2008-2009 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
S. Arkhipov

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.

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.

Textbook:
Bredon's "Topology and Geometry".

2008-2009 TOPICS COURSES, including CROSS-LISTED COURSES

MAT1002HF
NEVANLINNA THEORY
(Fields Institute program)
M. Ru

Diophantine approximation is a tool to study rational points on algebraic varieties defined over a number field. On the other hand, Nevanlinna theory studies holomorphic curves in complex algebraic varieties, especially it studies how well a holomorphic curve intersects divisors in a complex algebraic variety. It has been observed by Osgood, Vojta and others that there is a striking correspondence between statements in Nevanlinna theory and in Diophantine approximation. The mini-course will cover: Roth’s theorem and Schmidt’s subspace theorem; Diophantine equations and approximation; the theory of global and local heights; Faltings’ theorem on abelian varieties; the classical theory of Nevanlinna on meromorphic functions; The Ahlfors-Cartan theory of holomorphic curves; holomorphic curves in Abelian varieties; the complex hyperbolicities and the general case of Lang’s conjecture.

MAT 1011HS
INTRODUCTION TO LINEAR OPERATORS
G. Elliott

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:

• Complex analysis
• Real analysis

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
INTRODUCTORY NUMERICAL 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.

MAT1104HF
INTRODUCTION TO ARAKELOV GEOEMTRY
(Fields Institute program)
H. Gillet

The course will begin, following a review of the "non-arithmetic" theory, with a study of Arakelov's intersection theory on arithmetic surfaces as developed in Faltings. We will then develop arithmetic intersection theory for varieties of arbitrary dimesion. The main results include the relationship between arithmetic intersection theory, heights and height pairings, the arithmetic Bezout theorem, the arithmetic Hilbert-Samuel formula, and the arithmetic Riemann-Roch theorem. The course will continue with the K-theory of Hermitian vector bundles for general arithmetic varieties and the characteristic classes for this bundles, the determinant of cohomology and Quillen metrics, and the arithmetic Grothendieck-Riemann-Roch theorem. Some applications to problems in number theory will be discussed.

MAT 1120HS
LIE ALGEBRAS
B. Szegedy

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
LIE GROUPS AND HAMILTONIAN DYNAMICAL SYSTEMS
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.

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
(Fields Institute program)
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 1200HS (MAT415H1S)
INTRODUCTION TO ALGEBRAIC NUMBER THEORY
K. Murty

Topics to be covered are: algebraic number fields, ring of integers, group of units, the ideal class group, reciprocity laws, analytic methods. Some additional advanced topics will be discussed depending on time. The format of the course will be interactive and all students will be expected to participate in the discussions.

Textbook:
Problems in Algebraic Number Theory, Second Edition by: M. Ram Murty and J. Esmonde, Springer Verlag, Graduate Texts in Mathematics 190.

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.

Literature:
- 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.

MAT1312HF
JET SPACES
(Fields Institute program)
P-M. Wong

This is an outline of a mini-course in complex geometry. There are six chapters planned. The content in the rst four chapters are mostly very well known. There will be no time for complete proofs only outline and motivations will be given. Explicit references of the theorems will be given. I shall also try to provide examples to explain the ideas and clarify the concepts. A reasonable amount of details will be given in the last two chapters.

1. A brief introduction to Hermitian and Kähler Geometry. A quick outline of the basic notions of Hermitian connection and curvature of vector bundles. The concepts of holomorphic bisectional curvature, Ricci curvature and holomorphic sectional curvature. Schwarz Lemma. Chern classes of vector bundles.
2. A brief introduction to Complex Analysis and Algebraic geometry. Brief outline of the concepts of coherent sheaves and sheaf cohomologies. Characterizations of Stein manifolds via vanishing theorem, via strictly plurisubharmonic exhaustions and the Stein embedding theorem. The concepts of ample, big and nef bundles. Kodaira vanishing theorem and embedding theorem. Riemann-Roch theorem for coherent sheaves.
3. A brief introduction to Complex Finsler Geometry and Intrinsic Metrics. Intrinsic metrics, mainly the Kobayashi and the Caratheodory metric, will be introduced. Positive currents and Lelong numbers. A Finsler characterization of ample bundles and big bundles will be given.
4. A brief introduction to Nevanlinna Theory. A brief account of Nevanlinna Theory. Jensen Formula. First Main Theorem. Crofton Formula. Second Main theorem. Integrated form of Schwarz Lemma.
5. Holomorphic Jet Bundles. The basic theory of jet bundles will be introduced with reasonable amount of details and examples.
6. Applications to Hyperbolic Geometry. The results in the rst ve chapters will be applied to resolve problems in complex hyperbolic geometry.

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:

1. Cartan formalism: fibred bundles, connections and curvatures.
2. Riemannian metrics: Levi-Civita connections, geodesics, exponential map.
3. Riemannian curvature: Defintions, fundamental identities, variation of energy, Jacobi fields, Riemannian submersions, spaces with constant curvature, Riemannian submanifolds, relation between curvature and topology, Weyl tensor.

Textbook:
Gallot, Hulin and Lafontaine, "Riemannian geometry".

Prerequisites:
Manifolds, differential forms, group theory, some algebraic topology (fundamental groups).

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:

1. Definition of symplectic manifolds. Basic examples. Darboux's theorem. Cotangent bundles. Coadjoint orbits. Hamiltonian mechanics.
2. Compatible complex structures. Kahler manifolds.
3. Group actions on symplectic manifolds. Moment maps. Symplectic reduction. Toric manifolds.
4. Introduction to pseudoholomorphic curves. Introduction to Lagrangian Floer thoery.

Textbook:
Cannas da Silva, Lectures on Symplectic Geometry, Springer Lecture Notes in Mathematics

Prerequisites:
Knowledge of basic definitions and ideas in differentiable manifolds.

MAT 1345HS
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.

MAT 1355HS
COURSE ON RESOLUTION OF SINGULARITIES
(Fields Institute Program)

Module 1: Resolution of singularities for functions
Instructor: E. Bierstone

Background: examples, blowing-up and strict transform. Crucial exercises on transformation of differential operators by blowing up, semicontinuity of order of vanishing, normal crossings. Desingularization of spaces vs. desingularization of ideals. Motivating examples, marked ideals. Elementary proof of resolution of singularities.

Module 2: Resolution of singularities for foliations
Instructor: Felipe Cano
Note: Professor Cano's module will be taught from Jan 19 to Feb 13 on TTh 1:30 - 3.

Bloc I (1 week): Basic concepts on singular foliations and reduction of singularities.
a. Singular foliations and vector fields in dimension two.
b. Blowing-up vector fields. simple singulaities.
c. Separatrices and integral curves. Briot-Bouquet Theorem.
d. Seidenberg's result on desingularization of vector fields.
e. Camacho-Sad theorem of existence of separatrices.

Bloc II (1 week): Some applications of the reduction of singularities of codimension one foliations.
a. Simple singularities in codimension one. Behavior under blow-up.
b. The statement of reduction of singularities in dimension three. Consequences on the existence of invariant hypersurfaces.
c. About the dicriticalness.
d. Singular Frobenius I.
e. Singualr Frobenius II.

Bloc III (1 week): Technics for the reduction of singularities.
a. The reduction of the singularities of surfaces as a model for low dimensional problems.
b. The main invariants used in the control: Multiplicities, Newton polygons and resonancies.
c. Reduction of the singularities of codimension one foliations in dimension three.
d. The valuative aproach for vector fields. The birational problem of reduction of singularities. Globalization.
e. Local Uniformization of Vector Fields.

Module 3: Resolution of singularities of real analytic vector fields
Instructor: D. Panazzolo

Main Topics: Normal forms and classification of singularities of vector fields. Parametrized normal forms and bifurcations of limit cycles in analytic families of planar vector fields. The Newton polyhedron and the blowing-up of a vector field. Various recent results on resolution of singularities for vector fields: one-parameter familes in dimension two and real analytic three dimensional vector fields.

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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:

• complex tori
• line bundles on complex tori
• theta functions
• algebraizability, Appel-Humbert theorem
• Siegel space and A_g
  

Algebraic theory:

• abelian varieties
• cohomology and base change
• the theorem of the cube
• the dual abelian variety
• the Poincare bundle
• polarizations
• structure of End (A), Rosati involution
• moduli spaces
• complex multiplication
• Shimura varieties of PEL type

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 1403HS
COURSE ON TOPICS IN O-MINIMALITY
(Fields Institute program)

Module 1: O-minimality and Hardy fields
Instructor: C. Miller

Primary material: Hardy field theory as it relates to o-minimality; the growth dichotomy; and basic properties of the polynomially bounded case. More advanced topics and applications will be presented, as time permits, based on interests and preparation of the participants.

Module 2: Construction of o-minimal structures from quasianalytic classes
Instructor: J.-P. Rolin

Note: Professor Rolin's module will be taught from Mar 23 to Apr 17 on MW 1:30 - 3.

In a first part, we recall the definition of quasianalytic algebras of functions or germs of functions and give several examples of such algebras. Then we prove that convenient quasianalytic algebras generate o-minimal expansions of the real field. One important ingredient of the proof is the technique of resolution of singularities as discussed by Edward Bierstone in his course.

Module 3: Pfaffian closure and model completeness results for Pfaffian chains
Instructor: P. Speissegger

The goal of my lectures is to outline a proof of Gareth O. Jones'recent result that the expansion of the real field by a Pfaffian chain and the exponential function is model complete. The proof combines both Wilkie's model-theoretic and Lion and Speissegger's geometric approaches to proving model completeness.

Prerequisites for this course are basic differential geometry, basic model theory and the material covered by Chris Miller in his course.

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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.

MAT1448HF
TOPICS IN SET-THEORETIC TOPOLOGY
F. Tall

This course will explore a variety of set-theoretic methods, as applied to point-set (also known as “general”) topology. These methods are useful in set-theoretic applications to a number of mathematical fields, not just topology. The topology we use will be easy to learn; the set theory will be considerably more challenging. Indeed, an underlying theme will be that simple-to-state topological problems require sophisticated set theory to solve.
Particular topics could include applications of Martin’s Axiom, cardinal invariants, applications of elementary submodels, applications of large cardinals via forcing, normality vs. collectionwise normality (including the solution of the Normal Moore Space problem), etc. Some significant unsolved problems in general topology will be discussed.

Prerequisites:
MAT 409 or equivalent introductory set theory course; a basic knowledge of point-set topology, as in the Topology Problems course or MAT 327. Some acquaintance with the method of forcing is highly desirable.

MAT 1507HF
ASYMPTOTIC AND PERTURBATION METHODS
C. Sulem

Local Methods

1. Classification of regular/ singular points of linear ODEs
2. Approximate solutions near regular, regular-singular
3. Irregular singular points, irregular point at infinity
4. Asymptotic series
5. Some examples of nonlinear differential equations.

Asymptotic expansion of integrals

1. Introduction
2. Laplace method
3. Method of stationary phase
4. Steepest descent\cr
   

Perturbation methods

1. Examples
2. Regular and singular perturbation theory

Global Analysis

1. Boundary layer theory
2. More elaborated examples
3. WKB theory : Formal expansion, conditions for validity, geometrical optics (10.1-10.3)
4. Multiple scale analysis for ODEs:
5. Resonance and secular behavior, damped oscillator
6. Multiple scale analysis for PDEs

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

1. Variational principles;
2. Lagrangian mechanics on manifolds;
3. Oscillations;
4. Rigid bodies.
   

Hamiltonian Mechanics

1. Symplectic geometry;
2. Canonical formalism; Hamilton-Jacobi equation;
3. Perturbations;
4. Connection to quantum 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;

Reference:
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.

MAT1723HF (APM421H1F)
QUANTUM MECHANICS
J. Colliander

MAT1839HF
OPTIMAL TRANSPORTATION
R. McCann

This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in dynamical systems, geometry, physics, and nonlinear partial differential equations. The basic problem is to find the most efficient structure linking two continuous distributions of mass --- think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation. Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, periodic orbits for dynamical systems, long time asymptotics in kinetic theory and nonlinear diffusion, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, convexity, elliptic regularity, dynamical systems and fluid mechanics, not to mention physics, economics, and geometry.

Depending on interest and preparation of the students, we will go more or less deeply into certain aspects of the theory, such as geometry and geometric evolution equations including Ricci flow, the question of when the solution to a fully nonlinear degenerate elliptic equation is smooth, or into applications such as atmospheric and economic modelling.

Textbook:
Cedric Villani "Topics in Optimal Transportation" Providence: AMS 2003 GSM/58 ISBN 0-8218-3312-X

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 1844HS
COURSE ON MULTISUMMABILITY AND QUASIANALICITY
(Fields Institute Program)

Module 1: Basic multisummability
Instructor: R. Sch�fke

Abstract: TBA


Module 2: Resurgent functions
Instructor: D. Sauzin

• Reminders on Gevrey asymptotics and Borel-Laplace transform (cf. Sch�fke's course). The Nevanlinna theorem. Examples (Airy, Ei(z), Erf (z), Stirling).
• The definition of "resurgence". Analytic continuation in the Borel plane and stability by convolution. Application to non-linear dynamics. The example of the saddle-node.
• Ecalle's "Alien calculus". The definition of "alien derivations". The "bridge equation".
  

Module 3: Non-oscillatory trajectories
Instructor: F. Sanz

Note: Professor Sanz's module will be taught from Feb 23 to Mar 20 on
MW 10 - 11:30.

The course deals with the qualitative study of oscillatory and non-oscillatory trajectories of real analytic vector fields, mainly in dimension three. In the first part of the course, we describe several kinds of asymptotic behaviour that such transcendental objects can have: axial spiraling, asymptotic linking, separation by projection.
In the second part, we will study non-oscillatory trajectories that belonging to new o-minimal structures, an application of the contents of the courses given by J.-P. Rolin, R. Sch�fke and F. Cano.

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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

The course will introduce students to the beautiful mathematics behind the captivating images of Julia sets and the Mandelbrot set. The basic reference for the course is a book of J. Milnor "Dynamics in One Complex Variable". The subject is large and growing rapidly, and we will describe some of the modern developments. In particular, we will discuss the questions of algorithmic computability of Julia sets, which have been developed in our own work with M. Braverman. The exposition will be essentially self-contained, we will only assume familiarity with basic Complex Analysis. The course will be accessible to advanced undergraduates.

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.

STA2047HS
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
J. Quastel

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.

Reference:
Durrett, Probability: Theory and Examples


STA 2211HS
GRADUATE PROBABILITY II
B. Virag

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

Reference:
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.)

Last updated: January 6, 2009