Department of Mathematics

TENTATIVE 2012-2013 Graduate Courses Descriptions

Core Graduate Courses | Cross-Listed and Topics Courses

Core Courses

MAT 1000HF (MAT 457H1F)
REAL ANALYSIS I
A. Burchard
(View Timetable)

Measure Theory: Lebesque measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.

Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L-spaces, Holder and Minkowski inequalities.

Textbook: 
Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley

References:
Elias Stein and Rami Shakarchi: Measure Theory, Integration, and Hilbert Spaces
Eliott H. Lieb and Michael Loss: Analysis AMS Graduate Texts in Mathematics, 14 (either edition)
H.L. Royden: Real Analysis, Macmillan, 1988.
A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, 1975.

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MAT 1001HS (MAT 458H1S)
REAL ANALYSIS II
A. Burchard
(View Timetable)

Fourier analysis: Fourier series and transform, convergence results, Fourier inversion theorem, L-theory, estimates, convolutions.

More functional analysis:  Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.

Textbook: 
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.

References:
Katznelson: Harmonic Analysis, published by Dover or Cambridge Press
S.D. Promislow: A First Course in Functional Analysis, Wiley, 2008.
Elias Stein and Rami Shakarchi: Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis) (Book 4)
Lieb and Loss: Analysis 2nd edition, Graduate studies in Mathematics, AMS

MAT 1002HS (MAT 454H1S)
COMPLEX ANALYSIS
I. Graham
(View Timetable)

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, Riemann surfaces.
5. Modular functions and the Picard Theorems.
6. Other topics are possible, like product theorems, elliptic functions, and non-isolated removability theorems.

Recommended prerequisites: A first course in complex analysis and a course in real analysis. Measure theory is not required.

Main References:
T. Gamelin, Complex Analysis
W. Rudin, Real and Complex Analysis, 2nd or 3rd edition
D. Sarason, Complex Function Theory

Additional References:
L. Ahlfors: Complex Analysis, 3rd Edition
Stein and Shakarchi: Complex Analysis

MAT 1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
A. Nachman
(View Timetable)

This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools. A key theme will be the development of techniques for studying non-smooth solutions to these equations.

Textbook: 
L.C. Evans, Partial Differential Equations

MAT 1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
J. Colliander
(View Timetable)

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, and fixed point theorems.  One important 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.

Other topics in PDE will also be discussed.

Reference: 
Lawrence Evans: Partial Differential Equations

MAT 1100HF
ALGEBRA I
R.-O. Buchweitz
(View Timetable)

Basic notions of linear algebra: brief recollection. The language of Hom spaces and the corresponding canonical isomorphisms. Tensor product of vector spaces.

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.

Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, and unique factorization domains.

Modules: Modules and algebras over a ring, tensor products, modules over a principal ideal domain

Recommended prerequisites are a full year undergraduate course in Linear Algebra and one term of an introductory undergraduate course in higher algebra, covering, at least, basic group theory. While this material will be reviewed in the course, it will be done at "high speed", assuming that you have already some familiarity with the basics.  You will be very well prepared indeed, if you have no difficulties reading and understanding the book, listed here under "Other References", M. Artin: Algebra that the author wrote for his undergraduate algebra courses at MIT.

Textbooks:
Lang: Algebra, 3rd edition
Dummit and Foote: Abstract Algebra, 2nd Edition

Other References:
Jacobson: Basic Algebra, Volumes I and II.
Cohn: Basic Algebra
M. Artin: Algebra.

MAT 1101HS
ALGEBRA II
J. Repka
(View Timetable)

Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.

Commutative Rings: Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties. structure of semisimple algebras, application to representation theory of finite groups.

Textbooks:
Dummit and Foote: Abstract Algebra, 3rd Edition
Lang: Algebra, 3rd Edition.

Other References:
Jacobson: Basic Algebra, Volumes I and II.
Cohn: Basic Algebra
M. Artin: Algebra.

MAT 1300HF
TOPOLOGY I
M. Gualtieri
(View Timetable)

Local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.

Differential forms: exterior algebra, forms, pullbacks, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.

Prerequisites:  linear algebra; vector calculus; point set topology

Textbook:  
John M. Lee: Introduction to Smooth Manifolds

MAT 1301HS
TOPOLOGY II
M. Gualtieri
(View Timetable)

Fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.

Homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.

Textbook:  
Allen Hatcher, Algebraic Topology

Recommended Textbooks:
Munkres, Topology
Munkres, Algebraic Topology

2012-2013 GRADUATE COURSES (CROSS-LISTED AND TOPICS)

 

MAT 1011HS (MAT436H1S)
INTRODUCTION TO LINEAR OPERATORS
G.A. Elliott
(View Timetable)

Topics (and cross-listed):
The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced).  The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now.  Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books).

Prerequisites: 
Elementary analysis and linear algebra (including the spectral theorem for self-adjoint matrices).

Textbook: 
Gert K. Pedersen, Analysis Now

Recommended references:
Paul R. Halmos, A Hilbert Space Problem Book
Mikael Rørdam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras
Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

 

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MAT1016HF/MAT437H1F
TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS
G. A. Elliott
(View Timetable)

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

 Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

 Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, K-theory became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context.  (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classification of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications.

Prerequisites:
An attempt will be made to supply the necessary prerequisites when needed (rather few, beyond just elementary algebra and analysis).

Textbook:
Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras
 
Recommended References:
Edward G. Effros, Dimensions and C*-algebras
Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras
Kenneth R. Davidson, C*-Algebras by Example

MAT 1044HF
RANDOM MATRICES AND POTENTIAL THEORY
T. Bloom
(View Timetable)

We will develop the basics of potential theory and random matrix theory and the relation between them. The main goal will be to establish large deviation results for the empirical measure of the eigenvalues of Hermitian and other matrix ensembles.

Prerequisite:  Graduate level complex analysis or equivalent

Reference: T. Ransford, Potential Theory in the Complex Plane

 

MAT 1045HS
ERGODIC THEORY
K. Khanin
(View Timetable)

The course is an introduction to some of the basic notions and methods of the ergodic theory. It covers such notions as minimality, topological transitivity, ergodicity, unique ergodicity, weak mixing and
mixing. These notions will be explained by examining simple concrete examples of dynamical systems such as translations and automorphisms of tori, expanding maps of the interval, Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, and the Birkhoff ergodic theorem will be presented. We also plan to outline the thermodynamic formalism, entropy theory, and the theory of Lyapunov exponents.

Prerequisites: Knowledge of real analysis, basic topology, measure theory, and probability theory.

References:
I. Kornfeld, S. Fomin, Ya. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982.
Ya. Sinai, Topics in Ergodic Theory, Princeton Mathematical Series, 44. Princeton University Press, Princeton, NJ, 1994.

 

MAT1062HS
INTRODUCTORY NUMERICAL METHODS FOR PDE
M. Pugh
(View Timetable)

We will study numerical methods for solving partial differential equations that commonly arise in physics and engineering. We will pay special attention to how numerical methods should be designed in a way that respects the mathematical structure of the equation.

Parabolic PDE w/ finite difference methods
- explicit and implicit discretizations in 1-d consistency, stability, and convergence in 1-d boundary conditions in 1-d multi-dimensional problems

Elliptic PDE
- variational formulations and finite element methods

Hyperbolic PDE
- CFL stabilty condition
- nonlinear conservation laws, shock capturing

Special topics
- pseudospectral methods

Prerequisites: 
You should be familiar with the material that would be taught in a serious undergraduate PDE course. Sample programs will be provided in matlab. If you know matlab, great! If you don't, you're expected to be sufficiently comfortable with computers that you can learn matlab on the fly. Which isn't actually hard at all, unless you hate computers.

References:
My lecture notes from Winter 2008 and Winter 2009 and likely "Numerical Methods for Evolutionary Differential Equations" by Uri Ascher or " Numerical Methods for Scientists and Engineers" by Hamming.

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MAT 1063HS
INTRODUCTION TO GEOMETRIC FLOWS
I.M. Sigal
(View Timetable)

In this course we will study  mean curvature and  Ricci flows. If time permits, we will also describe the curvature flow of networks of plane curves. We will give careful definitions of these flows, present
existence results and results on formation of singularities (e.g. collapse to a point and neck-pinching) and soliton dynamics. We will also introduce main techniques, such as parabolic existence theory, maximum principles and monotonicity (entropy) formulae, Lyapunov functionals, probabilistic and other techniques. We will explain all needed notions from Differential Geometry and Partial Differential Equations, but knowledge of these subjects on an introductory level is required for this course.

Prerequisites:
Elementary ordinary and partial differential equations, Fourier analysis, elementary analysis and theory of functions and elementary differential geometry of surfaces.

Textbook:
Mantegazza, Carlo: Lecture Notes on Mean Curvature Flow, 2nd printing, Birkhauser, 2012

References:
K. Ecker, Regularity theory for mean curvature flow, Birkhaeuser, 2004, ISBN 08 176 32433
P. Topping, Lectures on the Ricci flow, London math society lecture notes series 325, Cambridge Univ Press,  2006; ISBN 0-521-68947-3 (available on internet)
Original papers

MAT1075HF
INTRODUCTION TO MICROLOCAL ANALYSIS
V. Ivrii
(View Timetable)

1. Pseudodifferential operators:
   • Symbols, Quantization, Calculus.
   • Oscillatory Front sets, coherent states, microlocalization.
   • L²-estimates, Garding inequalities.
2. Fourier Integral Operators:
   • Oscillatory solutions, relation to classical dynamics.
   • Lagrangian distributions, phase functions and amplitudes, Lagrangian manifolds.
   • Canonical graphs and Fourier Integral Operators.
   • Metaplectic operators.
   • Oscillatory solutions near and beyond caustics.
   • Maslov Canonical Operator.
3. Propagation of singularities.
   • Fourier Integral Operators Approach.
   • Heisenberg approach.
   • Coherent States approach.
   • Energy estimates approach.
4. Applications to Spectral Asymptotics.
   • Tauberian Approach.
   • Poisson Relations, Spectrum and Length Spectrum.
   • Sharp spectral asymptotics.

Prerequisites: musts are real analysis (something similar to MAT 1000HF and MAT 1001HS) and PDE (something similar MAT 1060HF, APM 351Y or APM 346H).  Assets are Advanced ODE and Analysis on manifolds (something like MAT 257Y).

Course material can be viewed at http://www.math.toronto.edu/ivrii/Grads/GCMA.php 

MAT 1110HS
LINEAR ALGEBRAIC GROUPS
F. Herzig
(View Timetable)

This course will be an introduction to the theory of linear algebraic groups over an algebraically closed field.  Algebraic groups are algebraic varieties equipped with a group structure such that the group operations are given by maps of algebraic varieties. An algebraic group is called linear if it can be embedded as a closed subgroup of a general linear group. This includes, for example, special linear, symplectic, and orthogonal groups. The subject has many parallels with the classical theory of (compact) Lie groups and Lie algebras.

Outline:
- varieties, affine algebraic groups, quotients
- Lie algebras
- diagonalisable, unipotent, and solvable algebraic groups
- Borel subgroups, parabolic subgroups
- Weyl group, Bruhat decomposition
- structure theory of reductive groups

Prerequisites:
basic algebraic geometry (affine and projective varieties over an algebraically closed field), graduate algebra

References:
Springer, Linear algebraic groups, Birkhauser
Borel, Linear algebraic groups, Springer GTM
Humphreys, Linear algebraic groups, Springer GTM

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MAT 1120HS
LIE GROUPS AND LIE ALGEBRAS
V. Kapovitch
(View Timetable)

Prerequisites:
basic group theory and basic differential topology (both at the undergraduate level)

Textbook:
Broecker and Dieck: Representations of Compact Lie Groups

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MAT 1126HS
LIE GROUPS AND HAMILTONIAN PDEs
B. Khesin
(View Timetable)

Prerequisites:
A basic course (or familiarity with main notions) of symplectic geometry would be helpful.

References:
B. Khesin and R. Wendt: “The geometry of infinite-dimensional groups,” Ergebnisse der Mathematik und Grenzgebiete 3.Folge, 51, Springer-Verlag, 2008, xviii+304pp,
see http://www.math.toronto.edu/khesin/papers/Lecture notes.pdf
A. Pressley and G. Segal: “Loop Groups,” Clarendon Press, Oxford (1986)

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MAT 1128HF
INTERACTING PARTICLE SYSTEMS AND STOCHASTIC PDEs
J. Quastel
(View Timetable)

This course will study some the most basic interacting particle systems such as stochastic Ising models and exclusion processes, building towards analysis of stochastic partial differential equations, such as the stochastic Fisher-KPP equation, stochastic heat equations and stochastic Burgers equation.

Prerequisite is a graduate course in probability (more advanced background material such as stochastic differential equations and Gibbs measures will be introduced as needed.)

No textbook.

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MAT 1128HS
THE SOUND OF SPARSE RANDOM GRAPHS
B. Virag
(View Timetable)

This is a course about eigenvalue of various models of sparse random graphs.  We will cover the basic theory of random Schrodinger operators (boxes of Z^d with randomly weighted loops added at each vertex), as well as Erdos-Renyi graph models and random regular graphs.

Grading.  Based on problems and student projects/presentations.

MAT 1190HS
SCHEMES
S. Kudla
(View Timetable)

This course will provide a basic introduction to the theory of schemes with an emphasis on the arithmetic examples.

Topics will include:

Basic denitions: Spec, affine schemes, schemes, morphisms, fiber products and base change, reduced and nonreduced schemes, the functor of points, Proj, separated, proper and projective morphisms, tangent spaces, regular schemes, flat, etale and smooth morphisms, coherent sheaves and their cohomology, Kaehler differentials, blowing up, examples, curves, surfaces and arithmetic surfaces, groups schemes.

Prerequisites:
One year of algebra at a graduate level, e.g. MAT 1100HF and MAT 1101HS. Some familiarity with algebraic number theory, basic algebraic geometry (e.g., the first chapter of Hartshorne) and sheaves will be assumed.

References:
Q. Liu, Algebraic geometry and arithmetic curves.
D. Eisensbud and J. Harris, The geometry of schemes.

Additional references:
R. Hartshorne, Algebraic geometry, Chapters II and III.
J. Silverman, Advanced topics in the arithmetic of elliptic curves.
D. Mumford, The red book of varieties and schemes. LN 1358.

 

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MAT 1194HS
ALGEBRAIC CURVES AND PLANE GEOMETRY
A. Khovanskii
(View Timetable)

1. Algebraic curve, Riemann surface of algebraic function, meromorphic functions and meromorphic 1-forms on algebraic curve.
2. Genus and Euler characteristic of algebraic curve, Riemann–Hurwitz formula, degree of divisor (f) and degree of divisor ( ω) of meromorphic function f and of meromorphic 1-form ω on an algebraic curve with given genus.
3. Generic algebraic curve P(x, y) = 0, (x, y) ∈ (C^*)² with fixed Newton polygon Δ. It’s Euler characteristic and genus. Space of holomorphic 1-forms. Pick’s formula (from an elementary geometry of integral convex polygons).
4. Weil reciprocity law.
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”’s 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 and odd degree.
12. Harnack inequality.
13. Estimate of the index of real polynomial vector field.
14. Petrovskii Theorem about real algebraic curve of degree six.

        Prerequisites: basic knowledge of complex analysis, no prerequisites in algebra or geometry.

        Suggested book:
        Philip A. Griffiths: Introduction to Algebraic  Curves

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MAT 1202HF (MAT417H1F)
ANALYTIC NUMBER THEORY
J. Friedlander
(View Timetable)

1. Introduction to some of the famous problems and theorems of the subject
2. Simple tools from elementary number theory, algebra and analysis
3. Dirichlet's theorem on primes in arithmetic progressions
4. Prime Number Theorem
5. Prime number theorem for arithmetic progressions
6. A selection, as time permits, of some subset of the following topics:
     a) further zeta-function theory
     b) L-functions and character sums
     c) exponential sums and uniform distribution
     d) Hardy-Littlewood-Ramanujan method
     e) sieve methods
     f) further theory of prime distribution

Prerequisites:
1. A half-year course in complex variables such as MAT 334 or MAT 354.
2. A course in groups, rings, fields, such as MAT 347.
3. A half year course course in introductory number theory such as MAT 315

Texts:
There is no formal text. The following books are useful references.

A) General Analytic Number Theory
     1. H. Davenport, Multiplicative number theory, 3rd ed. (revised by H.L. Montgomery) Graduate Texts in Mathematics, Vol. 74 Springer-Verlag 2000
     2. H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, Vol. 53. 2004
     3. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory I. Classical theory,Cambridge Studies in Advanced Mathematics, 97, Cambridge 2007.

B) More specialized texts
     4. J. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, Vol 57, 2010.
     5. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed, (revised by D. R. Heath-Brown) Clarendon Press, Oxford 1986.
     6. R. C. Vaughan, The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Mathematics,Vol. 125, Cambridge 1997

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MAT 1302HS (CSC 2413HS/APM 461H1S)
COMBINATORIAL THEORY
S. Tanny
(View Timetable)

We will cover topics in enumerative combinatorial theory, including: basic counting tools and direct counting methods; combinatorial proof techniques for many kinds of identities; binomial and multinomial coefficients; generating functions; and recursions, including nested recursions.

Prerequisites:
Some familiarity with the enumeration material usually covered in an introductory combinatorics course, such as MAT344, will be very useful. We will review this material as required, but at a fairly rapid pace. Students lacking this background should be prepared to put in the appropriate additional effort to familiarize themselves with it. Basic differential equations, linear algebra and some (very modest amount of) group theory will also be assumed.

 

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MAT 1312HS
QUASI-ISOMETRIC RIGIDITY
R. Young
(View Timetable)

One of the key ideas of geometric group theory is that of the quasi-isometry class of a group, which captures its large-scale geometry. Quasi-isometries are very flexible; for example, any two groups that act on the same space sufficiently nicely are quasi-isometric. Despite this flexibility, many properties are preserved by quasi-isometries, and this course will explore some of these results and the methods behind them.

This course will be arranged in a modular fashion, with different parts of the course being (mostly) independent. We will start with a quick overview of some of the basic notions, then proceed to more advanced material. The exact topics to be covered will depend on student interest, but some possible topics include:

* Quasi-isometries of nilpotent groups
* Coarse differentiation and the rigidity of Sol
* Gromov's polynomial growth theorem
* Quasi-isometries of symmetric spaces and buildings

Prerequisites: The core sequence. Some familiarity with riemannian metrics will be helpful.

References: Will vary according to modules.

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MAT 1314HS
INTRODUCTION TO NON-COMMUTATIVE GEOMETRY
G.A. Elliott
(View Timetable)

Some of the most basic objects of study in Connes's non-commutative geometry---for instance, the non-commutative tori---will be considered from an elementary point of view. In particular, various aspects of the structure and classification of these objects will be studied, and a comparison made between the properties of these objects and the properties of the underlying geometrical systems. Some indication will be given of their use in index theory.

Prerequisite: the spectral theorem

References:
A. Connes, Noncommutative Geometry, Academic Press, 1994.
J. Gracia-Bondia, J.C. Varilly, and H. Figueora, Elements of Noncommutative Geometry, Birkhauser, 2000.
Y. Kawahigashi and D.E. Evans, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998.
M. Rordam, F. Larsen, and N.J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000.
G.K. Pedersen, Analysis Now, Springer, 1989.

 

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MAT 1342HF (MAT464H1F)
DIFFERENTIAL GEOMETRY
R. Rotman
(View Timetable)

Topics: Riemannian metrics, Levi-Civita connection, geodesics, curvature, Gauss equations, convexity, Complete manifolds and Hopf-Rinow theorem, Jacobi fields, Rauch comparison and variations of energy.

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

References:
Riemannian geometry by Do Carmo.

 

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MAT 1350HF
HOMOLOGICAL ALGEBRA
P. Selick
(View Timetable)

Categories, functors and natural transformations; additive and abelian categories; adjoint functors; Tor, Ext and more general derived functors; algebraic limits; spectral sequences; classifying spaces.

References:
P.J. Hilton and U. Stammbach: A Course in Homological Algebra, Graduate Texts in Mathematics, Springer Verlag
Charles A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics
P. Selick: An Introduction to Homotopy Theory, Fields Institute Monograph Series

Prerequisite:  solid undergraduate mathematics program

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MAT 1360HS
CAUCHY RIEMANN MANIFOLDS
J. Bland
(View Timetable)

 

An introductory course to the study of Cauchy Riemann (CR) manifolds.

Cauchy Riemann manifolds arise naturally as the boundary of complex manifolds, and the geometry and analysis on the CR manifold is closely related to the geometry and analysis of the complex manifold which it bounds. An important class of examples arises in the study of holomorphic line bundles over compact Riemann surfaces, and Seifert - fibred three manifolds.

Alternatively, CR manifolds arise as contact manifolds with metrics on the hyperplane distribution.

This course will study the geometry and analysis on CR manifolds, relating it to that of the complex manifold which it bounds.

Prerequisites: A good background in differentiable manifolds including the de Rham complex of differential forms, Stoke's thereom, Frobenius integrability. A good background in complex analysis of one variable.

References:
Sorin Dragomir, Giuseppe Tomassini "Differential geometry and analysis on CR manifolds" Progress in Mathematics, Birkhauser.
Al Boggess: ``CR Manifolds and the Tangential Cauchy-Riemann Complex'', Studies in Advanced Mathematics, CRC Press 1991.

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MAT1404HF (MAT409H1F)
INTRODUCTION TO MODEL THEORY AND SET THEORY
S. Todorcevic
(View Timetable)

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

Prerequisite:
MAT357H1

Textbook:
Discovering Modern Set Theory, Vol I and II (by W.Just and M. Weese) AMS Graduate Studies in Mathematics, Vol. 8.

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MAT1448HS
TOPICS IN SET THEORETIC TOPOLOGY
W. Weiss
(View Timetable)

The course will focus upon applications of Set Theory to Topology, often called Set-theoretic Topology. We begin with a general study of cardinal functions. We will then study compact zero-dimensional spaces in detail. We will discuss independence results and the effect of non-traditional axioms. There will also be time to investigate areas suggested by the student participants.

Prerequisite: MAT409/MAT1404 or equivalent knowledge of axiomatic and combinatorial set theory.

Reference Texts:
1. K. Kunen & J. Vaughan, eds. Handbook of Set-theoretic Topology, North Holland.
2. J. van Mill & G. M. Reed, eds. Open Problems in Topology, North Holland
3. Elliott Pearl, ed.  Open Problems in Topology II, North Holland
For most students, these books are much too expensive to purchase. It will not be necessary to do so.

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MAT1450HF
FORCING
S. Todorcevic
(View Timetable)

This will be a course concentrated on developing the technique of Forcing to a level of sophistication  necessary for successful applications of this method. We shall show not only applications of this methods in providing independence results in set theory but also uses of this method in proving ZFC-results outright i.e, results that do not depend on additional axioms of set theory. While students will profit from texts listed below, we will be distributing set of notes that will closely follow our exposition.

Textbooks:
1. K. Kunen, Independence results in Set Theory, North Holland 1980.
2. T. Jech, Set Theory, Springer 2003.

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MAT1508HS (APM 446H1S)
APPLIED NON-LINEAR EQUATIONS
I. M. Sigal
(View Timetable)

In this course we will study partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. We will touch upon questions of existence, long-time behaviour, formation of singularities, and pattern formation. We will also address questions of existence of static, traveling wave, self-similar, topological and localized solutions and their stability.

Specifically we consider the Allen-Cahn equation (material science), harmonic map flow, Ginzburg-Landau equation (condensed matter physics - superfluidity and superconductivity), Cahn-Hilliard (material science, biology), Free boundary problem (material sciences - Stefan problem, Hele-Shaw flow), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), Gross-Pitaevskii equation (Bose-Einstein condensation), and Chern-Simmons equations (particle physics and quantum  Hall effect).

The course will be relatively self-contained, but familiarity with elementary ordinary and partial differential equations and Fourier analysis will be assumed.

Course Syllabus

• Allen-Cahn equation and mean-curvature flow
• Nonlinear Schroedinger and Gross-Piatevskii equations
• Keller-Segel equations of chemotaxis
• Ginzburg-Landau equations of superconductivity
• Chern-Simmons equations
• Hele-Shaw models of interface dynamics
• Elements of elasticity theory (dislocations and disinclinations, topological defects)
• DNA bubbles
• Models of cellular differentiation
• Traveling waves in neural networks

Prerequisites:
Elementary ordinary and partial differential equations, Fourier analysis, Elementary analysis and theory of functions or physics equivalent of these.

References:
Papers and internet sources;
K. Ecker, Regularity theory for mean curvature flow, Birkhaeuser, 2004, ISBN 08 176 32433

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MAT1700HS (APM426H1S)
GENERAL RELATIVITY
S. Alexakis
(View Timetable)

The course will provide a thorough overview of Einstein's  general theory of relativity, from a mathematical standpoint. There will be particular emphasis on the geometric content of the theory and its physical significance.

The course will start with an overview of the necessary geometric background, on the tensor calculus, metrics, curvature, geodesics, and the Einstein equations. We will next cover more global aspects of the theory, including the causal structure of space times and the geometry of special solutions (Minkowski and Schwarzschild). We end with the Penrose singularity theorems and (time permitting) the cosmic censorship conjectures and the geometry of black holes.

Prerequisites:
Some familiarity with basic differential or Riemannian geometry is desirable but not required. All students should be familiar with advanced multivariable calculus.

References:
We will mainly follow Robert Wald's "General relativity", but will also use J. L. Synge's "Relativity: the general theory".

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MAT1723HF (APM 421H1F)
MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS
D. Egli
(View Timetable)

Goals: The goal of this course is to give an introduction to the physics of quantum mechanics and to explain the key mathematical concepts that lie at the heart of it. Because the mathematics is rather deep we will arrive quickly at topics that are at the forefront of active research, such as Bose-Einstein condensation.

Mathematical rigour: We will try to as self-contained as possible and rigorous whenever rigour is instructive. When doing the whole proof is too time-consuming we will give at least the key ideas of the proof.

Syllabus :

• Schrödinger equation
• Quantum observables
• Spectrum and evolution
• Perturbation theory
• Motion in electro-magnetic field
• The second quantization
• Density matrices
• Open Systems

Prerequisites:
For this course it is desirable to have some familiarity with elementary functional and complex analysis. Some knowledge of ODEs and PDEs would be helpful.

Reference:
S. Gustafson and I. M. Sigal: Mathematical Concepts of Quantum Mechanics, 2^nd edition, Springer, 2005.

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MAT1856HS (APM 466H1S)
MATHEMATICAL THEORY OF FINANCE
L. Seco
(View Timetable)

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.

Prerequisites:
APM 346H1, STA 347H1

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      MAT1901HS
      ALGEBRAIC AND GEOMETRIC THEORY OF QUADRATIC   
      FORMS
      N. Karpenko
      (View Timetable)

Following [1, Part 1], we develop the basics of the theory of quadratic forms over arbitrary fields.  In the second half of the course we briefly introduce the Chow groups and then apply them to get some of more advanced results of [1, Part 3].

Here is the program in more details:

1. Bilinear forms.
2. Quadratic forms.
3. Forms over rational function fields.
4. Function fields of quadrics.
5. Forms and algebraic extensions.
6. u-invariants.
7. Applications of the Milnor conjecture.
8. Chow groups.
9. Cycles on powers of quadrics.
10. Izhoboldin dimension.

References:
1. R. Elman, N. Karpenko, A. Merkurjev.
The Algebraic and Geometric Theory of Quadratic Forms.
American Mathematical Society Colloquium Publications.
56. American Mathematical Society, Providence, RI, 2008. 435 pp.

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STA2047HS
STOCHASTIC CALCULUS
J. Quastel
(View Timetable)

Brownian motion, stochastic integrals, stochastic differential equations, diffusions, Cameron-Martin-Girsanov formula, diffusion approximations, applications.  The course willl be mathematically rigorous and self-contained.

Prerequisites:
No explicit prerequisites, but to understand ther material, it is necessary to have a good understanding at the advanced undergrduate level of at least one of the following: Probability, Real Analysis, Differential Equations, Mathematical Finance.

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STA2111HF
GRADUATE PROBABILITY THEORY I
D. Brenner
(View Timetable)

STA 2111H is a course designed for Master's and Ph.D. level students in statistics, mathematics, and other departments, who are interested in a rigorous, mathematical treatment of probability theory using measure theory. Specific topics to be covered include: probability measures, the extension theorem, random variables, distributions, expectations, laws of large numbers, Markov chains.

Prerequisites:
Students should have a strong undergraduate background in Real Analysis, including calculus, sequences and series, elementary set theory, and epsilon-delta proofs. Some previous exposure to undergraduate-level probability theory is also recommended.

 

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STA2211HS
GRADUATE PROBABILITY THEORY II
B. Virag
(View Timetable)

STA 2211H is a follow-up course to STA 2111F, designed for Master's and Ph.D. level students in statistics, mathematics, and other departments, who are interested in a rigorous, mathematical treatment of probability theory using measure theory. Specific topics to be covered include: weak convergence, characteristic functions, central limit theorems, the Radon-Nykodym Theorem, Lebesgue Decomposition, conditional probability and expectation, martingales, and Kolmogorov's Existence Theorem. 

 

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Students requiring individual instruction in mathematical topics should consult with the Mathematics Graduate Office.