): http://www.math.psu.edu/tabachni/courses/symplectic.pdf
D. McDuff and D. Salamon: "Introduction to symplectic topology" (Oxford Math. Monographs, 1998)
A. Perelomov: "Integrable systems of classical mechanics and Lie algebras" (Birkhauser, 1990)

Prerequisite: A basic course in symplectic geometry (or familiarity with its main notions).


MAT 1404HF (MAT 409H1F)
INTRODUCTION TO MODEL THEORY AND SET THEORY
W. Weiss

We will introduce the basic principles of axiomatic set theory, leading to the undecidability of the continuum hypothesis. We will also explore those aspects of infinitary combinatorics most useful in applications to other branches of mathematics.

Prerequisite: an introductory real analysis course such as MAT 357H

References:
W. Just and M. Weese: Discovering Modern Set Theory, I and II, AMS.
K. Kunen: Set Theory, Elsevier.


MAT 1430HS
SET THEORY: FORCING
S. Todorcevic

This will cover two of the standard themes of an advanced set theory course, forcing and large cardinals. For example, we will go over some of the most important forcing constructions including the Easton product construction and iterated forcing constructions such as finite and countable support iterations. We shall also examine how these constructions get amplified with the large cardinal assumptions.

References:
K. Kunen: Set Theory, An Introduction to Independence Proofs, North Holland 1980.
A. Kanamori, The Higher Infinite, Springer 2005.


MAT 1507HF (APM 441H1F)
ASYMPTOTIC AND PERTURBATION METHODS
C. Sulem

Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained coordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering.)

Reference:
Carl M. Bender and Steven A. Orszag: Advanced Mathematical Methods for Scientists and Engineers.


MAT 1508HS
REACTION-DIFFUSION EQUATIONS AND THEIR APPLICATIONS IN BIOLOGY, CHEMISTRY, PHYSICS AND MATERIAL SCIENCES
I.M. Sigal

The reaction diffusion equation is the workhorse of applied mathematics. These equations are used for modeling physical, chemecal, biological and sociological phenomena. They play an important role in material science and engineering. Recently these, or more precisely related, equations have played a crucial role in solving Poincaré conjecture.

In this course we cover the most central of these equations. Here are some examples of the equations we consider with the main areas of their applications: Allen-Cahn equation (condensed matter physics, material science; in condensed matter physics it is known as the Ginzburg-Landau equation), Cahn-Hilliard (material science, biology), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology) and Keller-Segel equations (biology).

We will also consider the equations which play an important role in geometry: the mean-curvature flow and Ricci flow. The latter was used in particular in the solution of the Poincaré conjecture, while the former plays an important role also in image recognition.

Prerequisites: Some familiarity with elementary ordinary and partial differential equations. Some familiarity with elementary theory of functions would be helpful.


MAT 1700HS (APM 426H1S)
GENERAL RELATIVITY
W. Abou-Salem

Special relativity. The geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein's equation. Cosmological consequences: the big bang and inflationary universe. Schwarschild stars: the bending of light and perihelion procession of Mercury. Black hole dynamics. Gravitational waves.

Prerequisites:
Thorough knowledge of linear algebra and multivariable calculus. Some familiarity with partial differential equations, topology, differential geometry, and/or physics will prove helpful.

Reference:
R. Wald, General Relativity, University of Chicago Press


MAT 1711HF
CONFORMAL FIELD THEORY IN TWO DIMENSIONS I
K. Hori

Objective: Conformal field theory in two dimensions is the starting point for perturbative formulation of string theory and also an important method in modern condensed matter physics. From the mathematics side, CFT is already a background for many of recent development, such as geometry of loop space and loop groups, geometric representation theory, mirror symmetry, twisted K-theory, geometric Langlands duality, etc. The course aims at an accessible introduction of this important subject in mathematical physics.

Outline:

0. QFT - introduction

1. Free field theory
 1.1 Free scalar fields
      massless scalar and conformal invariance
      T-duality
      the case with boundary: Dirichlet/Neumann/mixed
 1.2 Free Fermions
      massless Dirac fermion and conformal invariance
      Majorana fermions
      twisted boundary conditions, NS and R sector
      the case with boundary
 1.3 Boson-fermion correspondence
      GSO projection
      comment on interacting theories
      the case with boundary

2. Conformal Field Theory from Phase Transitions
    Magnetic phenomena and Ising models
      mean field theory and Landau's theory
    2d Ising model
      Onsager's solution
      free field realization
      order/disorder operators
    Other models

3. Infinite conformal symmetry
    Conformal symmetry and anomaly
    Ward identity for conformal symmetry
    The Virasoro algebra
    Representation theory of Virasoro algebra --- preview
    Solution of unitary minimal models
    example --- critical Ising model

Prerequisite: Solid background in quantum mechanics and statistical mechanics.

References:
C. Itzykson, H. Saleur and J.-B. Zuber (Editors), "Conformal Invariance and Applications to Statistical Mechanics".
C. Itzykson and J.-M. Drouffe, "Statistical Field Theory"



MAT 1723HF (APM 421H1F)
MATHEMATICAL CONCEPTS OF QUANTUM MECHANINCS AND QUANTUM INFORMATION
I. M. Sigal

The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some topics which are at the forefront of active research. Among the latter topics we cover Bose-Einstein condensation and quantum information. Both of these have witnessed an explosion of research in the last decade and both involve deep and beautiful mathematics.

We will try to be as self-contained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively time-consuming we give an idea of the proof, if such exists, and/or explain the mathematics involved without providing all the details.

Prerequisites for this course: some familiarity with elementary ordinary and partial differential equations. Knowledge of elementary theory of functions and operators would be helpful.

Syllabus:

• Schroedinger equation
• Quantum observables
• Spectrum and evolution
• Density matrix
• Bose-Einstein condensation
• Quasiclassical asymptotics
• Approximate methods
• Hartree-Fock theory
• Open systems and Lindblad evolution
• Quantum entropy
• Quantum channels and information processing
• Quantum Shannon theorems

References:
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2005

In covering information theory we will follow on-line material, papers and the books

A.S. Holevo, Statistical Structure of Quantum Theory, Springer, 2001.
A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam, The Netherlands: North Holland.


MAT 1739HS
CONFORMAL FIELD THEORY IN TWO DIMENSIONS II
K. Hori

1. Representation theory of some infinite dimensional Lie algebras
     Introduction --- Finite dimensional simple Lie algebras
     The Virasoro algebra
     affine Lie algebras
     Super-Virasoro algebras

2. WZW models
    The classical theory
    Knizhnik-Zamolodchikov equations
    The space of conformal blocks and Verlinde formula
    The geometry of loop groups
    Gauged WZW models

3. Superconformal Field Theories
    Spectral flows
    N=2 Landau-Ginzburg models
    N=2 minimal models
    Calabi-Yau sigma models
    Mirror symmetry
    Superstring compactifications

Prerequisite: Some familiarity with finite dimensional simple Lie algebras and their representation theory. Some familiarity with elementary notions in differential geometry (such as fibre bundles and connections).

References:
C. Itzykson, H. Saleur and J.-B. Zuber (Editors), "Conformal Invariance and Applications to Statistical Mechanics".
A. Pressley and G. Segal, "Loop Groups".
V. Kac, "Infinite Dimensional Lie Algebras".


MAT 1750HF
COMPUTATIONAL MATHEMATICS
M. Shub

Computational Linear Algebra. Here the standard material is worked through but from a complexity theoretic and dynamical systems point of view. This course includes linear algebra topics such as in Trefethen and Bau "Numerical Linear Algebra", Gaussian elimination and iterative methods for solving linear systems, the QR algorithm, and QR with shifts and other iterative methods for solving eigenvalue problems. An emphasis will be put on condition numbers and complexity and the dynamics of the algorithms.

Textbook:
Lloyd N. Trefethen and David Bau, III: Numerical Linear Algebra, SIAM, 1997.

Prerequisites: calculus on manifolds and basic linear algebra.


MAT 1751HS
TOPICS IN COMPUTATIONAL MATHEMATICS
M. Shub

Expansion of the material covered in MAT 1750HF, including a study of Newton's method and the complexity of solving non-linear equations as well as a general theory of complexity of real computation as in the book "Complexity and Real Computation" by Blum, Cucker,Shub and Smale.

Textbook:
Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale: Complexity and Real Computation, Springer-Verlag, New York, 1997.

Prerequisite: MAT 1750HF or permission of the instructor.


MAT 1844HF
INTRODUCTION TO DYNAMICS
C. Pugh

Textbook: Brin and Stuck, Introduction to Dynamical Systems, Cambridge University Press.

Content: I will cover much of the text, including structural stability, the Smale horeshoe, the Anosov ergodicity theorem, and an introduction to complex dynamics.

Intent: There are 16 members of the department in the dynamics group. We intend to give a two semester core course that prepares students for research in dynamics and ergodic theory. This year is a trial of the concept. Konstantin Khanin will give the ergodic theory component of the course in the spring semester, MAT 1045HS. If you want to study dynamics and/or ergodic theory here you should take the course.


MAT 1856HS (APM 466H1S)
MATHEMATICAL THEORY OF FINANCE
L. Seco

Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.


STA 2111HF
GRADUATE PROBABILITY I
Instructor TBA

Random variables, expected value, independence, laws of large numbers, random walks, martingales, Markov chains.

Prerequisite: measure theory (may be taken at the same time) or permission of the instructor.

Textbook:
Durrett, Probability: Theory and Examples


STA 2211HS
GRADUATE PROBABILITY II
Instructor TBA

Weak convergence, central limit theorems, stable laws, infinitely divisible laws, ergodic theorems, Brownian motion.

Textbook:
Durrett, Probability: Theory and Examples

COURSE IN TEACHING TECHNIQUES

The following course is offered to help train students to become effective tutorial leaders and eventually lecturers. It is not for degree credit and is not to be offered every year.

The goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems.

Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.

The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later (private) assessment.

COURSES FOR GRADUATE STUDENTS FROM OTHER DEPARTMENTS

(Math graduate students cannot take the following courses for graduate credit.)

(These courses are used as reading courses for engineering and science students in need of instruction in special topics in theoretical mathematics. These course numbers can also be used as dual numbers for some third and fourth year undergraduate mathematics courses if the instructor agrees to adapt the courses to the special needs of graduate students. A listing of such courses is available in the 2006-2007 Faculty of Arts and Science Calendar. Students taking these courses should get an enrolment form from the graduate studies office of the Mathematics Department. Permission from the instructor is required.)