2005-2006 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-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.

1. Point set topology: Topological spaces and continuous functions, connectedness and compactness, Tychonoff's theorem and the Stone-Cech compactification, metric spaces, countability and separation axioms.
2. Homotopy: Fundamental groups, Van Kampen theorem, Brouwer's theorem for the 2-disk. Homotopy of spaces and maps, higher homotopy groups.
3. The language of category theory.
4. A word about the classification of surfaces.

1. More on category theory, the fundamental theorem of covering spaces.
2. Homology: Simplicial and singular homology, homotopy invariance, exact sequences, excision, Brouwer's theorem for the n-disk, Mayer-Vietoris, degrees of maps, CW-complexes, the topology of Euclidean spaces, Borsuk-Ulam.
   
3. Cohomology: Cohomology groups, cup products, cohomology with coefficients.
4. Topological manifolds: Orientation, fundamental class, Poincare duality.

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.

We will pick and choose from the following texts:
Gerald Folland: Introduction to Partial Differential Equations
Lawrence Evans: Partial Differential Equations
Elias Stein and Guido Weiss: Introduction to Fourier Analysis on Euclidean Spaces
Michael Taylor: Partial Differential Equations
Francois Treves: Basic Linear Partial Differential Equations


MAT 1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
R. Jerrard

This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, several fixed point theorems, and nonlinear semigroup theory. A recurring theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.

The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.

2005-2006 CROSS-LISTED COURSES

1. Theorem on existance and uniqueness of solutions:
   • Vector fields and ODE's,
   • Contracting map principle,
   • Theorem on existence and uniqueness,
   • Differentiability of solutions with respect to initial conditions and parameters,
   • Flow defined by an ODE.
2. Linear systems:
   • Exponentials of linear operators in Euclidean space,
   • Canonical form of a linear operator acting in real Euclidean space,
   • Phase portrets of a linear system in plane.
3. Stability of solutions:
   • Definition of stability and asymptotic stability,
   • Theorems on stability and instability in terms of eigenvalues of the linearized system.
4. Differentiable manifolds:
   • Local charts,
   • Tangent space,
   • Vector fields and ODE's on differentiable manifolds.
5. Differential forms and Stokes' formula:
   • Exterior forms on Rⁿ,
   • Differential forms on manifolds,
   • Exterior product and differentiation of differential forms,
   • Stokes' formula,
   • Stokes' formula in R³,
   • Vortex field of a vector field on R²ⁿ⁺¹,
   • Hydrodynamical lemma
6. Newton equations and Hamiltonian systems:
   • Newton equations for N-body problem,
   • Angular momentum and Kepler's laws for two body problem,
   • Hamiltonian system for N-body problem,
   • Liouville's Theorem on conservation of the volume in the phase space.
7. Poisson's brackets of Hamiltonian functions:
   • Poisson's bracket of vector fields,
   • The Jacobi identity,
   • Poisson's bracket of Hamiltonian functions,
   • Conservation laws for a Hamiltonian system.
8. Liouville theorem on integrable systems:
   • Action of the flows defined by 1st integrals,
   • Discrete subgroups in Rⁿ,
   • Hamiltonian system on unvariant tori
9. Canonical formulism:
   • The vortex form of a Hamiltonian system,
   • Poincare-Cartan's integral invariant,
   • Canonical change of variables,
   • Generating function,
   • Hamilton-Jacobi equations.
10. Action-angle variables:
    • Action-angle variables in the case of 1 degree of freedom,
    • Action-angle variables of an integrable system.
11. Kolmogorov's theorem:
    • Theorem on averages,
    • Averages of small perturbations,
    • Statement of Kolmogorov's theorem.

Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers.

Prerequisite a course in groups, rings and fields.




MAT 1194HF (MAT 449H1F)
INTRODUCTION TO ALGEBRAIC GEOMETRY, ALGEBRAIC CURVES AND PLANE GEOMETRY
A. Khovanskii

I will proof main theorems of the Theory of Algebraic Curves, and will consider its applications to Classical Plane Geometry and to geometry of Real Algebraic Curves:

1. Algebraic curve, Riemann surface of algebraic function, meromorphic functions and meromorphic 1-forms on algebraic curve.
2. 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.
3. Generic algebraic curve with fixed Newton polygon. Its Euler characteristic and genus. Space of holomorphic 1-forms. Pick formula (from an elementary geometry of integral convex polygons).
4. A. Weil theorem.
5. Abel Theorem.
6. Geometry of nondegenerate and degenerate cubic. Group structure.
7. Pascal and Menelaus Theorems as corollaries from Abel Theorem, solutions of Poncelet problem and "butterfly" problem using Abel Theorem.
8. Riemann-Roch theorem.
9. Jacobi theorem.
10. Euler-Jacobi formula. Pascal Theorem as a corollary from Euler-Jacobi formula.
11. Real smooth algebraic curves of even an odd degree.
12. Harnack inequality.
13. Estimation of the index of real polynomial vector field.
14. Petrovskii Theorem about real algebraic curve of degree six.

Prerequisite: I will not assume any special knowledge, but acquaintance with complex analysis in one variable will be very useful.

Recommended Literature: Griffiths P.A., Introduction to Algebraic Curves, Translations of mathematical monographs 76, American Mathematical Society, Providence, 1989


MAT 1196HF (MAT 445H1F)
REPRESENTATION THEORY
F. Murnaghan

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.

Prerequisite: A strong background in abstract algebra, particularly in group theory and linear algebra.

References:
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Volume 42, Springer-Verlag
B. Simon, Representations of Finite and Compact Groups, Graduate Studies in Mathematics, Volume 10, AMS
S. Weintraub, Representation Theory of Finite Groups, Graduate Studies in Mathematics, Volume 59, AMS (2003)
W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, Volume 129, Springer-Verlag
James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Volume 9, Springer-Verlag
Curtis and Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, 1962
J.L. Alperin and R.B. Bell, Groups and Representations, Graduate Texts in Math. 162 (1995), Springer
Daniel Bump, Lie Groups, Graduate Texts in Math. 225 (2004), Springer
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Graduate Texts in Math. 222 (2003), Springer


MAT 1200HS (MAT 415HS)
ALGEBRAIC NUMBER THEORY
H. Kim

Introduce techniques of modern number theory via p-adic numbers, adeles, ideles and harmonic analysis on locally compact groups. The main goal will be Tate's proof of the analytic continuation and the functional equation of Hecke's L-functions, and their applications such as class number formulas, Dirichlet's theorem on arithmetic progressions, Tchebotarev's density theorem. This course is the first half of a year-long course in algebraic number theory. The second half, entitled "Class field theory," will be given in the fall of 2006.

Recommended prerequisite: Galois theory, some measure theory and point-set topology.

References: Algebraic Number Theory by S. Lang,
Fourier Analysis on Number Fields by D. Ramakrishnan and R.J. Valenza


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


MAT 1342HS (MAT 464H1S)
DIFFERENTIAL GEOMETRY
G. Mikhalkin

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 conjecture 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.


MAT 1404HF (MAT 409H1F)
SET THEORY
F.D. Tall

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.

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


MAT 1638HS (APM 436H1S)
FLUID MECHANICS
J. Colliander

Boltzmann, Euler and Navier-Stokes equations. Viscous and non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propagation. Analysis of one dimensional gas flow.

Prerequisite: APM351Y1 (Partial Differential Equations)


MAT 1700HS (APM 426H1S)
GENERAL RELATIVITY
R. McCann

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. Graviational 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)
FOUNDATIONS OF QUANTUM MECHANICS
R. Jerrard

• brief review of classical mechanics
• postulates of quantum mechanics: states and observables, Schroedinger equation
• Hilbert spaces: basic properties, bounded and unbounded operators, spectral theorem for self-adjoint operators, Stone's Theorem
• the uncertainty principle
• exact solution of the Schroedinger equation in some special cases
• spectral theory of Schroedinger operators
• some aspects of scattering

Reference:
S. Gustafson and I.M. Sigal: Mathematical Concepts of Quantum Mechanics will be used as reference, but is not required.


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.

2005-2006 TOPICS COURSES

Prerequisite: the spectral theorem.

Reference:
Yasuyuki Kawahigashi and David E. Evans, Quantum symmetries on operator algebras, Oxford University Press, 1998.


MAT 1062HF
TOPICS IN LINEAR PARTIAL DIFFERENTIAL EQUATIONS II
V. Ivrii

This course covers more advanced topics in PDE than those covered by MAT 1060H, as given in 2004-2005. Students are assumed to have attended this course or the undergraduate courses APM 346H or MAT 351Y and are familiar with methods of solving basic PDEs.

• Theory of distributions.
  • Classes D, E, S and their dual D', E', S'. Basic operations. Fourier transform.
  • Sobolev spaces H^s on R^d.
  • Paley-Wiener theorem.
• Hyperbolic linear equations with constant coefficients
  • Solution of the Cauchy problem for constant coefficient operators via Fourier transform.
  • Hyperbolicity, Gårding hyperbolicity, strong hyperbolicity.
  • Fundamental solution of the Cauchy problem. Support of solutions. Characteristic cone.
  • Singular support of solutions: strongly hyperbolic case. Examples (wave equation, Maxwell, elasticity)
  • Singular support of solutions: non-strongly hyperbolic case, conical refraction. Examples (crystall optics, elasticity, magnetic hydrodynamics).
• Introduction to pseudo-differential operators.
  • Pseudo-differential operators.
  • Calculus.
  • L²-boundedness.
  • Application of pseudo-differential operators to elliptic equations. Parametrices.
• Strictly hyperbolic equations and systems.
  • Energy estimates. Existence theorem for Cauchy problem.
  • Holmgren uniqueness theorem.
  • Propagation of singularities. Examples (partially the same as before but in inhomogeneous media).
  • Construction of oscillatory solutions.
• Strictly hyperbolic equations and systems near boundary.
  • Energy estimates, existence and uniqueness of solutions.
  • Construction of oscillatory solutions.
  • Reflection and refraction. Compensation waves.
  • Gliding and grazing waves.
  • Rayleigh waves.
  • Crawling waves.

I will not follow any specific book, but rather pick from here and there, including such old books as Courant and Hilbert, Partial Differential Equations. I will distribute handouts.


MAT 1120HF
CLIFFORD ALGEBRAS AND LIE GROUPS
E. Meinrenken

This course will be an introduction to Clifford algebras, Dirac operators, and their applications to Lie groups and representation theory. Possible topics include:

1. Clifford algebras
2. Spin groups
3. Spinor modules
4. Dirac operators
5. Weil algebras
6. Kostant-Dirac operators
7. Duflo theorem

Prerequisites: Linear algebra; some basic knowledge of the theory of Lie groups and Lie algebras would be helpful.

References: I plan to provide lecture notes for this course. See also Shlomo Sternberg: Lie algebras. Lecture Notes, Harvard 2004.


MAT 1126HF
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.

Prerequisite: Familiarity with basic symplectic geometry is advisable.

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

Plan:

1. Main notions in Lie groups, Lie algebras, adjoint and coadjoint representations: A Lie group and its Lie algebra, adjoint representation, group adjoint and coadjoint orbits.
2. The Euler equations as equations of the geodesic flow: Least action principle, the Euler top, the Euler equation of ideal fluid.
3. The Hamiltonian framework for the Euler equations: Equations on the dual Lie algebra, Poisson structures.
4. Examples: SO(3) (motion of a rigid body), E(3) (Kirchhoff equations: body in a fluid), SDiff(M) (ideal hydrodynamics), MHD (magnetohydrodynamics), Virasoro algebra (the KdV and Camassa-Holm equations), and Landau-Lifschits equation.
5. Conservation laws for fluids: First integrals for ideal, barotropic, and compressible fluids. Relation to symplectic structures on the spaces of curves and polygons in R³. Vortex approximations in 2D.
6. Steady solutions in 2D and 3D: Bernoulli function, related variational problems, restrictions on topology of steady solutions, Arnold's stability criterion.
7. Topology bounds the energy of a field: An ergodic interpretation of helicity (asymptotic Hopf invariant), energy estimates, Sakharov--Zeldovich problem.
8. Other applications (time permitted): Differential geometry of diffeomorphism groups: diameter and curvatures; generalized flows, Kolmogorov's approach, Brenier's model with Coulomb interaction, etc.

On one hand Toric Varieties are very useful by themselves and because of their relations with all Modern Mathematics. On the other hand 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.

Prerequisite a course in complex analysis.

Recommended Literature: Toroidal Embeddings, by G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Springer Lecture Notes 339, 1973.
Introduction to Toric Varieties, by W. Fulton, Princeton University Press 1993.


MAT 1199HF
INTRODUCTION TO MODULAR FORMS
V. Blomer

A modular form is a holomorphic function that lives on the upper half plane and enjoys certain symmetry properties. The theory of modular forms is central in many branches of mathematics, and appears in particular in number theory, representation theory, algebraic geometry, and complex analysis. This course will give an introduction to modular forms from a mainly number theoretic point of view, including

• classical modular forms for congruence subgroups
• Hecke operators and newforms
• theta functions and quadratic forms
• L-functions
• arithmetic properties of Fourier coefficients of cusp forms and, if time permits,
• Galois representations
• half-integral weight modular forms and Shimura's correspondence
• Maass wave forms

Prerequisite: the spectral theorem.

References:
Alain Connes, Noncommutative geometry, Academic Press, 1994.
Yasuyuki Kawahigashi and David E. Evans, Quantum symmetries on operator algebras, Oxford University Press, 1998.
M. Rordam and N.J. Lausten, An Introduction to K-Theory for C*-algebras, Cambridge University Press, Cambridge, 2000.


MAT 1355HS
RESOLUTIONS OF SINGULARITIES
E. Bierstone

According to Grothendieck, resolution of singularities is "the most powerful tool we have for studying algebraic and analytic varieties". Our goal in this course is an understandable constructive proof of canonical resolution of singularities in characteristic zero. We will develop techniques of differentials, blowing up, etc. that play important roles in desingularization and many other questions.

Recommended background:
An introductory course in either algebraic geometry, several complex variables or commutative algebra. If you have questions about the course or about your preparation, please discuss them with the instructor.


MAT 1450HS
SET THEORY: RAMSEY THEORY
S. Todorcevic

The course will cover some of the basic results of high-dimensional Ramsey theory as well as some of its applications to functional analysis and topological dynamics.

Prerequisite: No special prerequisites are needed.

References:
R.L.Graham, B.L.Rotshild, and J.H. Spencer, Ramsey Theory, Wiley-Interscience 1990.
S.A.Argyros and S.Todorcevic, Ramsey methods in analysis, Birkhauser 2005.


MAT 1500YY (MAT 1501HF/MAT 1502HS)
APPLIED ANALYSIS (with Applications to Geometry, Physics, Engineering and Biology)
I.M. Sigal

In this course we describe underlying concepts and effective methods of analysis and illustrate them on various problems arising in applications to Geometry, Physics (Quantum Mechanics), Engineering (material sciences, computer vision) and Biology (chemotaxis, gene expression). The course consists of two relatively independent parts which cover the following topics:

Part I

1. Spectral analysis.
2. Calculus of variations, optimization, and control (including applications to interface motion and pattern analysis).
3. Calculus of maps (contraction mapping principle, method of successive approximations, derivatives of maps, inverse and implicit function theorems) and bifurcation theory.

1. Gradient and Hamiltonian systems.
2. Stochastic calculus (key notions of probability theory, Markov chains. and their applications, stochastic integrals and differential equations).
3. Some mathematical models of computer vision and Biology.

Prerequisites: Vector calculus, linear algebra, elementary differential equations. It is also desirable to have some knowledge of elementary analysis involving theory of integration and point set topology.

Textbook:
L.Jonsson, M. Merkli, I.M. Sigal and F.Ting: Lectures on Applied Analysis, 2005.

Additional references:
G. Folland: Real Analysis,
E. Lieb and M. Loss: Analysis, AMS Press.
R. McOwen: Partial Differential Equations.
Papers


MAT 1751HF
TOPICS IN COMPUTATIONAL MATHEMATICS
M. Shub

We will study the problem P versus NP over various fields and the complexity theory of numerical analysis. Some special subjects of study which may be included are: the role of the condition number in numerical analysis as a complexity measure, linear programming over the rationals and reals, the location of zeros of systems of polynomial equations in n-variables, dynamical systems related to numerical algorithms, algebraic complexity theory and topological lower bounds.

Prerequisite: MAT 1750H or a reasonable acquaintance with the first nine chapters of the textbook listed below.

Textbook:
Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale: Complexity and Real Computation, Springer-Verlag, 1998, ISBN 9 780387 982816


MAT 1839HS
OPTIMAL TRANSPORTATION AND NONLINEAR DYNAMICS
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.

Text:
Cedric Villani "Topics in Optimal Transportation" Providence: AMS 2003. GSM/58 ISBN 0-8218-3312-X $59 ($47 AMS members)


MAT 1843HS
BIFURCATION, SYMMETRY-BREAKING, AND PATTERN FORMATION
M. Golubitsky

This one semester introductory course discusses the typical ways that solutions to differential equations change as a parameter is varied; how these theorems change when the differential equations have symmetry, and how such ideas can be used in a variety of applications.

1. Steady-state and Hopf bifurcation (the origins of multiplicity and oscillation);
2. Equivariant bifurcation theory (also called spontaneous symmetry-breaking) with preliminaries on group representation theory;
3. Periodic solutions with spatio-temporal symmetry, and applications (such as, to rotating waves and the rhythms of animal gaits);
4. Pattern formation (such as in reaction-diffusion systems, the Taylor-Couette experiment, and networks of ODEs).
5. Intermittency through heteroclinic cycles; symmetric chaos.

Textbooks:
M. Golubitsky, I.N. Stewart and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol. II., Applied Mathematical Sciences 69. Springer-Verlag, New York, 1988.
M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Birkhauser, Basel, 2003. (The paperback edition.)

Prerequisites:
Linear algebra and a first course in ODEs (one that includes linear stability implies stability for systems). Some small familiarity with groups would be useful. Scientists and engineers are welcome.


MAT 1845HF
DYNAMICS IN DIMENSION TWO
C. Pugh

This is an introductory course about qualitative dynamical systems. It includes much of the background material possessed by current dynamics researchers, but by presenting things in dimension two, I hope that all the ideas and most of the proofs can be comprehended by staring at the right picture. Obviously, this is the "geometric point of view."

Here are some of the topics I intend to cover:

1. Flows and cascades (iterations of a diffeomorphism), concepts such as orbits, invariance, recurrence, etc.
2. Poincaré-Bendixson Theory in the plane, the sphere, and what else happens on general surfaces.
3. Stable Manifold Theory (It's easier in dimension two.)
4. Hartman-Grobman linearization - maybe some of the Sternberg linearization.
5. Structural Stability and Peixoto's Density Theorem.
6. The closing lemma (which is much simpler in dimension two).
7. Morse-Smale dynamics.
8. Structural stability and Ω-stability.
9. Smale's horseshoe and the Birkhoff-Smale Theorem.
10. Anosov diffeomorphisms and Franks' characterization of them.
11. Axiom A dynamics in dimension two.
12. Lyapunov exponents.
13. Katok's Theorem about positive entropy in dimension two implying the existence of a horseshoe.

References: The course will be taught from the instructor's own notes. The following books are for background:
Robinson, Clark: Dynamical Systems
Hasselblatt and Katok: A First Course in Dynamics
Hartman: Ordinary Differential Equations


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
J. Rosenthal

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


STA 2211HS
GRADUATE PROBABILITY II
J. Rosenthal

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

FIELDS INSTITUTE PROGRAM COURSES

Stochastic (or Schramm) Loewner Evolution (SLE) is a family of conformally invariant random processes conjectured to describe the scaling limit of various combinatorial models arising in Statistical Mechanics and Conformal Field Theory, such as Loop Erased random walk, Self-avoiding random walk, percolation, and the Ising model. SLE proved to be an important link between Complex Analysis, Probability, and Theoretical Physics. SLE has also been used by Lawler, Schramm, and Werner to verify the Mandelbrot's conjecture about the dimension of the Brownian Frontier. We start with a careful discussion of the necessary background from Stochastic Analysis and the Geometric Function Theory. Then we move to the proof of Mandelbrot's conjecture. Other topics that might be covered are the dimension properties of the SLE and the proof of the Smirnov's theorem about the critical limit of percolation.

Prerequisites: basics of complex analysis and probability theory.

Note: The topic of the course is closely related to the Fall 2005 program in the Fields institute.

References:

1. "Random Planar Curves and Schramm-Loewner Evolution" by Wendelin Werner. (http://arxiv.org/abs/math.PR/0303354)
2. "Conformal Restriction and Related Questions" by Wendelin Werner. (http://arxiv.org/abs/math.PR/0307353)
3. Conformally Invariant Processes in the Plane by Greg Lawler

In the first part of the course we shall discuss the renormalization approach to Herman theory and prove rigidity theorem for circle diffeomorphisms with Diophantine rotation numbers. We then extend the whole construction to circle diffeomorphisms with break points. Finally, in the third part of the course we use renormalizations to prove a KAM-type theorem for area-preserving twist maps.

Prerequisite: no special prerequisites required.

Reference:
W. de Melo, S. van Strien, One-dimensional dynamics, Springer-Verlag, Berlin, 1993.


MAT 1844HF
RENORMALIZATION IN ONE-DIMENSIONAL DYNAMICS
M. Yampolsky

Renormalization ideas entered one-dimensional dynamics in the late 1970's with the discovery of Feigenbaum universality. After seminal works of Sullivan, and Douady and Hubbard it has revolutionized the field. The course will serve as a self-contained introduction to this beautiful subject, only familiarity with complex analysis will be assumed.

We will describe two main examples of renormalization in dynamics - unimodal maps and critical circle maps. For the latter we will outline the construction of the renormalization theory, culminating with Lanford universality. For the former we will explain the connection of renormalization to self-similarity of the Mandelbrot set.

Prerequisite: Permission from the instructor.


MAT 1846HS
SEVERAL GEMS OF COMPLEX DYNAMICS
M. Yampolsky

In this course we plan to present a few beautiful and fundamental results of modern complex dynamics. The course will be structured as a series of mini-courses, which will be loosely related to each other. We plan to make presentation self-contained, and accessible to a graduate student with knowledge of basic complex analysis and differential geometry, and interest in dynamics.

Some of the theorems we plan to cover are:

• Discontinuities in the dependence of Julia sets on parameters; Lavaurs theorem.
• Theorem of Shishikura on Hausdorff dimension of the boundary of the Mandelbrot set.
• A. Epstein's proof of Fatou-Shishikura bound on the number of non-repelling cycles.

A common theme in the above results is the study of perturbations of parabolic orbits. Further topics may include computability of Julia sets, properties of Siegel disks, or other themes suggested by the audience.

Prerequisite: Permission from the instructor.

Prerequisite: Permission from the instructor.


MAT 1847HS
HOLOMORPHIC DYNAMICS
M. Lyubich

The central theme of this course will be the Rigidity Conjecture in Holomorphic Dynamics that asserts that any two rational maps (except one special class of maps covered by torus automorphisms) which are topologically conjugate must be conjugate by a Mobius transformation. This Conjecture is intimately related to the Mostow Rigidity phenomenon in hyperbolic geometry. In the quadratic case, it is related to the MLC Conjecture asserting that the Mandelbrot set is locally connected. After covering necessary background in basic holomorphic dynamics and renormalization theory, recent advances in the problem will be discussed.

References:
J. Milnor, Dynamics in one complex variable. Vieweg.
C. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies, v. 142.

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 2005-2006 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.)