Some topics to be covered:
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
S. Arkhipov
Optimistic plan:
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.
Textbook:
Bredon's "Topology and Geometry".
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MAT 1011HS
INTRODUCTION TO LINEAR OPERATORS
G. Elliot
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)
Textbook:
Gert K. Pedersen, Analysis Now, Revised Printing, Graduate Texts in Mathematics, Springer, 1989
Recommended references:
Paul R. Halmos, A Hilbert Space Problem Book, Second Edition, Graduate Texts in Mathematics, Springer, 1982
Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras, London Mathematical Society Student Texts 49, Cambridge University Press, 2000
MAT 1044HF
POTENTIAL THEORY (One Variable)
T. Bloom
This part of the course will develop potential theory in the plane. We will discuss subharmonic, harmonic functions, Green’s functions, equilibrium measure.
Reference:
T. Ransford, Potential Theory in the Complex Plane
Several variables (pluripotential theory).
This part of the course will develop the basic theory of Bedford and Taylor on plurisubharmonic functions and the complex Monge Ampere equation.
Reference:
M. Klimek, Pluripotential Theory
Prerequisites:
Advanced undergraduate or graduate course on:
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
finite-difference methods
consistency, stability, and convergence in 1-d
boundary conditions in 1-d
multi-dimensional problems
Hyperbolic PDE
finite-difference methods
CFL stabilty condition
nonlinear conservation laws and weak solutions
consistency and convergence in 1-d
Elliptic PDE
variational formulations and finite element methods
Special topics
spectral and pseudospectral methods
Prerequisites:
You should be familiar with the material
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. (This isn't hard, unless you hate computers.) Engineers, physicists,
chemists, etc are welcome!
MAT 1103HF
INTRODUCTION TO AGEBRAIC GEOMETRY, ALGEBRAIC CURVES
AND PLANE GEOMETRY
A. Khovanskii
Algebraic curve, Riemann surface of algebraic function,
meromorphic functions and meromorphic 1-forms on algebraic curve.
Genus and Euler characteristic of an algebraic curve,
Riemann--Hurwitz formula, degree of divisors of meromorphic
functions and of meromorphic 1-forms on algebraic curves with
given genus. Generic algebraic curve with fixed Newton polygon.
It's Euler characteristic and genus. Space of holomorphic
1-forms. Pick formula (from an elementary geometry of integral
convex polygons). A.Weil reciprocity law. Abel Theorem. Geometry
of nondegenerate and degenerate cubic. Group structure. Pascal and
Menelaus Theorems as corollaries from Abel Theorem, solutions of
Poncelet problem and "butterfly" problem using Abel Theorem.
Riemann--Roch theorem. Jacobi theorem. Euler--Jacobi formula.
Pascal Theorem as a corollary from Euler--Jacobi formula. Real smooth algebraic curves of even an odd degree. Harnack inequality.
Estimate of the index of real polynomial vector field. Petrovskii
Theorem about real algebraic curve of degree six.
MAT 11120HS
LIE ALGEBRA
S. Balazs
Lie algebras are fundamental in the theory of Lie Groups (groups with a differential structure on them). Informally speaking, a Lie algebra is an "infinitesimal" version of a Lie group. Lie algebras arise in several areas of mathematic and physics. The most well known example for a Lie algebra is the three dimensional euclidean space with the cross product (which is related to the orthogonal group). A classification of Lie algebras in general is out of reach however a certain class (containing the most important examples) called the semi-simple Lie algebras can be classified. This course focuses mostly on semi-simple Lie algebras, their representations and their classification. Another goal of the course is to provide background information and
motivation for the topic.
Prerequisites:
Linear algebra and some knowledge in abstract algebra
MAT 1121HF
THE GEOMETRY OF INFINITE-DIMENSIONAL LIE GROUPS
B. Khesin
I. Introduction and main notions.
1. Lie groups and Lie algebras.
2. Adjoint and coadjoint orbits.
3. Central extensions.
4. The Lie--Poisson (or Euler) equations for Lie groups.
5. Symplectic reduction.
II. Infinite-dimensional Lie groups: their geometry, orbits, and dynamical systems.
1. Affine Kac--Moody Lie algebras and groups.
2. The Virasoro algebra and group. The Korteweg-de Vries equation.
3. Groups of diffeomorphisms. The hydrodynamical Euler equation.
4. Groups of (pseudo)differential operators. Integrable KP-KdV hierarchies.
5. The double loop (or elliptic) Lie groups and Lie algebras. Calogero--Moser systems.
III. Applications of groups.
1. Poisson structures on moduli spaces of flat connections and holomorphic bundles.
2. The Lagrangian formalism and the Chern--Simons action functional.
3. The classical and holomorphic linking numbers.
References:
1. B. Khesin and R. Wendt "The geometry of infinite-dimensional groups", Springer (2008), to appear
2. A. Pressley and G. Segal: "Loop Groups", Clarendon Press, Oxford (1986)
Prerequisites:
A basic course (or familiarity with main notions) of
symplectic geometry would be helpful.
MAT 1190HS
INTRODUCTION TO SCHEMES
G. Mikhalkin
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.
MAT1190HS
TORIC VARIETIES AND NEWTON POLYHEDRA
A. Khovanskii
I will present the Theory of Toric Varieties and its applications. On one hand the Toric Varieties are very useful by themselves. On the other hand they link the Algebraic Geometry with the theory of Convex Polyhedra. This link provides an elementary view of many examples and phenomena in algebraic geometry. It makes everything much more computable and transparent. One can consider this course as an introduction to Algebraic Geometry.
Recommended Literature:
1."Introduction to Toric Varieties", by W.Fulton, Prinston University Press 1993.
2."Toroidal Embeddings, by G.Kempf, F.Knudsen, D. Mamford,B Saint--Donat, Springer Lecture Notes 339, 1973.
3. "Basic Algebraic Geometry", by I.R.Shafarevich, Springer--Varlag, 1977.
MAT1191HF
TRANSCENDENTAL METHODS IN ALGEBRAIC GEOMETRY
Y-T, Siu
Applications of L^2 \partial-estimates and multiplier ideal sheaf tech
niques to problems in algebraic geometry such as the effective Nullstellensatz, the Fujita conjecture on effective global generation and very ampleness of line bundles, the effective Matsusaka big theorem, the deformational invariance of plurigenera, the finite generation of the canonical ring, and the abundance conjecture. Will also discuss hyperbolicity problems and the application of algebraic-geometric techniques to partial differential equations through multiplier ideal sheaves, especially the global regularity of the complex Neumann problem, the existence of Hermitian-Einstein and Kähler-Einstein metrics, and the global nondeformability of irreducible compact Hermitian symmetric manifolds.
MAT 1197HS
REPRESENTATIONS OF REDUCTIVE P-ADIC GROUPS
F. Murnaghan
Basic theory of representations of reductive p-adic groups, with an emphasis on proofs for general linear and classical groups. Some course notes will be provided. The course may also include material from the book of Bushnell and Henniart on the local Langlands conjecture for GL(2).
Prerequisites:
Some basic representation theory (preferably of compact topological groups, although knowledge of representation theory of finite groups should be sufficient); background in semisimple Lie algebras and root systems would also be useful.
MAT 1199HF
INTRODUCTION TO AUTOMORPHIC FORMS AND L-FUNCTIONS
H. Kim
Topics to be covered:
1. classical automorphic forms and their L-functions, following closely Bump's book, chapter 1
2. Tate's thesis (Hecke's L-functions), from Bump's book, 3.1 and 3.2
References:
Automorphic Forms and Representations by Daniel Bump Lectures on Automorphic L-functions by J. Cogdell, H. Kim and R. Murty
Prerequisites:
complex analysis and some knowledge of algebraic number theory
MAT 1202HS (MAT417H1S)
THEORY OF CHARACTER SUMS
V. Blomer
Character sums, like Gauss sums or Kloosterman sums, play a major role in all branches of number theory. This course presents the underlying algebraic and analytic theory. Highlights include a proof of the Riemann hypothesis for curves over finite fields. On the side we will see elliptic curves, L-functions and various arithmetic applications.
References:
- H. Iwaniec, E. Kowalski, Analytic Number Theory, AMS 2004
- W. Schmidt, Equations over finite fields: an elementary approach,
Kendrick Press 2004
- S. Konyagin, I. Shparlinski, Character sums with exponential functions and their applications,
Cambridge 1999
Prerequisites:
Interest in number theory, good knowledge of algebra, analysis, number theory. Fourth year undergraduate students are welcome.
MAT 1314HS
INTRODUCTION TO NONCOMMUTATIVE GEOMETRY
R. Ponge
Noncommutative geometry is a rapidly growing subject launched by Alain Connes in the 80s. A main aim is to translate the tools of differential geometry into the operator theoretic language of quantum mechanics. More precisely, there is a natural duality between spaces and algebras (e.g. by Gelf'and theorem the algebras of continuous functions on a compact space allows us to exhaust all the so-called C^*-algebras that are commutative). Thus noncommutative geometry bypasses the fact that noncommutative spaces hardly makes sense by instead considering noncommulative algebras which formally plays the role of the algebra of functions on a "ghost" noncommutative space. This allows us to treat a variety of problems that cannot be treated using classical differential geometry.
The aim of this course is to provide an introduction to some tools and methods of noncommutative geometry with a special emphasis on the connections with index theory. The first few weeks of the course will be devoted to a review of some background on spectral theory and pseudodifferential operators.
Prerequisite:
K
knowledge of basic notions of measure theory and functional analysis as in the Real Analysis book of Gerald Folland.
References:
- A. Connes: "Noncommutative Geometry", Academic Press, 1994.
- J. Gracia-Bonda, J.C. Varilly, H. Figueora: "Elements of Noncommutative Geometry", Birkhauser, 2001.
MAT 1342HS
DIFFERENTIAL GEOMETRY
F. Rochon
The course will cover the following topics:
Prerequisites:
Manifolds, differential forms, group theory, some algebraic topology (fundamental groups).
Textbooks:
Gallot, Hulin and Lafontaine, "Riemannian geometry".
MAT 1344HS
SYMPLECTIC GEOMETRY
J. Kamnitzer
This will be an introductory course in symplectic geometry. We will
cover the basic definitions and results in symplectic geometry with an emphasis on connections to other areas such as Lie groups and complex geometry. The topics that will be covered are:
Textbook:
Cannas da Silva, Lectures on Symplectic Geometry, Springer
Lecture Notes in Mathematics
Prerequisites:
Knowledge of basic definitions and ideas in
differentiable manifolds.
MAT 1345HF
HOMOLOGICAL ALGEBRA
S. Arkhipov
MAT 1360HS
ABELIAN VARIETIES
S. Kudla
Abelian varieties are higher dimensional generalizations of elliptic curves. They have a rich and fascinating geometry and play a fundamental role in many of the most important recent developments in number theory.
This course will provide a basic introduction to their theory followed by a sketch of more advanced aspects. We will mostly follow the classic treatment in Mumford, Abelian varieties.
Topics may include:
Analytic theory:
Algebraic theory:
Prerequisites:
Basic algebraic geoemtry, e.g. the first chapter of Hartshorne.
Basic differential geometry, e.g., differential forms, complex manifolds, deRham cohomology.
Familiarity with sheaves and sheaf cohomology would be helpful.
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
DESCRIPTIVE SET THEORY
S. Todorcevic
This will be a course in classical descriptive Set Theory up to the second level of projective hierarchy. We shall, however, also cover some more recently developed themes such as the rough classification of Borel equivalence relations.
Textbook:
A. S. Kechris, Classical Descriptive Set Theory, graduate texts in math. Springer 1995.
MAT 1507HF
ASYMPTOTIC AND PERTURBATION METHODS
C. Sulem
Local Methods
Asymptotic expansion of integrals
Perturbation methods
Global Analysis
Textbook:
Advanced Mathematical Methods for Scientists and Engineers; Asymptotic
Methods and Perturbation Theory}by Carl M. Bender and Steven A. Orszag, 1st edition, McGraw-Hill, 1978, 2nd ed., Springer 1999.
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 1705HF
MATHEMATICAL METHODS OF CLASSICAL MECHANICS
V. Ivrii
Lagrangian Mechanics
Hamiltonian Mechanics
Prerequisites
Any course in Analysis on Manifolds (f.e. MAT 251);
Good course in Ordinary Differential Equations (f.e. MAT267
preferable but MAT244 would be sufficient)
Assets (nice to have but not required)
Any course in Differential Geometry (f.e. MAT 251);
Knowledge of Newtonian Mechanics;
Textbook
Mathematical Methods of Classical Mechanics (Graduate Texts in
Mathematics) by V. I. Arnold, Springer Verlag (1997).
Misc
Course will be accesable to senior undergraduate students, including those in physics. We will run our wiki.
MAT 1844HF
INTRODUCTION TO DYNAMICS
C. Pugh
I will cover much of the text, including structural stability, the Smale horseshoe, 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. 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.
Textbook:
Brin and Stuck, Introduction to Dynamical Systems,
Cambridge University Press, and instructor's notes.
MAT 1846HS
GEOMETRIC FUNCTION THEORY
I. Binder
This course will serve as an introduction to the Geometric Function Theory, a classical area of Analysis which has seen a significant development recently due to the discovered connections to other areas of Mathematics, such as Mathematical Physics and Complex Dynamics. We will start with the distortion theorems for conformal maps and the properties of planar harmonic measure, will discuss the Potential Theory, extremal length method, and Loewner Evolution. We will finish with a modern application of the Geometric Functions Theory to the Mathematical Physics, the Stochastic Loewner Evolution.
Textbooks:
1.Garnett, John B. Marshall, Donald E. "Harmonic measure." New Mathematical Monographs, 2. Cambridge University Press, Cambridge, 2005. xvi+571 pp. ISBN: 978-0-521-47018-6; 0-521-47018-8
2. Pommerenke, Ch. "Boundary behaviour of conformal maps." Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 299. Springer-Verlag, Berlin, 1992. x+300 pp. ISBN: 3-540-54751-7
Prerequisite:
MAT 1001HS (COMPLEX ANALYSIS) or equivalent
MAT 1847HS
HOLOMORPHIC DYNAMICS
M. Yampolsky
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 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.
STA2047HF
STOCHASTIC CALCULUS
J. Quastel
Brownian motion, sample path properties, quadratic variation, stochastic integrals, martingales, stochastic differential equations, connection with partial differential equations, approximation by Markov Chains, applications.
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
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 1499HSThe 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.
(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.)
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Last updated: May 12, 2008