Department of Mathematics

TENTATIVE 2011-2012 Graduate Course 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
S. Alexakis
(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: 
Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces
Lieb and Loss, Analysis 2nd edition, Graduate studies in Mathematics, AMS

References:
Katznelson, Harmonic Analysis, published by Dover
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.
S.D. Promislow, A First Course in Functional Analysis, Wiley, 2008.

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.

Main References:
Stein and Shakarchi: Complex Analysis
L. Ahlfors: Complex Analysis, 3rd Edition
T. Gamelin, Complex Analysis
W. Rudin, Real and Complex Analysis, 2nd or 3rd edition

Additional References:
Remmert: Theory of Complex Functions
Remmert: Classical Topics in Complex Function Theory
Needham: Visual Complex Analysis

MAT 1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
M. Pugh
(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
V. Ivrii
(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.

More information can be found at

http://weyl.math.toronto.edu:8888/MAT1061-2012S-wiki/

Reference: 
Lawrence Evans: Partial Differential Equations

MAT 1100HF
ALGEBRA I
D. Bar-Natan
(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

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. Kamnitzer
(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
Y. Karshon
(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
R. Rotman
(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

2011-2012 GRADUATE COURSES (CROSS-LISTED AND TOPICS)

JEB 1433HS
MATHEMATICS OF MEDICAL IMAGING
A. Nachman
(View Timetable)

Topics to be covered: 

1. Magnetic Resonance Imaging: the Bloch Equation.
2. Introduction to Compressed Sensing.
3. Inverse boundary value problems: Electric Impedance Tomography.
4. Inverse problems with interior data: Current Density Impedance Imaging.
5. Inverse scattering methods: progress on quantitative Ultrasound imaging.
6. Open problems.

 

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MAT 1007HS
TOPICS IN COMPLEX VARIABLES: INTRODUCTION TO
HARMONIC ANALYSIS AND APPLICATIONS
M. Goldstein
(View Timetable)

Objective: The course is designed for students interested in analysis, partial differential equations and mathematical physics. The objective of this course is to discuss some basic collection of tools from modern harmonic analysis and their applications. The emphasis will be put on the description of the ideas and techniques involved. We will sketch the proofs explaining the principal details and omitting the small ones. That will allow to cover a fairly large variety of topics including for instance applications of harmonic analysis to the analytic number theory.

Texts:
"Harmonic Analysis : Real Variables Methods" by E.Stein, Princeton University;
"Introduction to Hp spaces", by P. Koosis, Cambridge University Press, 1998;
R. Vaughan, "The Hardy - Littlewood method", Cambridge University Press, 1981. Press 1993.

Lecture Notes:
Detailed handwritten notes of the lectures will be posted weekly on the web.

Syllabus: The following material to be covered in this course:

1. Fourier series and Fourier integrals
2. Oscillatory integrals
3. Fourier transform of measure supported on hyper-surface of non-vanishing Gaussian curvature
4. Applications: asymptotic formula for number of lattice points in ellipsoid with a review of the recent complete solution of the problem in dimension $\ge$ 4.
5. Poisson formula representation of functions harmonic in the unit disk. Non-tangential limits and Fatou theorem
6. Subharmonic functions, F. Riesz representation theorem for subharmonic functions, Cartan estimate for analytic and subharmonic functions
7. F. and M. Riesz theorem
8. Harmonic conjugate. Hilbert transform and M. Riesz theorem
9. Blaschke product and other applications of Jensen formula
10. Calderon-Zygmund theory of singular integrals

 

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MAT 1011HF (MAT495H1F)
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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MAT1016HS
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

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MAT1102HS
TOPICS IN THE THEORY OF GROUPS:  GEOMETRIC GROUP THEORY
R. Young
(View Timetable)

This course aims to be an introduction to some important topics in geometric group theory.  Geometric group theory is an extremely broad field; in this course, we will lean toward the study of negatively and non-positively curved groups, but with various detours along the way.

Outline:
Basics

• quasi-isometries
• volume growth, number of ends, Stallings's Theorem Hyperbolic groups
• geometry of hyperbolic space
• $\delta$-hyperbolicity and its many equivalent definitions Algorithmic questions in geometric group theory
• Dehn's algorithm and the Dehn function
• small cancellation and random groups Non-positively curved groups
• CAT(0)-spaces and the link condition for cube complexes
• subgroups of non-positively curved groups and Brady-Bridson Morse theory
  

Prerequisites: 
core topology course and first half of algebra core course.  Some differential geometry (may be taken at same time) would be helpful.

References:
M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature.
É. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov. (English translation available online)

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MAT 1103HF
TOPICS IN ALGEBRA I: SYMMETRIES I - FINITE GROUPS
R.-O. Buchweitz
(View Timetable)

The aim of the course will be to give a modern treatment of the classification of finite subgroups of various linear groups such as SO(3), O(3), SU(2), GL(2,C), their projective counterparts, and of some related arithmetic groups such as GL(2,Z), SL(2,Z). Especially the classification of the finite subgroups of GL(2,C) is surprisingly difficult to find in the literature, but recent developments in Singularity Theory, such as the McKay correspondence, allow for a very succinct and "visual" classification.

Mathematical tools to be developed: Some Euclidean and spherical geometry, Root systems and reflection groups, Linear representation theory of groups, Geometry and algebra of quaternions, The McKay Correspondence and Coxeter-Dynkin diagrams, Goursat's Lemma for subgroups of products.

Prerequisites:
A good undergraduate algebra course or, concurrently, the graduate core course in algebra.

References:
There is no comprehensive textbook covering the material of this course. Useful sources for the content matter:

---A good textbook for the basic material, especially algebraic treatment of Euclidean geometry and finite symmetries, is:
Artin, M.: Algebra (2nd Edition).
Addison Wesley 2010. 560 pp.
ISBN: 978-0132413770

--- Classification of finite subgroups of SO(3), SU(2) and arithmetic ofquaternions:
Conway, J.H. and Smith, D.A.: On Quaternions and Octonions:
Their Geometry, Arithmetic and Symmetry.
A.K.Peters, Natick, Massachussetts, 2003. 159 pp.
ISBN 1-56881-134-9

--- An excellent source book for complex numbers, quaternions, and beyond:H.-D. Ebbinghaus et al.: Numbers
Graduate Texts in Mathematics 123
Corrected 3. printing (or later)
Springer-Verlag New York Inc.1995 418 pp.
ISBN 0-387-97497-0

--- Geometry and finite symmetry groups, especially those under
consideration here:
Coxeter, H.M.S.: Regular Polytopes,
Methuen & Co. Ltd, London,
First published in 1948,
xx + 321 pp.

and

Coxeter, H.M.S.: Regular Complex Polytopes,
Cambridge University Press, Cambridge,
First published in 1971,
xiv + 210 pp.

The course material is also covered on various web sites that we will discuss during the course.

 

MAT1104HS
TOPICS IN ALGEBRA II: THE MOD p REPRESENTATION
THEORY OF p-ADIC  GROUPS
F. Herzig
(View Timetable)

This course will give an introduction to the smooth representation theory of a p-adic reductive group (like $GL_n(\BQ_p)$) over $\overline\BF_p$. The study was begun by Barthel-Livné for the group  $GL_2(F)$ where F is a finite extension of $\BQ_p$. Breuil completed the classification of irreducible representations of $GL_2(\BQ_p)$ (having a central character), and related these representations to two-dimensional representations of $Gal(\overline\BQ_p / \BQ_p)$ over $\overline\BF_p$. This was one of the starting points for the investigation of the p-adic Langlands correspondence for $GL_2(\BQ_p)$. At this point it is not clear how the p-adic Langlands correspondence generalises to other groups. It is one of the goals of the course to explain some of the recent progress in the classification of irreducible $\overline\BF_p$-representations for other groups, particularly for $GL_n(F)$.

Some of the topics to be covered (time permitting):

• the work of Barthel-Livné for $GL_2(F)$,
• irreducible $\overline\BF_p$-representations of $GL_n(\BF_p)$,
• parabolic induction and compact induction,
• unramified Hecke algebras and the Satake transform,
• generalised Steinberg representations,
• ordinary parts,
• classification of irreducible representations in terms of supercuspidal ones.

Prerequisites:
A good command of graduate algebra, including solid knowledge of the representation theory of finite groups over the complex numbers. Some familiarity with p-adic numbers.

References:
L. Barthel, R. Livne, Irreducible modular representations of $GL_2(F)$ of a local field, Duke Math. J. 75, 1994, 261-292.
C. Breuil, Sur quelques repr´esentations modulaires et p-adiques de $GL_2(\BQ_p)$  I, Compos. Math. 138, 2003, 165-188.
M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque 331 (2010), 335-381.
F. Herzig, A Satake isomorphism in characteristic p, Compos. Math. 147 (2011), no. 1, 263-283.
F. Herzig, The classification of irreducible admissible mod p representations of a p-adic $GL_n$, preprint.

MAT 1191HS
TOPICS IN ALGEBRAIC GEOMETRY: INTRODUCTION TO D-MODULES
S. Arkhipov
(View Timetable)

Our goal is to develop the techniques of algebraic D-modules to be used in the framework of geometric representation theory. We plan to cover the following topics:

1. The ring of algebraic differential operators on a smooth affine scheme. D-modules:  Left and right D-modules. Examples: The case of an affine space and calculations in coordinates. The sheaf of differential operators on a smooth algebraic variety. Standard filtration by degree. Quasiclassical limit: differential operators and functions on the cotangent bundle.
2. Sheaves of D-modules. Coherent D-modules. Inner Hom and tensor product. Operations on D-modules: direct and inverse image. The cases of a closed embedding, an open embedding and of a smooth surjective map. Kashiwara theorem. Push-forward to a point.
3. Good filtrations on D-modules. Characteristic variety. Bernstein inequality. Holonomic D-modules. Verdier duality.
4. Preservation of holonomicity under operations on D-modules. Regular holonomic D-modules on curves. Regular holonomic D-modules in general.  Preservation of regularity under operations on D-modules.
5. Solutions of a D-module. Constructible sheaves. Perverse sheaves. Riemann-Hilbert correspondence.
6. Applications of the theory. D-modules on the flag variety for a simple algebraic group. Beilinson-Bernstein proof of Kazhdan-Lusztig conjectures.
7. D-modules approach to nearby cicles, vanishing cycles.

References:
A. Borel. Algebraic D-modules. Perspectives in Mathematics, 2. Academic Press, Inc., Boston, MA,
1987. xii+355 pp

J. Bernstein. Lectures on D-modules. Available at:
http://www.math.uchicago.edu/~mitya/langlands.html

Prerequisites:
Basic graduate algebra course, the language of modern algebraic geometry:
schemes, quasicoherent sheaves etc, basic homological algebra and sheaf theory.

 

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MAT 1194HS
ALGEBRAIC CURVES: GALOIS THEORY AND RIEMANN SURFACES
A. Khovanskii
(View Timetable)

Numerous unsuccessful attempts to solve various algebraic and differential equations ''in explicit form" led mathematicians to believe that explicit solutions for these equations simply do not exist. Galois theory explains why an algebraic equation is usually not solvable by means of radicals. It is very understandable and has a lot of other applications.  Liouville theory explains why the integral of an elementary function usually is not an elementary function. Liouville theory is also very elegant and understandable.

According to my Topological Galois Theory for functions in one complex variable, there are topological restrictions on the way the Riemann surface of a function representable by radicals  or by quadratures covers the complex plane. If the function does not satisfy these restrictions, then it is not representable by radicals or byquadratures. Besides their geometric clarity the topological results on nonsolvability  are stronger than the algebraic results.

In the course I plan to present Liouville Theory, Galois Theory, and Topological Galois Theory. I also plan to include some very new results, found by my graduate student Yuri Burda.

Prerequisites:
A basic knowledge of complex analysis  and  algebra.

Possible references:
A.Khovanskii "On solvability and unsolvability of equations in explicit form",  Russian Math. Surveys 59, no. 4 (2004), 661-736.

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MAT 1195HF
MATHEMATICAL ASPECTS OF CRYPTOGRAPHY
R. Venkatesan
(View Timetable)

We will study a number of papers related to design, algorithms and security analysis of cryptographic primitives based on hard problems in number theory, elliptic curves, and other domains such as codes and lattices.  Dixons algorithm, Number field sieve, Pollard Rho, Bit security of some primitives.  Attacks on Knapsacks and RSA variants, Authentication protocols and use of Zero-Knowledge primitives, Schemes for cloud scenarios.   Brief look at complexity issues and the construction of hash functions, MACS, and Ciphers, and attacks on them.

        Prerequisites: Students should have some introduction to number theory, and elliptic curves.

        Useful references:
        http://www.amazon.com/Introduction-Modern-Cryptography-Principles-Protocols/dp/1584885513/ref=sr_1_1?ie=UTF8&qid=1314023370&sr=8-1
        http://www.amazon.com/Elliptic-Curves-Cryptography-Mathematics-Applications/dp/1420071467/ref=sr_1_3?s=books&ie=UTF8&qid=1314023661&sr=1-3
         http://www.amazon.com/Introduction-Cryptography-Discrete-Mathematics-Applications/dp/1584886188/ref=sr_1_3?s=books&ie=UTF8&qid=1314023761&sr=1-3#_

 

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MAT 1198HS
AUTOMORPHIC FORMS AND REPRESENTATION THEORY II: AUTOMORPHIC REPRESENTATIONS AND THE COMPARISON OF TRACE FORMULAS
J. Arthur
(View Timetable)

The trace formula is a powerful identity, which applies to the automorphic representations of a reductive group G over a number field F. For example, G could be one of the split groups GL(N), SO(2n+1), Sp(2n), or SO(2n) that represent the four infinite families of type A,B,C or D, from the classification of complex simple Lie algebras. The power comes in the comparison of trace formulas on different groups.

This will be a course on how the comparison works. We shall see that it leads to reciprocity laws for the fundamental arithmetic information hidden in automorphic representations. In particular, we shall see how the comparison can lead to a classification of automorphic representations of orthogonal and symplectic groups. I will try to tailor the level of the course to the background of those who take it.

References:
J. Arthur, Eisenstein series and the trace formula, in Automorphic Forms,
Representations and L-functions, Proc. Sympos. Pure Math. 33 (1979), Part 1,
Amer. Math. Soc., 253-274.

R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. Journal
51 (1984), 611-650 (especially the last three sections).

J. Arthur, The Endoscopic Classification of Representations: Orthogonal and
Symplectic Groups, to appear.

MAT 1200HF (MAT415H1F)
TOPICS IN ALGEBRAIC NUMBER THEORY
S. Kudla
(View Timetable)

This course will cover the basic theory of algebraic numbers.

Part I: Algebraic number fields: basic theory

• review of some commutative algebra
• localization, integrality
• Dedekind rings, fractional ideals, the ideal class group
• extensions, rings of integers, norms and traces,
• splitting of primes (1)
• discrete valuation rings, completions, $\BZ_p$, $\BQ_p$
• splitting of primes (2), ramified and unramified extensions
• the trace form, different and discriminant
• Galois extensions, Frobenius elements
• examples:
  imaginary quadratic fields, reduced quadratic forms and the class number
  real quadratic fields, Pell's equation and the fundamental unit
  cyclotomic fields

Part II: Algebraic number fields: global theory

• the product formula
• units and S-units, the regulator map
• finiteness of the class number
• Minkowski's unit theorem
• ideles and adeles

Part III: Some more advanced topics will be discussed as time allows.

• zeta functions
• L-functions
• class number formulas
• Artin L-functions
• class field theory, reciprocity laws
• non-abelian class field theory

References:
S. Lang, Algebraic Number Theory.
D. Marcus, Number Fields.
Z. I. Borevich and I. R. Shafarevich, Number Theory.

Prerequisites:
A solid knowledge abstract algebra (e.g., Dummit and Foote, MAT347, MAT1100-1101) is essential. For the later part of the course, notions from topology and the theory of analytic functions will be used.

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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
TOPICS IN GEOMETRY: HAMILTONIAN GROUP ACTIONS
L. Jeffrey
(View Timetable)

A symplectic manifold (a manifold equipped with a nondegenerate closed two-form) is the natural mathematical generalization of the phase space considered in classical mechanics. Hamiltonian group actions are a special case of Hamiltonian flows, which are a natural generalization of Hamilton's equations. Coadjoint orbits are natural examples of symplectic manifolds equipped with Hamiltonian group actions.

The course treats the following topics.

• Moment maps; symplectic quotients
• The symplectic structure on coadjoint orbits
• The Atiyah-Guillemin-Sternberg convexity theorem
• Delzant's theorem and introduction to toric geometry from
• the symplectic point of view
• Geometric quantization; applications to representation theory (survey)
• Equivariant cohomology and applications to symplectic geometry:
  (a) the localization theorem of Berline-Vergne, the Duistermaat-Heckman theorem;
  (b) Recent results on cohomology rings of symplectci quotients, obtained using localization (survey)
• An infinite dimensional symplectic quotient: the moduli space of flat connections on a Riemann surface (following Atiyah-Bott 1982 and Goldman 1984)

Suggested Prerequisites:
None in particular, but MAT 1300HF as a prerequisite or co-requisite may be helpful.  Students unsure of their suitability can contact the instructor.

References:

1. M. Audin, Torus actions on symplectic manifolds. Second revised edition. Progress in Mathematics, 93. Birkhauser Verlag, Basel, 2004. viii+325 pp. ISBN: 3-7643-2176-8
2. N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag (Grundlehren v. 298) (1992), chap. 7.
3. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press (1984).
4. V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-spaces, Birkhauser, 1994.
5. V. Guillemin, S. Sternberg, Supersymmetry and equivariant de Rham theory
6. V. Guillemin, E. Lerman,  S. Sternberg, Symplectic fibrations and multiplicity diagrams

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MAT 1313HS
SEMINAR IN GEOMETRY: RIEMANN SURFACES
M. Gualtieri
(View Timetable)

This course provides an overview of the main properties of Riemann Surfaces, including the classification of Riemann surfaces themselves as well as the study of vector bundles over Riemann surfaces.  Topics will include the theory of divisors and the Jacobian, Abelian varieties, the Riemann-Roch formula, the Birkhoff-Grothendieck classification, the moduli space of stable bundles, Abel's theorem, embeddings of curves, and applications to integrable systems. 

Prerequisites:
A course on complex analysis and the first term of the core course in geometry/topology, i.e. MAT1300.

There is NO OFFICIAL TEXTBOOK, though I would recommend Donaldson's notes (freely available) as well as Gunning's classic texts from Princeton University Press.

 

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MAT 1342HF (MAT464H1F)
DIFFERENTIAL GEOMETRY
V. Kapovitch
(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 1347HF
INTEGRABLE SYSTEMS
B. Khesin
(View Timetable)

Syllabus:

1. Hamiltonian systems, first integrals, the Arnold-Liouville theorem, tori.
2. Examples: Newton equation, billiards, the geodesic flow on an ellipsoid
3. Lie algebras, the Lie-Poisson bracket; examples: a rigid body, the Toda lattice
4. Bihamiltonian systems, the Lenard-Magri scheme, the Lax formulation; example: the Korteweg-de Vries (KdV) equation
5. Introduction to the inverse scattering; example: KdV, nonlinear Schroedinger (NLS)
6. Introduction to solitons
7. Near-integrability and non-integrability: a glimpse of the KAM theory,
8. Topics and generalizations:

• the Kadomtsev-Petviashvili (KP) hierarchy on PDE's
• spectral parameter and spectral curve
• pentagram map
• contact integrability and null geodesics

 

Prerequisite:
Some familiarity with the main notions of classical mechanics or symplectic geometry will be useful, but not required 

References:
Palais, R.S. "The symmetries of solitons." Bull. Am. Math. Soc., New Ser. 34, No.4, 339-403 (1997).
Segal, G. in the book  "Integrable Systems."  Oxford Sci. Publ. 1999
 Moser, J. "Various aspects of integrable Hamiltonian systems."  Progr. Math. 8, Birkhauser, Boston, Mass.,1980, pp. 233-289.

 

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MAT1350HF
TOPICS IN ALGEBRAIC TOPOLOGY I:  RATIONAL HOMOTOPY THEORY
M. Amann / V. Kapovitch
(View Timetable)

We shall provide an introduction to Rational Homotopy Theory---a comparatively easily computable version of homotopy theory, which is suited for geometric applications. Using the theory of minimal Sullivan models we shall review the categorical translation of the rational homotopy type of a (simply-connected) topological space to the realm of commutative differential graded algebras. In particular, this will enable us to compute the rationalized homotopy groups of a large variety of spaces.

In the second part of the course we shall focus on prominent topics like formality, ellipticity or group actions with a view towards current research. Whenever possible we intend to discuss geometric applications, as to homogeneous spaces or Kahler manifolds, for example.

A certain familiarity with basic concepts from algebraic topology like (co)homology theory, fundamental groups/homotopy groups, CW-complexes, fibrations etc. is required.

We shall mainly build upon the textbooks [1] and [2].

References:
Y. Felix, S. Halperin, and J.-C. Thomas. Rational homotopy theory, volume 205 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001.
Y. Felix, J. Oprea, and D. Tanre. Algebraic models in geometry, volume 17 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2008

 

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MAT 1360HF
COMPLEX MANIFOLDS
J. Bland
(View Timetable)

An introduction to complex manifolds: vector bundles, complex line bundles, Hermitian connections, curvature, Kahler metrics, Kodaira embedding theorem, Hodge theory. 

Textbook:
Kodaira: ''Complex manifolds and Deformation of complex structures''
Griffiths and Harris: ''Principles of Algebraic Geometry''.

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 in one variable.

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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.

MAT1430HS
SET THEORY: COMBINATORIAL SET THEORY
S. Todorcevic
(View Timetable)

The course will concentrate on topics from combinatorial set theory. This will include partition calculus on ordered sets, combinatorial analysis of large cardinals, and some applications to other areas of mathematics.

Prerequisite:
MAT409H

References:
Corresponding articles in Handbook of Set Theory, Springer 2010
P. Erdos, A. Hajnal, A. Mate and R. Rado, Combinatorial set theory, North Holland 1984.
S. Todorcevic, Introduction to Ramsey spaces, Princeton Univ. Press, 2010.

 

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MAT1499HS
TEACHING LARGE MATHEMATICS CLASSES
J. Repka
(View Timetable)

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.
This course is not for degree credit.

 

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MAT1502HF
SCHRAMM-LOEWNER EVOLUTIONS AND LATTICE MODELS
I. Binder
(View Timetable)

Invention of Schramm-Loewner evolution brought up a significant progress in both from physical and mathematical understanding of two-dimensional lattice models of Statistical Mechanics. In the course, we will discuss the recent progress in the rigorous understanding of Universality and Conformal Invariance phenomena for the  lattice models. The topics will include:

1. Percolation, Ising, Potts, and FK-cluster models.
2. Schramm-Loewner evolution: definition and geometric properties.
3. Convergence of the interface to Schramm-Loewner evolution for percolation and Ising models.

References:
Gregory F. Lawler, Conformally Invariant Processes in the Plane Geoffrey Grimmett, Percolation.
Rodney J. Baxter, Exactly Solved Models in Statistical Mechanics

Prerequisite:
Core graduate courses in Complex Analysis and Probability; or instructor's permission.

 

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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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MAT1711HS
TOPICS IN QUANTUM MECHANICS: MATHEMATICS APPLICABLE TO QUANTUM INFORMATION
M.-D. Choi
(View Timetable)

We will look into the down-to-earth structure of the non-commutative analysis as needed in recent development of quantum information, in connection with mysterious quantum computers.  This will cover basic aspects of quantum channels, entanglements, entropies,  error corrections and other  topics.    We will also investigate in detail the unexplored mathematical structure of numerical ranges, and tensor products, so as to realize the new meanings of the old values, as well as to seek the new values of the old meanings.

Students taking this course for credit are required to hand in a written report on a related topic based on the student's background and interest. No background knowledge of physics or computer science is required in this course.

Related Information:  Most of my earliest research work (of 70's) has been used intensively in the recent development of quantum information theory.  In particular, my paper  "Completely positive linear maps on complex matrices", (Linear Algebra Appl. 10 (1975), pp. 285-290) has become the pioneering research work in the field.  It has been cited in more than 500 research papers (as shown in Google Scholars 2011 April).

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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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MAT1847HS
HOLOMORPHIC DYNAMICS
M. Yampolsky
(View Timetable)

The course will serve as an introduction to the beautiful field of iteration of analytic maps of one complex variable. We will cover such topics as local properties of periodic orbits of analytic maps; the structure of the Julia set and the Fatou set of a rational map; the dynamics of polynomials, and the Mandelbrot set. 

Prerequisite:
A familiarity with Complex Analysis will be essential for understanding the material of the course; otherwise, the exposition will be self-contained.

Reference:
Much of the course will rely on the textbook "Dynamics in One Complex Variable" by J. Milnor.

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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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STA2111HF
GRADUATE PROBABILITY THEORY I
J. Rosenthal
(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
J. Rosenthal
(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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STA4247HS
POINT PROCESSES, NOISE AND STOCHASTIC ANALYSIS
B. Virag
(View Timetable)

Introduction to the theory of point processes - Poisson and compound processes, point processes with repulsion and attraction. Brownian motion, white noise. Stochastic integration and stochastic differential equations. 

Topics:

• Poisson processes
• Determinantal and permanental processes
• Random analytic functions and their zeros
• Brownian motion, construction and path properties
• White noise and other noises
• Skorokhod's embedding
• Blackwell's proof of the CLT and Donsker's theorem
• Kolmogorov-Chentsov theorem
• Law of iterated logarithm
• Levy's modulus of continuity
• Stochastic integrals (first with no theory), Ito's formula, change of variables
• L2 theory of stochastic integration
• Cameron-Martin formula

 

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