2006-2007 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.

Some topics to be covered:

1. The Fourier Transform. Distributions.
2. Sobolev spaces on Rⁿ. Sobolev spaces on bounded domains. Weak solutions.
3. 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.
4. 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.


MAT 1100YY
ALGEBRA
V. Blomer

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.

A selection of topics from:

1. Set theory: Zorn's Lemma, axiom of choice, well-ordered sets, cardinals, ordinals
2. Point set topology: Metric spaces, topological spaces, compactness, separation properties, connectedness, paracompactness, CW complexes
3. Homotopy theory: homotopy, fundamental group, covering spaces, Van Kampen's theorem
4. Homological algebra: categories and functors, chain complexes, homology, exact sequences, Snake Lemma, Mayer-Vietoris
5. Homology theory: Eilenberg-Steenrod homology axioms; singular homology theory; cellular homology, cohomology, cup and cap products, applications of homology (Brouwer fixed-point theorem, vector fields on spheres, Jordan Curve Theorem), other homology theories (simplicial and Cech homology)
6. Manifolds: classification of surfaces, de Rham cohomology, orientation, Poincare Duality

2006-2007 CROSS-LISTED COURSES

In the title of this course the term "real" stands not only as a reference to the field of real numbers but also as a reference to the real (geometric) world. In this course in addition to real varieties we shall also treat complex varieties (and sometimes even varieties defined over some function fields), but we shall primarily focus on geometric aspects of algebraic geometry. We'll pay special attention to topology of real and complex algebraic varieties, in particular to the topics related to Hilbert's 16th problem and to vanishing cycles of complex singularities. We introduce the so-called "patchworking" technique for constructing real algebraic from simpler pieces. The patchworking serves as a link from this course to the second half-course "Tropical Geometry".

Prerequisites: acquaintance with some basic Algebraic Geometry and Geometric Topology. New Ph.D. students are welcome, especially if they like Geometry. Please contact the instructor.


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

We will cover a selection of topics in enumerative combinatorics, such as more advanced methods in recursions, an analysis of some unusual self-referencing recursions, binomial coefficients and their identities, some special combinatorial numbers and their identities (Fibonacci, Stirling, Eulerian), and a general approach to the theory of generating functions.

Prerequisite:
Linear algebra.

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


MAT 1340HF (MAT 425H1F)
DIFFERENTIAL TOPOLOGY
A. Nabutovsky

Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic, Hopf degree theorem. Additional topics may vary.

Textbook:
Victor Guillemin and Alan Pollack, Differential Topology

Prerequisites: MAT 257Y (second year Analysis course), MAT 240H (second year Algebra course) and MAT 327H (introductory Topology course).


MAT 1342HS (MAT 464H1S)
DIFFERENTIAL GEOMETRY
J. Bland

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.

References:
Manfredo Perdigao de Carmo: Riemannian Geometry.
J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry, Elsevier, 1975.

Prerequisite: Introductory differential geometry course such as MAT 363H.


MAT 1404HF (MAT 409H1F)
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

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


MAT 1508HS (APM 446H1S)
APPLIED NONLINEAR EQUATIONS
A. Burchard
Nonlinear partial differential equations and their physical origin. Fourier transform; Green's function; variational methods; symmetries and conservation laws. Special solutions (steady states, solitary waves, travelling waves, self-similar solutions). Calculus of maps; bifurcations; stability, dynamics near equilibrium. Propogation of nonlinear waves; dispersion, modulation, optical bistability. Global behaviour solutions; asymptotics and blow-up.

Prerequisite:
APM346H1/APM351Y1


MAT 1700HS (APM 426H1S)
GENERAL RELATIVITY
P. Blue

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

The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some topics which are at the forefront of active research, such as Bose-Einstein condensation, control of chemical reactions and quantum information. We will try to be as self-contained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively time-consuming we give an idea of the proof, if such exists, and/or explain the mathematics involved without providing all the details.

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

Syllabus:

• Schroedinger equation
• Quantum observables
• Spectrum and evolution
• Density matrix and open systems
• Bose-Einstein condensation
• Quasiclassical asymptotics
• Path integral
• Approximate methods
• Hartree-Fock theory
• Control of chemical reactions
• Quantum entropy
• Quantum channels and information processing
• Basic notions of quantum field theory

References:
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Springer
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam, The Netherlands: North Holland.


MAT 1856HS (APM 466H1S)
MATHEMATICAL THEORY OF FINANCE
P. Olivares

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.

2006-2007 TOPICS COURSES

Topics will be chosen from among the following:

The multidimensional Fourier transform. Reconstruction from Fourier transform samples, Nyquist's theorem, Poisson summation.

The Radon transform. Reconstruction from Radon transform samples.

How does MRI work? The Bloch Equation in Magnetic Resonance Imaging; connection to the analytic development of the Inverse Scattering Transform (also used in the exact solution of integrable nonlinear partial differential equations).

Current Density Impedance Imaging.

(time permitting) Partial differential equations techniques for Ultrasound Imaging and Terahertz Tomography.

Other reference:
"Magnetic resonance imaging: Physical Principles & Sequence Design". EM Haake, BW Brown, MR Thompson, R Venkatesan, J Wiley 1999.

Marking scheme: 75% project, 25% homework.


MAT 1035HF
INTRODUCTION TO C*-ALGEBRAS
M.-D. Choi

This course is concerned with the basic aspects of C*-algebras. During the first half of the semester, the lectures will be devoted to a systematic investigation of the finite-dimensional case---namely, the theory of n \times n complex matrices. Through many simple concrete examples we may describe various phenomena arising from the interplay of normed structure, order structure and algebraic structure. Later in the semester, we will continue to explore more striking facts pertinent to the infinite-dimensional case. Based on the background of students, various topics of related interest will be pursued.


MAT 1045HS
INTRODUCTION TO (SMOOTH) ERGODIC THEORY
G. Forni

The course is an introduction to some of the basic notions and methods of the theory of dynamical systems. It covers notions of topological dynamics and ergodic theory such 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, topological Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, the Von Neumann mean ergodic theorem and the Birkhoff ergodic theorem will be presented. Time permitting we will outline either entropy theory (topological entropy and Kolmogorov-Sinai entropy) and/or the subadditive ergodic theorem, the theory of Lyapunov exponents and the Oseledets theorem (with examples).

Prerequisites: Knowledge of real analysis, basic topology and measure theory. Some knowledge of functional analysis would be useful.


MAT 1052HF
HARMONIC ANALYSIS, DIOPHANTINE EQUATIONS AND INEQUALITIES
M. Goldstein


Waring's Problem. Weyl's Inequality. The Hardy-Littlewood Circle Method. Continued Fractions and Approximation by Rationals. Roth's Theorem.

References:
1. Natanson, M., Additive Number Theory,
2. Montgomery, H., Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis
3. Schmidt, W., Diophantine Approximations

Prerequisites: Introductory courses in complex analysis (e.g. MAT 354H) and real analysis (e.g. MAT 357H).


MAT 1062HS
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
C. Sulem

GOAL: The goal is to introduce some basic numerical techniques for solving ordinary and partial differential equations. The course is intended to students in Applied Mathematics and Engineering who wish to learn numerical methods useful for their research.

1. Introduction: Interpolation and Approximation techniques. Numerical differentiation and integration
2. ODEs: (a) Initial value problems. Euler's method, Predictor-Corrector, Runge-Kutta schemes. Convergence, consistency, stability. (b) Boundary value problems. Numerical solution of algebraic equations.
3. PDEs: (a) Classification. (b) Parabolic equations: explicit/implicit methods. Stability. Convergence theory. Finite difference methods; Collocation and Spectral methods. (c) Hyperbolic equations: Method of characteristics. Elliptic equations.

Galois Theory belongs to algebra. It is very understandable and has a lot of applications. For example it explains why algebraic equations are usually not solvable by means of radicals.

About 30 years ago I constructed a topological version of Galois Theory for functions in one complex variable. According to it, there are topological restrictions on the way the Riemann surface of a function representable by radicals covers the complex plane. If the function does not satisfy these restrictions, then it is not representable by radicals. Beside its geometric clarity the topological results on nonsolvability are stronger than the algebraic results. They have a lot of generalizations.

In the course I plan to present Galois Theory in details, to discuss topological results on nonsolvability by radical and their generalizations, to give an introduction to a multidimensional version of the theory.

Prerequisites:
The spectral theorem

References:
M. Rordam, K-Theory for C*-Algebras
H. Lin, An Introduction to the Classification of Amenable C*-Algebras
A. Connes, Noncommutative Geometry



MAT 1126HF
INTRODUCTION TO NONHOLONOMIC MECHANICS AND GEOMETRY
B. Khesin

Nonholonomic mechanics describes the geometry of systems subordinated to nonholonomic constraints, i.e systems whose restrictions on velocities do not arise from the constraints on the configuration space. The best known examples of such systems are a sliding skate and a rolling ball, as well as their numerous generalizations.

We start with an introduction to the Euler-Lagrange equation and the Lagrange-d'Alambert principle. After defining Hamiltonian systems and their integrability we go through

1. main nonholonomic examples (integrable and not);
2. reduction of symmetries in nonholonomic mechanics;
3. main concepts of the nonholonomic (i.e. sub-Riemannian) differential geometry: geometry of distributions, their curvatures, Carnot-Caratheodory metrics and their geodesics, singular geodesics;
4. relation to geometric mechanics and control theory.

1. A. M. Bloch "Nonholonomic mechanics and control" Interdisc. Appl. Math., vol.24. Springer-Verlag, New York, 2003, 483 pp.
2. R. Montgomery "A tour of subriemannian geometries, their geodesics and applications" Mathem. Surveys and Monographs, vol.91, AMS, Providence, RI, 2002, 259 pp.

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

• Affine varieties: Nullstellensatz, Zariski topology, ideal theory in Noetherian rings, Krull dimension, examples
• projective varieties: graded rings and ideals, P^n, examples, cones, quadrics, Grassmannians
• morphisms: regular functions, localization, local rings, rational functions, morphisms, products, examples
• rational maps: dominant and birational morphisms, birational equivalence, blowing up, examples
• nonsingular varieties: singular and nonsingular points, regular local rings, completions, analytic isomorphism, curve singularities,
• nonsingular curves: function fields, valuations, discrete valuation rings, fundamental theorem on nonsingular curves and function fields, examples.
• intersection theory: dimensions, Hilbert polynomials, intersection multiplicities, Bezout's theorem

This course is intended to introduce students to a recent technique in Algebraic Geometry based on application of the moment map and toric degenerations. One of the simplest examples of the moment map is the logarithm map that takes a point of the complex torus C*^n to the point in Rⁿ obtained by taking the logarithm of the absolute value coordinatewise . The images of holomorphic subvarieties of C*ⁿ under this map are called amoebas.

If one modifies this moment map by taking the logarithm with base t and lets t to go to infinity then the amoebas tend to some piecewise- linear polyhedral complexes in R^n. The dimension of these limiting complexes is equal to the dimension of the original varities. It turns out that such complexes can be considered as algebraic varieties over the so-called tropical semifield. The term "tropical semifield" appeared in Computer Science and, in the current context, refers to the real numbers augmented with the negative infinity and equipped with two operations, taking the maximum for addition and addition for multiplication.

In the course we consider applications of both the amoebas themselves and the resulting tropical geometry. One area where amoebas turn out to be useful is Topology of Real Algebraic Varieties, in particular, problems related to Hilbert's 16th problem. Using amoebas we show topological uniqueness of a homologically maximal curve in the real torus R*² and deduce a partial topological description for hypersurfaces in R*ⁿ for n>2. Applications of tropical geometry include construction of real algebraic varieties with prescribed topology (patchworking) as well as enumerative geometry.

A typical problem in enumerative algebraic geometry is to compute the number of curves of given degree and genus and with a given set of geometric constraints (e.g. passing through a point or another algebraic cycle, being tangent to such cycle, etc.). For a proper number of geometric constraints one expects a finite number of such curves. Even in the cases when this number is not finite there exists a way to interpret the answer to such problem as a (perhaps fractional or negative) Gromov-Witten number. Tropical geometry can be used for computation of these numbers. In this course we'll compute such numbers for arbitrary genus and degree when the ambient space is a toric surface and for genus 0 (and arbitrary degree) if the ambient space is a higher-dimensional toric variety. In addition we consider real counterparts of the enumerative problems, in particular, the Welschinger invariant, and do some computations for them.

Prerequisites: MAT 1194HF (MAT 449H1F) or some basic knowlege of Topology and Geometry. Contact the instructor for permission.

MAT 1197HF
GEOMETRY OF FLAG VARIETIES FOR SEMI-SIMPLE ALGEBRAIC GROUPS AND REPRESENTATION THEORY
S. Arkhipov

Below are the main topics to be covered in the course.

1. Recollections: Borel subgroups and maximal tori in a semi-simple algebraic group G, the weight and root lattices of a given maximal torus T ⊂ G, combinatorics of the root system for T ⊂ B ⊂ G, the Weyl group W of G, the unipotent radical U ⊂ B.
2. Finite dimensional representation theory of G: dominant weights, classification of finite dimensional representations of G, semi-simplicity.
3. The actions of G on the set of Borel subgroups in G and on the set of maximal unipotent subgroups in G. The flag variety G/B and the base affine space G/U for G.
4. Algebras of regular functions on G, G/U and G/B as representations of G.
5. The action of a maximal torus T on G/B. Fixed points. Recollections: Morse theory. Schubert cells.
6. Enumeration and description of the orbits of B-action on G/B. Bruhat decomposition. Schubert cells as B-orbits.
7. Bruhat cell decomposition of the Flag variety G/B. Cohomology of G/B.
8. Bernstein-Gelfand-Gelfand description of multiplication in the cohomology ring of G/B. G-equivariant, B-equivariant and T-equivariant cohomology of G/B.
9. G-equivariant line bundles on G/B, classification, Borel-Weil-Bott theorem.
10. Highest weight modules over the Lie algebra \mathfrak g. Verma and contragradient Verma modules.
11. Kac-Weyl character formula for an irreducible representation of G: geometric proof.
12. Cotangent bundle to the Flag variety T^*(G/B). Representation theory description. The nilpotent cone of \mathfrak g. Springer-Grothendieck map. Regular functions on T^*(G/B) and on the nilpotent cone.
13. Conormal bundles to Schubert cells. Steinberg variety for G.
14. Convolution in Borel-Moore homology of the Steinberg variety.

Topics to be covered: Fundamentals of design; existence of triple systems; enumeration; computational methods; isomorphism and invariants; configuration theory; chromatic invariants; resolvability; directed, Mendelsohn, and mixed triples sytems.

Prerequisite(s):
Linear Algebra and an introductory course in combinatorics such as MAT 344HF are recommended.


MAT 1312HS
LIE GROUPS, SYMMETRIC SPACES, SUB-RIEMANNIAN GEOMETRY AND THE EQUATIONS OF PHYSICS
V. Jurdjevic

This course will be driven by two fundamental observations; the first observation is that the dual of the Lie algebra g of a Lie group G is a Poisson manifold and that much of the structure and the theory of Lie groups and their symmetric spaces can be directly deduced from this fact alone, and the second observation is that the orbit theorem with related accessibility results together with the Maximum Principle and the associated Hamiltonian formalism of optimal control theory are effective tools in carrying out the program suggested by the first observation. The symplectic outlook on Lie groups and their homogeneous spaces offers an additional advantage in that it ellucidates the importance of Lie groups for the equations of applied mathematics.

The course will be a synthesis of the following topics:

1. Symmetric Spaces, Sub-Riemannian Problems and Principal Bundles.
2. Kepler's Problem and Non-Euclidean Geometry.
3. Kirchhoff's Elastic Problem, Elasticae and Mechanical Tops.
4. The Rolling Sphere Problems.
5. Integrable Hamiltonian Systems. Kowalewski- Lyapunov Integrability Criteria.
6. Complex Symplectic Geometry. Real Forms. Applications to Geometry and Mechanics.
7. Problems of Quantum Control.
8. Schroedinger's Equation and Infinite Dimensional Hamiltonian Systems. Solitons.

Much of this material will be taken from [J2] and the graduate monograph of mine in progress that will be ( I hope) finished by the time the course is offered.

Prerequisites:

• The orbit theorem and the theory of Lie determined systems (Chapters 2 and 3 of [J1]).
• The Maximum Principle of optimal control. ([J1] and [AS]).
• Basics from differential and symplectic geometry. Some knowledge of principal bundles and their connections is also desirable. For instance, the first few pages of [S] on the Geometry of G-structures are enough.

References:

• [AS] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Point of View, Encyclopedia of Mathematical Sciences, vol. 87, Springer-Verlag, New York, 2004.
  
• [H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
  
• [J1] V. Jurdjevic, Geometric Control Theory, Studies in Advanced Math, vol. 52, Cambridge Univ. Press, New York, 1997.
  
• [J2] V. Jurdjevic, Hamiltonian Systems on Complex Lie groups and their Homogeneous Spaces, Memoirs AMS Number 838 178 (2005).
  
• [S] S. Sternberg, Lectures on Differential Geometry, Prentice- Hall, Inc, Englewood Cliffs, N.J., 1964.

• Elliptic differential operators,
• K-theory
• The Atiyah-Singer index theorem
• Cohomological formulas
• Applications

This is a continuation of 06-1350 and the abstract remains the same:

An "Algebraic Knot Theory" should consist of two ingredients

1. A map taking knots to algebraic entities; such a map may be useful, say, to tell different knots apart.
2. A collection of rules of the general nature of "if two knots are related in such and such a way, their corresponding algebraic entities are related in such and such a way". Such rules may allow us, say, to tell how far a knot is from the unknot or how far are two knots from each other.

(If you have seen homology as in algebraic topology, recall that its strength stems from it being a functor. Not merely it assigns groups to spaces, but further, if spaces are related by maps, the corresponding groups are related by a homomorphism. We seek the same, or similar, for knots.)

The first ingredient for an "Algebraic Knot Theory" exists in many ways and forms; these are the many types and theories of "knot invariants". There is very little of the second ingredient at present, though when properly generalized and interpreted, the so-called Kontsevich Integral seems to be it. But viewed from this angle, the Kontsevich Integral is remarkably poorly understood.

The purpose of this class will be to understand all of the above.


MAT 1355HF
RESOLUTION OF SINGULARITIES AND APPLICATIONS IN ANALYSIS AND IN GEOMETRY
P. Milman

In this course we will derive various applications of resolution of singularities to the classical-type inequalities of Analysis. We will also examine some applications to problems in Geometry. The course will include a proof of a `mini' variant of desingularization that will suffice for these applications.

Prerequisites: Standard undergraduate material (first 3 years) of the math specialist program including the implicit function theorem (and related material) as well as some familiarity with the basic algebraic and geometric notions such as polynomials, analytic functions, ideals, rings etc., affine and projective spaces.


MAT 1435HF
ANALYTIC METHODS IN COMBINATORICS
B. Szegedy

The goal of this course is to highlight some powerful analytic methods that prove to be extremely successful in modern combinatorics. A considerable part of the course will focus on the so-called Regularity Lemma (by Szemeredi) and recent generalizations for hypergraphs (by Gowers, Rodl, Nagle, Skokan, Schacht, Kohayakawa). We will also discuss applications of harmonic analysis.

Prerequisites: Knowledge of graph theory and harmonic analysis.


MAT 1450HS
SET THEORY: STRUCTURAL RAMSEY THEORY AND DYNAMICS OF GROUPS OF AUTOMORPHISMS
S. Todorcevic

This course is a natural continuation of the 2005-06 MAT 1450HF course, but will not be bounded to those who took it as the subject matter is related but not too much dependent. The stress this time will be on finite Ramsey theory, or more precisely finite structural Ramsey theory, and its relationship to Fraisse theory of homogeneous structure and topological dynamics of groups of automorphisms.

Prerequisites: Basic core courses in mathematics.

Textbooks:
Graham-Rothschild-Spencer, Ramsey Theory, 1990
My textbook "Introduction to Ramsey Spaces", which at the moment is available as draft but at that time may be even in press.


MAT 1502HS
STOCHASTIC CALCULUS
J. Quastel

Random walk, Markov chains, Martingales, Brownian motion, Stochastic Integrals, Markov processes and associated partial differential equations, Ito's formula, Stochastic differential equations, Cameron-Martin-Girsanov formula, relation between discrete and continuous time models, applications including Black-Scholes formula.

Prerequisite:
Probabilty or Real analysis.


MAT 1739HF
INTRODUCTION TO SUPERSYMMETRIC QUANTUM FIELD THEORIES
K. Hori

This course will give an accessible introduction to supersymmetric quantum field theories in low dimensions. Supersymmetry is relevant for many recent developments in various fields in mathemetics, including Gromov-Witten invariants, Donaldson invariants, Floer homology, various fixed point theorems, etc. No special background is assumed.

1. Gaussian integrals and Feynman diagrams, fermionic integrals.
2. Supersymmetric quantum mechanics and Morse theory.
3. Non-linear sigma models, Landau-Gizburg models, Linear sigma models.
4. Renormalization group flows, supersymmetric non-renormalization theorems.
5. Chiral rings and topological field theory.
6. Mirror symmetry.

1. P.Deligne, P.Etingof, D.S.Freed, L.C.Jeffrey, D.Kazhdan, J.W.Morgan, D.R. Morrison, E. Witten, editors, ``Quantum Fields and Strings: A Course for Mathematicians'' vol 1 & 2 (AMS, 1999).
2. E. Witten, ``Supersymmetry And Morse Theory'', J. Diff. Geom. 17 (1982) 661.

The second group of models deals with mechanisms through which networks of interacting biomolecules (proteins or genes) carry out the essential functions in living cells. Among the questions which are addressed here is how the genetic and biochemical networks withstand considerable variations and random perturbations of biochemical parameters. The complexity and high inter-connectedness of these networks makes the question of the stability in their functioning of special importance.

Finally we will discuss mathematical models of the dynamics of HIV-1 and of cancer growth.

The models above are expressed in terms of Markov chains and stochastic ordinary differential equations. In addition, in the first case, reaction-diffusion equations (e.g. Keller-Segel equations) and stochastic particle dynamics are used. This mathematical background together with its biological interpretation will be developed in the course.

Prerequisites for this course: some familiarity with elementary ordinary and partial differential equations and elementary probability theory. No knowledge of biology is required.


MAT 1900Y/1901H/1902H
READINGS IN PURE MATHEMATICS

Numbers assigned for students wishing individual instruction in an area of pure mathematics.


MAT 1950Y/1951H/1952H
READINGS IN APPLIED MATHEMATICS

Numbers assigned for students wishing individual instruction in an area of applied mathematics.


STA 2111HF
GRADUATE PROBABILITY I
B. Virag

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
J. Quastel

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

Textbook:
Durrett, Probability: Theory and Examples

FIELDS INSTITUTE PROGRAM COURSES

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.)