MAT 1000HF (MAT 457Y1Y)
REAL ANALYSIS I
Measure Theory: Lebesque measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.
Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L spaces, Holder and Minkowski inequalities.
Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces,
H.L. Royden: Real Analysis, Macmillan, 1988.
MAT 1001HS (MAT 457Y1Y)
REAL ANALYSIS II
Fourier analysis: Fourier series and transform, convergence results, Fourier inversion theorem, L theory, estimates, convolutions.
More functional analysis: Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.
Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces,
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.
S.D. Promislow, A First Course in Functional Analysis, Wiley, 2008.
MAT 1002HS (MAT 454H1S)
Stein and Shakarch: Complex Analysis
L. Ahlfors: Complex Analysis, 3rd Edition, McGraw-Hill, New York, 1966.
W. Rudin: Real and Complex Analysis, 2nd Edition, McGraw-Hill, New York,
Remmert: Theory of Complex Functions
Remmert: Classical topics in Complex function theory
Garnett and Marshall: Harmonic Measure
Needham: Visual Complex Analysis
Bierstone: Notes taken by Oleg Ivrii
PARTIAL DIFFERENTIAL EQUATIONS I
This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools. A key theme will be the development of techniques for studying non-smooth solutions to these equations.
L.C. Evans, "Partial Differential Equations"
PARTIAL DIFFERENTIAL EQUATIONS II
This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, and fixed point theorems. One important theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.
The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.
Other topics in PDE will also be discussed.
Lawrence Evans: Partial Differential Equations
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.
Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, and unique factorization domains.
Modules: Modules and algebras over a ring, tensor products, modules over a principal ideal domain,Textbooks:
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Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.
Commutative Rings: Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties. structure of semisimple algebras, application to representation theory of finite groups.Textbooks:
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, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.
John M. Lee: Introduction to Smooth Manifolds
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.
Allen Hatcher, Algebraic Topology
M. Greenberg and J. Harper, Algebraic Topology: A First Course
HARDY CLASSES OF ANALYTIC FUNCTIONS
(1) Harmonic functions on the unit disk
(2) Hardy classes Hin the unit disk
(3) Hilbert Transform
(5) Functions of Bounded mean Oscillations
Paul Koosis, Introduction to H spaces, with an appendix on Wolff’s proof of the corona
theorem Cambridge University Press, 1980.
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INTRODUCTION TO LINEAR OPERATORS
The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced). The intention will be to discuss a number (as many as conveniently possible) of the topics in Pedersen's textbook Analysis Now, reviewing more quickly those items with which people are familiar. Students will be encouraged to lecture on some of the material, and also work through (at least some of) the exercises in the book.
Prerequisites: Elementary analysis and linear algebra (including the spectral theorem for self-adjoint matrices)
Gert K. Pedersen, Analysis Now, Revised Printing, Graduate Texts in Mathematics, Springer, 1989
Paul R. Halmos, A Hilbert Space Problem Book, Second Edition, Graduate Texts in Mathematics, Springer, 1982
Mikael Rørdam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras, London Mathematical Society Student Texts 49, Cambridge University Press, 2000
PROBABILITY AND GEOMETRY ON GROUPS
Probability is one of the fastest developing areas of mathematics today, finding new connections to other branches constantly. One example is the rich interplay between large-scale geometric properties of a space and the behaviour of stochastic processes (like random walks and percolation) on the space. The obvious best source of discrete metric spaces are the Cayley graphs of finitely generated groups, especially that their large-scale geometric (and hence, probabilistic) properties reflect the algebraic properties. A famous example is the construction of expander graphs using group representations, another one is Gromov's theorem on the equivalence between a group being almost nilpotent and the polynomial volume growth of its Cayley graphs. The course will contain a large variety of interrelated topics in this area, with an emphasis on open problems.
Prerequisites: The core courses Real Analysis and Algebra are recommended. Former exposure to probability (e.g., the graduate probability courses) would be helpful.
Some references we will use:
S. Hoory, N. Linial and A. Wigderson: Expander graphs and their applications,
Bulletin of AMS, 2006,
M. Kapovich: Lectures on geometric group theory,
A. Lubotzky: Discrete groups, expanding graphs and invariant measures,
Progress in Math. 125, Birkhauser Verlag, Basel, 1994.
R. Lyons with Y. Peres: Probability on trees and networks,
W. Woess: Random walks on infinite graphs and groups, Cambridge University Press, 2000.
INTRODUCTION TO GEOMETRIC FLOWS
In this course we will study mean curvature, Ricci and harmonic map flows. We also plan to describe the curvature flow of networks of plane curves. We will give careful definitions of these flows, present existence results and results on formation of singularities (e.g. collapse to a point and neck-pinching) and soliton dynamics. We will also introduce main techniques, such as parabolic existence theory, maximum principles and monotonicity (entropy) formulae.
We will explain all needed notions from Differential Geometry and Partial Differential Equations, but knowledge of these subjects at an introductory level is required for this course.
K. Ecker: Regularity theory for mean curvature flow, Birkhaeuser, 2004; ISBN 08 176 32433
P. Topping, Lectures on the Ricci flow, London Math Society Lecture Notes, series 325, Cambridge Univ. Press, 2006; ISBN 0-521-68947-3.
INTRODUCTION TO MICROLOCAL ANALYSIS
The following topics will be covered: Tempered distributions and the Fourier transform,
Pseudodifferential operators on the Euclidean space and their basic properties, elliptic regularity, Microlocalization and the wave front set, Pseudodifferential operators on manifolds, Application to the wave equation (propagation of singularities theorem), complex power and the residue trace.
Prerequisites: Real Analysis (MAT1000HF and MAT1001HS)
Introduction to Microlocal Analysis, by Richard Melrose (available online: http://www-math.mit.edu/~rbm/iml90.pdf)
LIE GROUPS AND CLIFFORD ALGEBRAS
The plan of this course is to present the basic theory of Clifford algebras, with applications to Lie groups and Lie algebras. Detailed lecture notes will be provided. Topic include:
Prerequisites: Basic knowledge of linear algebra and differential geometry (vector fields, differential forms).
S. Sternberg: Lecture notes on Lie algebras. Available at
C. Chevalley: The algebraic theory of spinors and Clifford algebras (reprinted version), Springer 1997.
B. Lawson, L. Michelson: Spin geometry, Princeton University Press (1989).
RANDOM MATRIX THEORY
Snapshots of random matrix theory and its numerous applications. Topics include limiting eigenvalue distributions, connections to exclusion processes, free probability, orthogonal polynomials, random Schrodinger operators, random polynomials and the Riemann zeta function. Feynmann diagrams and random planar graphs. Dyson's Brownian motion.
INTRODUCTION TO ALGEBRAIC GEOMETRY AND COMMUTATIVE ALGEBRA
We will study the geometry of algebraic varieties over algebraically closed fields. The course will cover affine varieties, nullstellensatz, projective varieties, Zariski topology, sheaves of functions and modules, dimension, smoothness, and line bundles. All necessary commutative algebra results and concepts will be introduced and explained.
Prerequisites: undergraduate algebra (MAT 347Y) or graduate algebra (MAT 1100Y)
Daniel Perrin, Algebraic geometry, an introduction (Springer)
INTRODUCTION TO AUTOMORPHIC FORMS
The theory of automorphic forms has enjoyed remarkable progress in recent years. While the subject still has a long way to go, it is starting to realize some of its promise as a grand scheme to characterize basic objects from the arithmetic world and other areas of mathematics. The course will be a broad introduction to automorphic forms, automorphic representations and the Langlands programme.
It will have no formal prerequisites, other than a knowledge of the material in the core courses, and will be at the level of an advanced second year graduate course. Any of the Department's courses on Lie groups, Lie algebras, modular forms, algebraic number theory or representation theory of p-adic groups will provide very useful background, but will not be essential. I will try to tailor the course to what is most suitable for the audience.
J. Cogdell, H. Kim and R. Murty, Lectures on Automorphic L-functions, Fields Institute Monographs, vol. 5, 2004
S. Gelbart, An elementary intoduction to the Langlands program, Bulletin of the AMS, vol. 10, 1984, pp. 177-219
T. Bailey and A. Knapp (editors), Representation Theory and Automorphic Forms, Proceedings of Symposia of Pure Mathematics, AMS, vol. 61, 1997
AUTOMORPHIC FORMS AND THE TRACE FORMULA
This would be a continuation of the fall course above. The trace formula is one of the most powerful tools in the theory of automorphic forms. Some simple cases will be a part of the first course, but in the second half here, we shall study the general trace formula. We shall see that its terms serve as windows to fundamental questions in harmonic analysis and representation theory, and their application to the Langlands programme.
S. Gelbart, Lectures on the Arthur-Selberg trace formula, University Lecture Series, AMS, 1995
J. Arthur, An introduction to the trace formula, in Clay Mathematics Proceedings, AMS, 2005, pp. 1-263
Serre: Local Fields
THE ARITHMETIC THEORY OF MODULAR CURVES
This course will provide an introduction to the arithmetic theory of modular curves and the modular forms that live on them.
First we will cover the basic theory:
We will continue to more advanced topics, possibly including:
J.-P. Serre, A course in arithmetic
G. Shimura, Introduction to the arithmetic theory of automorphic forms.
S. Lang, Introduction to modular forms.
C.L. Siegel, Advanced analytic number theory.
A basic knowledge of complex analysis, some algebraic number theory and some algebraic geometry.
MAT 1302HS (APM 461H1S/CSC 2413HS)
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisite: MAT 224H1, recommended preparation MAT 344H1
COMPUTABILITY AND COMPLEXITY IN GEOMETRY, TOPOLOGY, AND DYNAMICS
A. Nabutovsky and M. Yampolsky
(offered by the Fields Institute as part of the Thematic Program on the Foundations of Computational Mathematics)
Part I: Computability and Complexity in Topology and Differential Geometry
Computability (Turing machines, recursive functions, degrees of unsolvability), algorithmic information theory, unsolvable problems in group theory, group homology, non-computability in topology, applications of non-computability and algorithmic information theory in geometric calculus of variations (``thick" knots, local minima of Riemannian functionals, geometry of moduli spaces arising in differential geometry).
Literature: S. Weinberger, ``Computers, Rigidity and Moduli", Princeton University Press, 2005 and papers of the instructor.
Prerequisites: Core graduate course in Topology. Familiarity with foundations of Riemannian Geometry is desirable but not necessary.
Part II: Computability and Complexity in Dynamics (M. Yampolsky)
Basic notions of computable analysis: computable real functions, computable sets in R. Dynamics of rational maps of the Riemann sphere: Fatou and Julia sets. Computability and complexity of Julia sets. Algorithms used to draw Julia sets versus theoretical complexity bounds. Constructing non-computable examples of Julia sets. Computability of filled Julia sets of polynomials. Computable properties and the topology of Julia sets. Open problems.
Literature: M. Braverman, M. Yampolsky, "Computability of Julia sets", Springer, 2008
MAT 1340HF (MAT 425H1F)
INTRODUCTION TO SMOOTH TOPOLOGY AND MAPPING DEGREE THEORY
References: J. Milnor: Topology from the Differential Viewpoint, a few original
papers will be suggested as additional material.
This graduate course will be an introduction to the broad topic of Morse theory. We begin with the classical approach to Morse theory, studying the topology of manifolds using functions defined on them, and then move on to the modern formulations of Bott, Smale, Witten, and Floer, and explore some of the modern applications, which touch upon several fields of intense current study. Topics will include:
Relevant textbooks and sources:
Prerequisites: The prerequisites are an understanding of the geometry of smooth manifolds, homology and cohomology, vector fields, and Sard's theorem (Mat327H1 or Mat425H1 or MAT427H1 or 464H1 or, ideally, the first term of 1300Y - any of these would be acceptable prerequisites.)
MAT 1342HF (MAT 464H1F)
Riemannian metrics, Levi-Civita connection, geodesics, curvature, Gauss equations, convexity, Complete manifolds and Hopf-Rinow theorem, Jacobi fields, Rauch comparison and variations of energy.
TextBook: "Riemannian geometry" by Do Carmo.
Prerequisites: Manifolds, differential forms, group theory, basic algebraic topology (fundamental groups)
FIBRE BUNDLES, CHARACTERISTIC CLASSES, AND K-THEORY
Topics may include: bundles and bundle operations; topological groups and group actions; classifying spaces; homotopy fibre; loop spaces; Stiefel-Whitney classes and Chern classes; other characteristic classes; axioms for generalized cohomology; K-theory; cohomology operations and Adams operations; Borel construction and equivariant cohomology.
Milnor and Stasheff: "Characteristic Classes", Princeton University Press
Husemoller: "Fibre Bundles", Springer Graduate Texts in Mathematics
Atiyah: "K-Theory", Oxford University Press, ISBN: 978-0-19-853276-7
Selick: "Introduction to Homotopy Theory", Fields Institute Monograph Series
Prerequisites: Basic algebraic topology: particularly cohomology.
HAMILTONIAN GROUP ACTIONS
This course is about Hamiltonian actions of compact Lie groups on symplectic manifolds. Applications include using polytopes and graphs to read “classical" geometric information (volumes; symplectic packings) and “quantum" information (dimension/multiplicities of group representations).
Prerequisites: Manifolds, differential forms, and (co)homology. Background on Lie groups and symplectic geometry will be useful.
Michele Audin, "Torus actions on symplectic manifolds".
Victor Guillemin, Eugene Lerman, and Shlomo Sternberg, "Symplectic Fibrations and Multiplicity Diagrams".
Viktor Ginzburg, Victor Guillemin, and Yael Karshon, "Moment Maps, Cobordisms, and Hamiltonian Group Actions".
ALGEBRAIC KNOT THEORY
Agenda: Understand the promise and the difficulty of the not-yet-existent “Algebraic Knot Theory”.
An “Algebraic Knot Theory” should consist of two ingredients:
(If you have seen homology in algebraic topology, recall that its strength stems from it being a functor. Not merely it assigns groups to spaces, but further, if spaces are related by maps, the corresponding groups are related by a homomorphism. We seek the same, or similar, for knots.)
The first ingredient for an “Algebraic Knot Theory” exists in many ways and forms; these are the many types and theories of “knot invariants”. There is very little of the second ingredient at present, though when properly generalized and interpreted, the so-called Kontsevich Integral seems to be it. But viewed from this angle, the Kontsevich Integral is remarkably poorly understood.
The purpose of this class is to understand all of the above.
Prerequisites: graduate topology and algebra
An introduction to complex manifolds: vector bundles, complex line bundles, Hermitian connections, curvature, Kahler metrics, Kodaira embedding theorem, Hodge theory.
Kodaira: ``Complex manifolds and Deformation of complex structures''
Griffiths and Harris: ``Principles of Algebraic Geometry''
A good background in differentiable manifolds including the de Rham complex of differential forms, Stoke's thereom, Frobenius integrability, and a good background in complex analysis in one variable.
MAT 1404HF (MAT 409H1F)
We will introduce the basic principles of cardinals, ordinals, axiomatic set theory, infinitary combinatorics, consistency and independence of the continuum hypothesis. In addition to the pure theory, the aim is to introduce students to the basic set-theoretic techniques applicable to other mathematical fields in which infinite sets play an essential role.
Prerequisite: an introductory real analysis course such as MAT 357H
W. Just and M. Weese: Discovering Modern Set Theory, I and II, AMS.
K. Kunen: Set Theory, Elsevier.
MAT 1700HS (APM 426H1S)
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.
Reference: R. Wald, General Relativity, University of Chicago Press
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MAT 1723HF (APM 421H1F)
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. In particular we will present an introduction to quantum information theory, which has witnessed an explosion of research in the last decade and which involves 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.
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics,
2nd edition, Springer, 2005
For material not contained in this book, e.g. quantum information theory, we will try to provide handouts and refer to on-line sources. Useful, but optional, books on the subject are
"Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum
Information (Paperback - Sep 2000), Cambridge University Press, ISBN 0 521 63503 9 (paperback);
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 1856HS (APM 466H1S)
MATHEMATICAL THEORY OF FINANCE
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.
Prerequisites: APM 346H1, STA 347H1
Brownian motion, sample path properties, quadratic variation, stochastic integrals, martingales, stochastic differential equations, connection with partial differential equations, approximation by Markov Chains, applications.
GRADUATE PROBABILITY II
Weak convergence, central limit theorems, stable laws, infinitely divisible laws, ergodic theorems, Brownian motion.
Reference: Durrett, Probability: Theory and Examples
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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.MAT 1499HS
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.)MAT 2000Y READINGS IN THEORETICAL MATHEMATICS
(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.)
Last updated: December 14, 2009