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. Analytic continuation, monodromy theorem, little Picard theorem.

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-Hoelder 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 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.

1. Some prior knowledge of point set topology will be useful. However, there will be a brief (approx. 6 weeks) discussion of point set topology (definitions, compact spaces, quotient spaces, etc.).
2. Homotopy theory: definition of homotopic maps, homotopy type of space, contractible spaces, retract, deformation retract, local deformation retract, fundamental group, higher homotopy groups. Calculation of the fundamental group of the circle, representative for its generator, proof of Van Kampen Theorem, Brouwer fixed point theorem for the two dimensional disc.
3. Theory of covering spaces: definition of covering space, group of covering transformations; criterion for lifing maps, calculation of the group of covering transformations, classification of the covering spaces over a given space, universal covering space.
4. Homology theory: construction of singular homotopy theory, its properties (homology of a point, homotopy property, long exact sequence, excision); calculation of the homology of the spheres and their generators, Brouwer fixed point theorem for the n-dimensional disc, Mayer-Vietoris sequence, Jourdan-Brouwer separation theorem, invariance of domain; CW complexes and their homology, Euler characteristic, homology of real and complex projective spaces and Grassmannians.
5. Cohomology theory: universal coefficient theorem, the cup product, limits.
6. Topological manifolds: orientation sheaf, orientation existence of a fundamental class, Poincare duality; cohomology ring of real and complex projective space. Borsuk-Ulam theorem.

GRADUATE COURSES

This course will cover basic techniques for solving partial differential equations on the computer, with emphasis on finite difference methods. Special attention will be paid to how the features of a good discretization reflect the mathematical properties of the PDE being solved.

• Parabolic equations
  • Explict and implicit discretizations
  • Consistency, stability, and convergence
  • Von Neumann stability (Fourier analysis)
  • Variational inequalities and free boundaries
  • Multi-dimensional problems
• Elliptic equations
  • The maximum principle
  • Solution of sparse linear systems
• Hyperbolic equations and conservation laws
  • CFL stability
  • Flux conservation and shock capturing
• Variational formulations and finite element methods
• Possible special topics (if time permits)
  • The Euler and Navier-Stokes equations of fluid dynamics
  • Adaptive mesh refinement techniques
  • Spectral and pseudo-spectral methods
  • Special techniques: random walkers, lattice gas

Bounded linear operators on Hilbert space, and applications. Necessarily, much of the attention will be on the finite-dimensional case, but we will bring out the new properties found in passing to general compact operators, and then to general bounded operators. Spectrum, numerical range. Spectral variation theorems for normal operators. Dilation and representation theorems.

The student need not have taken an advanced linear algebra course; the prerequisites are basic linear algebra, real analysis, and functional analysis. C*-algebras may be taken concurrently.

The material covered will depend partly on the needs and wishes of the students enrolled. Textbook: R. Bhatia: Matrix Analysis, Springer, 1997.

Text:
R. Bhatia: Matrix Analysis, Springer, 1997.


MAT 1016F
Structure of C*-Algebras
G.A. Elliott

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, to appear as part of a volume of the Encyclopaedia of Mathematics, Springer-Verlag, 2001.


MAT 1052S
Topics in Ordinary Differential Equations
M. Goldstein

Topics to be covered: Sturm-Liouville Equation; Analysis of the spectrum. Inverse problem; Marchenko-Gelfand-Levitan equation for inverse problem; Connection between Korteweg-DeVries equation and Sturm-Liouville equation; Solution of KdV equation.


MAT1062S
Topics in Partial Differential Equations I
P. Greiner

Elliptic and subelliptic partial differential equations with connections to Riemannian and subRiemannian geometry.

Prerequisite: Multivariate calculus.


MAT1063F/Y
Introduction to Microlocal Analysis
V. Ivrii

Microlocal Analysis is a theory of pseudo-differential and Fourier integral operators (PDOs and FIOs). PDOs are a generalization of (partial) differential operators (DO) and PDOs appear as parametrices (almost inverse) of elliptic Dos. FIOs appear as propagators of hyperbolic and similar equations. We will consider mainly semi-classical PDOs and FIOs.

A less formal description of (semi-classical) microlocal analysis is a semi-classical limit of quantum mechanics or a high-frequency limit of electrodynamics, etc. However, this course will not be physical.

This will be a half-year + half-year course with credits for each of them if necessary, and I plan also a third half-year course.

Course web page: http://www.math.toronto.edu/ivrii/GCMA.


MAT 1101F
Bruhat-Tits Trees and Buildings
M. Kapranov

The course will be an example-oriented elementary introduction into representation theory of p-adic groups. The topics to be discussed include the following: the Bruhat-Tits tree and its analogy with the Lobatchevsky plane; the action of the groups PGL₂(Q_q); apartments and horocycles; the Hecke operator; the unramified principal series representation; intertwining operators, spherical functions; the Bruhat-Tits building for the group GL_n over a p-adic field and its geometry (links of simplices, apartments, horocycles); relation to the unramified principal series for GL_n(Q_p).

Literature:
J.P. Serre, Trees, Springer, 1987.
K. Brown, Buildings, Springer, 1983.


MAT 1103S
Introduction to Iwahori-Hecke Algebras
T. Haines

The course consists of two parts. The first part is a treatment of the basic structure theorems of the Iwahori-Hecke algebra of a split p-adic group, "from the ground up". The Iwahori-Hecke algebra has two well-known presentations by generators and relations, the first discovered by Iwahori-Matsumoto and the second by Bernstein. From Bernstein's presentation one can deduce his description of the center of these algebras. I will cover this material with detailed, but elementary, proofs. Along the way I will discuss the related notions of affine Weyl groups, and the affine Hecke algebra of a root system. I will state some open problems of a purely algebraic/combinatoric nature.

The second half of the course is devoted to the fundamental results of Kazhdan-Lusztig concerning the relation of Hecke algebras to the geometry of Schubert varieties (material taken from [2] and [3] below). More precisely, I will introduce the Kazhdan-Lusztig (henceforth, K-L) polynomials and prove their basic properties. Then I will show how these polynomials can be described in terms of intersection cohomology of Schubert varieties. Also, I will demonstrate how the K-L polynomials explain the failure of Poincare duality on singular Schubert varieties.

At the end, I will explain the statements (without proofs) of certain major theorems in the area: the Kazhdan-Lusztig conjecture that the multiplicities of highest weight modules in Verma modules can be computed using K-L polynomials, and a related theorem of Lusztig that asserts the same for weight multiplicities in highest weight modules.

The material in the second half will require some knowledge of algebraic geometry, but prerequisites will be kept to a minimum. It will serve as an introduction to the intersection cohomology groups of Goresky-MacPherson-Deligne. Moreover, the connection between Hecke algebras and sheaves on flag varieties will serve as a useful illustration of Grothendieck's "function-sheaf" dictionary.

Main References:
J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29 (1990).
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53, 165-184 (1979).
D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math. 36 (1980).
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. IHES, no. 25.
R. Kottwitz, some unpublished notes.


MAT 1124S
Topics in Matrix Theory
M.-D. Choi

This course is concerned with major developments in matrix analysis, including topics on different sorts of means for non-commuting matrices, numerical ranges, almost commutativity, and spectral variations. The lectures will be devoted to developments of recent research interest, showing new meanings of classical results with unified approaches to various fields. No background in matrix analysis will be assumed.


MAT 1126F
Lie Groups and Fluid Dynamics
B. Khesin
This course deals with various problems in Lie theory, Hamiltonian systems, topology, geometry and analysis, motivated by hydrodynamics and magnetohydrodynamics. After defining the necessary notions in Lie groups, we discuss the dynamics of an ideal fluid from the group-theoretic and Hamiltonian points of view. We cover geometry of conservation laws of the Euler equation, topology of steady flows and their stability, relation of the energy and helicity of vector fields, geometry of diffeomorphism groups, as well descriptions of magnetohydrodynamics and of the Korteweg-de Vries equation in the Lie group framework.

References:
Arnold and Khesin: Topological Methods in Hydrodynamics, Appl. Math. Series, v. 125, Springer-Verlag, 1998/1999.
Marsden and Ratiu: Introduction to Mechanics and Symmetry, Texts in Applied Math., v. 17, Springer-Verlag, 1994/1999.


MAT 1190F
Introduction to Algebraic Geometry, Toric Varieties
A. Khovanskii

Toric varieties connect algebraic geometry with the theory of convex polyhedra. This connection provides an elementary way to see many examples and phenomena in algebraic geometry. It makes everything much more computable and concrete. I will not assume any special knowledge, but acquaintance with complex analysis in one variable will be very useful.

Recommended Literature:
W. Fulton: Introduction to Toric Varieties, Princeton University Press, 1993.
G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat: Toroidal Embeddings, Springer Lecture Notes 339, 1973.
I.R. Shafarevich: Basic Algebraic Geometry, Springer-Verlag, 1977.


MAT 1194F (MAT 449F)
Algebraic Curves
K. Consani

Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities.

A good part of the course will be devoted to the study of some topics on the arithmetic of elliptic curves.

References:
Hartshorne, R.: Algebraic Geometry, Springer-Verlag, New York, 1977.
Griffiths and Harris: Principles of Algebraic Geometry, Wiley, New York, 1978.
Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, Springer, 1984.
Silverman, J.: Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994.


MAT 1200S
Algebraic Number Theory
J. Scherk

A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; diophantine equations.


MAT 1303F (APM 461F/CSC 2413F)
Combinatorial Methods
E. Mendelsohn

Advanced topics in combinatorial theory chosen from a variety of areas, including enumeration, combinatorial designs, combinatorial identities, generating functions, graphs, finite geometries, coding theory, and combinatorial cryptography.


MAT 1312F
Variational Problems in Physics, Economics and Geometry
R. McCann

This is a research oriented topics course focusing on problems of current interest at the interface between mathematical analysis and applications in physics, economics and geometry. Topics include:

1. optimal transportation of mass in curved landscapes and with concave costs;
2. duality in game theory;
3. pattern formation;
4. metric geometry of probability measures;
5. nonlinear diffusion processes;
6. interfacial instabilities in two-phase fluids;
7. minimal and other surfaces with prescribed curvature;
8. fully nonlinear equations (Monge-Ampere type).

Each student may be expected to do some independent reading and present a research paper of current interest.


MAT 1314S
Introduction to Noncommutative Geometry
G.A. Elliott

An effort will be made to survey the combination of methods developed recently for approaching problems in topology, geometry, number theory, and physics which has come to be known as noncommutative geometry.

A simple geometrical structure which is fruitfully approached by the methods of operator algebras---thus yielding an example of non-commutative geometry in action!---is that of a foliation. An important example is the Kronecker foliation. The operator algebra associated to this foliation (in either the C*-algebra framework or the von Neumann algebra one) is familiar from other contexts.

References:
Alain Connes: Noncommutative Geometry, Academic Press, San Diego, 1994.
Jose M. Gracia-Bondia, Joseph C. Varilly, and Hector Figueroa: Elements of Noncommutative Geometry, Birkhauser, Boston, 2001.
Joachim Cuntz: Cyclic Cohomology and the Bivariant Chern-Connes Character, to appear as part of a volume of the Enyclopaedia of Mathematics, Springer-Verlag, 2001.


MAT 1341F
Differentiable Manifolds
E. Meinrenken

This course provides an introduction to smooth manifolds as basic objects in differential geometry. Topics include: vector fields, foliations and Frobenius' theorem, differential forms, vector bundles and fiber bundles, Lie groups and principal bundles, connections and curvature, characteristic classes.

Suggested References:
Conlon: Diffrentiable Manifolds.
Greub, Halperin, Vanstone: Connections, Curvature and Cohomology, Vol. I.


MAT 1343S (MAT 464S)
Riemannian Geometry
E. Meinrenken

Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.


MAT 1360S
Complex Manifolds
G. Maschler

This course is an introduction to the geometry of complex manifolds, with an emphasis on its various interactions with Riemannian geometry. Topics include:

Complex coordinates, almost complex structures, integrability

Hermitian and Kahler metrics, Hodge theory

Line bundles and divisors, Kodaira theorems

Basic results on complex surfaces

a selection from the following:

Canonical Kahler metrices and Gauduchon metrics

Supersymmetry and special holonomy

Self-dual metrics and twister spaces

Hermitian-Einstein metrics and stability

Distinguished minimal submanifolds

Recommended prerequisites are Differentiable Manifolds. Basic knowledge of Topology and Riemannian Geometry should prove helpful.

References:
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley and Sons, Inc., New York, 1994.
R.O. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics 65, Springer-Verlag, New York-Berlin, 1980.
M.J. Field, Several complex variables and complex manifolds I, II, London Mathematical Society Lecture Note Series 65-66, Cambridge University Press, Cambridge-New York, 1982.
S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, Wiley Classics Library, John Wiley and Sons, Inc., New York, 1996.
A.L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987.
D.D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.


MAT 1392S
Graphs, Knots and Binary Matroids
J. Mighton

Matroids and binary matroids are powerful generalizations of graphs that capture the fundimental properties of dependence common to graphs and matrices. A graph G is bipartite if its vertices can be partitioned into disjoint sets X and Y so that every edge in G has one endpoint in X and the other in Y. We will construct an isomorphism between binary matroids and bipartite graphs. Under this mapping, graphs, knots and groups may be represented as bipartite graphs: for finite groups and 3-connected graphs this reduction is complete. The bipartite graphs have many remarkable features (especially in the representation of duals and minors) which reveal new structure in diverse areas of mathematics.

This lecture is a preview of a graduate course I will offer in the math Department in the spring term. The course is intended to guide students to research in this area. Possible research topics arising from the course include: finding fast algorithms for the Jones Polynomial on special classes of knots and graphs (this is important as all computations in NP reduce to the Jones Polynomial); characterizing other invariants of knots, graphs and groups on the bipartite graphs; and reducing open problems in graph theory, knot theory, group theory and matroid theory to equivalent problems on the bipartite graphs. The course may be of interest to students in Computer Science as the bipartite graphs have applications in complexity theory.


MAT 1430F
Set Theory
F.D. Tall

Set theory and its relations with other branches of mathematics, ZFC axioms. Ordinal and cardinal numbers. Reflection principles. Constructible sets and the continuum hypothesis. Introduction to independence proofs. Topics from large cardinals and infinitary combinatorics.

Required Textbook:
W. Just and M. Weese: Discovering Modern Set Theory, I and II, AMS publication.
Recommended Textbook:
K. Kunen: Set Theory, Elsevier.


MAT 1500Y/MAT 1501F/MAT 1502S
Applied Analysis
I. Equations and Variational Calculus
II. Dynamical Systems and and Stochastic Analysis
I.M. Sigal

In this course we describe underlying concepts and effective methods of analysis and illustrate them on various problems arising in applications. The course consists of two relatively independent parts which cover the following topics:
Part I. Equations and Variational Calculus
1. Fourier and wavelet transforms.
2. Calculus of variations, optimization, and control (including applications to pattern analysis).
3. Calculus of maps (contraction mapping principle, method of successive approximations, inverse and implicit function theorems) and bifurcations.
4. Evolution equations (stationary solutions, solitary waves, solitons) with examples from physics, engineering, biology, chemistry.
Part II. Dynamical Systems and Stochastic Analysis
5. Finite and infinite dimensional dynamical systems (including stability theory of stationary solutions and travelling waves).
6. Asymptotic methods.
7. Stochastic calculus (key notions of probability theory, Markov chains and their applications, stochastic integrals and differential equations).
8. Numerical methods.

Textbook:
I.M. Sigal and M. Merkli: Lectures on Applied Analysis, Toronto (2000).
References: G. Folland: Real Analysis.
R. McOwen: Partial Differential Equations.
E. Lieb and M. Loss: Analysis, AMS Press.


MAT 1507F (APM 441F)
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 1508S (APM 446S)
Applied Nonlinear Equations
A. Tourin

The purpose of this course is to present the so-called theory of viscosity solutions and its broad range of applications in geometrical optics, image processing, finance and economics, front propagation, phase transition, deterministic and stochastic optimal control, differential games.

We will describe the class of nonlinear equations which are treated within the theory, present the basic theory and develop some of its applications to the fields mentioned above. The numerical analysis of these equations will be also covered. We will spend some time on the connections with other fields such as conservation laws, dynamical systems and the calculus of variations.

1. Theory of viscosity solution
2. Monotone numerical approximations
3. Image processing, Optimal control
4. Mathematical Finance, Economics
5. Front propagation, phase theory, reaction diffusion equations, bistability
6. Travelling waves, Allen-Cahn equation
7. Bifurcations
8. Conservation laws and vanishing viscosity
9. Calculus of variations, Dynamical systems.

When scientists seek to determine the properties of the interior of a body without the need to destroy, penetrate or even access the body, they analyze it's effect on (acoustic, electromagnetic or quantum mechanical) waves. Mathematically, wave propagation is modeled by Partial Differential Equations, and the class of problems considered consists in determining the coefficients of such an equation, assumed unknown inside a region, from knowledge of its solutions outside that region. These problems turn out to have a beautiful mathematical structure ( nonlinear harmonic analysis ) and are important in applications ranging from medical imaging, and seismology to nondestructive detection of structural flaws.

Much of the course will be devoted to the solution of the inverse boundary value problem of Calderon, since the breakthroughs achieved there have been central to the recent activity in the field. We will also discuss the connections to inverse scattering, multidimensional Borg-Levinson theorems and inverse obstacle problems.

Prerequisites and a possible time change will be discussed at the first meeting, Thurs. Sept. 13.


MAT 1700S (APM 426S)
General Relativity
J. Colliander

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 precession of Mercury. Black hole dynamics or gravity waves (as time permits).


MAT 1840F
Control Theory
V. Jurdjevic

Differential systems with controls and reachable sets. Non-commutativity. Lie bracket and controllability. Optimality and maximum principle. Hamiltonian formalism and symplectic geometry. Integrability. Applications to engineering, mechanics and geometry.


MAT 1845F
Complex Dynamics
M. Yampolsky

An introductory course on dynamical systems in one complex variable, accessible to students who have taken complex analysis. The course will be based on the new J. Milnor's book on the dynamics in one complex variable.


MAT 1856S (APM 466S)
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 2111F
Graduate Probability I
J. Quastel

A rigorous introduction to probability theory: Probability spaces, random variables, independence, characteristic functions, Markov chains, limit theorems.


STA 2211S
Graduate Probability II
J. Quastel

Continuation of Graduate Probability I, with emphasis on stochastic processes: Poisson processes and Brownian motion, Markov processes, Martingale techniques, weak convergence, stochastic differential equations.

Last updated: August 10, 2001