Core Graduate Courses  CrossListed Courses  Topics Courses  Timetable
CORE COURSES
MAT1000HF/MAT457H1F
REAL ANALYSIS I
Measure Theory: Lebesgue measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, RadonNikodym theorem.
Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^pspaces, Holder and Minkowski inequalities.
Textbook:
Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley 2nd edition, 1999
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.
MAT1001HS/MAT458H1S
REAL ANALYSIS II
I. UriarteTuero
Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, topological vector spaces, Schwartz space, distributions.
Functional Analysis: The main topic here will be the spectral theorem for bounded selfadjoint operators, possibly together with its extensions to unbounded and differential operators.
Textbook:
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.
Reference:
E. Lorch, Spectral Theory.
W. Rudin, Functional Analysis, Second Edition, Indian Edition (if available; the book is hard to get, although there is a pdf on line).
MAT1002HS/MAT454H1S
COMPLEX ANALYSIS
E. Bierstone
Review of holomorphic and harmonic functions (Chapters 14 in Ahlfors).
 Topology of a space of holomorphic functions: Series and infinite products, Weierstrass pfunction, Weierstrass and MittagLeffler theorems.
 Normal families: Normal families and equicontinuity, theorems of Montel and Picard.
 Conformal mappings: Riemann mapping theorem, SchwarzChristoffel formula.
 Riemann surfaces: Riemann surface associated with an elliptic curve, inversion of an elliptic integral, Abel’s theorem.
 Further topics possible; e.g., analytic continuation, monodromy theorem.
Recommended prerequisites:
Undergraduate courses in real and complex analysis.
Textbook:
L. Ahlfors, Compex Analysis, third edition, McGrawHill
Recommended references:
H. Cartan, Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover
D. Marshall, Complex Analysis, Cambridge Math. Textbooks
M.F. Taylor, Introduction to Complex Analysis, American Math. Soc., Graduate Studies in Math. 202
MAT1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
F. Pusateri
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.
Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN13 9780821849743
References:
 R. McOwen, Partial Differential Equations, (2nd ed),
Hardcover: 2003 Prentice Hall ISBN 0130093351,
Paperback: 2002 Pearson ISBN13 9780130093356  Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 9781461448082
MAT1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
C. Sulem
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 finitetime blowup of solutions and/or longtime 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.
Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN13 9780821849743
Reference:
 R. McOwen, Partial Differential Equations, (2nd ed),
Hardcover: 2003 Prentice Hall ISBN 0130093351,
Paperback: 2002 Pearson ISBN13 9780130093356  Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 9781461448082
MAT1100HF
ALGEBRA I
I. Varma
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, JordanHölder 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.
MAT1101HS
ALGEBRA II
F. Herzig
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.
Recommended textbooks:
Grillet: Abstract Algebra (2nd ed.)
Dummit and Foote: Abstract Algebra, 3rd Edition
Jacobson: Basic Algebra, Volumes I and II.
Lang: Algebra 3rd Edition
MAT1300HF
DIFFERENTIAL TOPOLOGY
M. Gualtieri
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:
Differential Topology, Victor Guillemin and Alan Pollack,
American Mathematical Society ISBN10: 0821851934, ISBN13: 9780821851937
MAT1301HS
ALGEBRAIC TOPOLOGY
R. Rotman
Fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, VanKampen'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 BorsukUlam, cohomology and deRham's theorem, products.
Textbook:
Allen Hatcher, Algebraic Topology
Recommended Textbooks:
Munkres, Topology
Munkres, Algebraic Topology
MAT1600HF
MATHEMATICAL PROBABILITY I
K. Khanin
The class will cover classical limit theorems for sums of independent random variables, such as the Law of Large Numbers and Central Limit Theorem, conditional distributions and martingales, metrics on probability measures.
References:
Lecture notes and a list of recommended books will be provided.
Recommended prerequisite:
Real Analysis I.
Textbook:
Durrett's "Probability: Theory and Examples", 4th edition
MAT1601HS
MATHEMATICAL PROBABILITY II
B. Landon
The class will cover some of the following topics: Brownian motion and examples of functional central limit theorems, Gaussian processes, Poisson processes, Markov chains, exchangeability.
References:
A list of recommended books will be provided.
Recommended prerequisites:
Real Analysis I and Probability I.
Textbook:
Durrett's "Probability: Theory and Examples", 4th edition
MAT1850HF
LINEAR ALGEBRA AND OPTIMIZATION
M. Pugh
This course will develop advanced methods in linear algebra and introduce the theory of optimization. On the linear algebra side, we will study important matrix factorizations (e.g. LU, QR, SVD), matrix approximations (both deterministic and randomized), convergence of iterative methods, and spectral theorems. On the optimization side, we will introduce the finite element method, linear programming, gradient methods, and basic convex optimization. The course will be focused on fundamental theory, but appropriate illustrative applications may be chosen by the instructor.
CROSSLISTED
MAT1011HF/MAT436H1F
INTRODUCTION TO LINEAR OPERATORS
G. Elliott
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 noncommutative 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 selfadjoint matrices).
Textbook:
Gert K. Pedersen, Analysis Now
References:
Paul R. Halmos, A Hilbert Space Problem Book
Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to KTheory for C*Algebras
Bruce E. Blackadar, Operator Algebras: Theory of C*Algebras and von Neumann Algebras
MAT1016HS/MAT437H1S
TOPICS IN OPERATOR ALGEBRAS: KTHEORY AND C*ALGEBRAS
G. Elliott
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 selfadjoint 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 Murrayvon 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 Kgroup (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 Kgroup. Also, Atiyah and Singer famously showed that Ktheory was important in connection with the Fredholm index. Partly because of these developments, Ktheory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)
The purpose of this course is to give a general, elementary, introduction to the ideas of Ktheory in the operator algebra context. (Very briefly, Ktheory generalizes the notion of dimension of a vector space.)
The course will begin with a description of the method (Ktheoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of Ktheory to study Bratteli's approximately finitedimensional C*algebrasboth to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*algebraswhat Bratteli called AF algebrasis closed under passing to extensions (a result that uses the Bott periodicity feature of Ktheory).
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 KTheory 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
MAT115HS/MAT448H1S
COMMUTATIVE ALGEBRA
S. Kudla
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.
MAT1200HS/MAT415H1S
ALGEBRIC NUMBER THEORY
J. Tsimerman
Dedekind domains, ideal class group, splitting of prime ideals, finiteness of class number, Dirichlet unit theorem, further topics such as counting number fields as time allows.

Prerequisites:
Solid knowledge of abstract algebra is essential (e.g. Dummit and Foote, MAT347, MAT11001101)Reference(s):
The main reference will be Problems in Algebraic Number Theory by J. Esmonde and Ram Murty Milne’s course notes.
MAT1302HS/APM461H1S
COMBINATORIAL THEORY
S. Kopparty
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisites:
Linear algebra, elementary number theory, elementary group and field theory, elementary analysis.
MAT1304HF/CSC2429HF
TOPICS IN COMBINATORICS: ERROR CORRECTING CODES
S. Kopparty
This course is about the theoretical computer science aspects of errorcorrecting codes. After an introduction to the classical results, we will see a number of modern topics — local testing and decoding, codes from expander graphs, Fourier analytic methods, list decoding, and connections to pseudorandomness and complexity theory.
MAT1304HS/CSC2429HS
TOPICS IN COMBINATORICS: ALGEBRAIC COMPLEXITY THEORY
S. Saraf
The goal of this course will be to understand to power and limitations of algebraic computation. Arithmetic circuits are a very natural model of computation for many natural algebraic algorithms such as matrix multiplication, computing fast fourier transforms, computing the determinant etc. The problem of proving lower bounds for arithmetic circuits is one of the most interesting and challenging problems in complexity theory. This course will discuss lower bounds for arithmetic circuits, both recent and classical. It will also discuss the very related problems of derandomizing polynomial identity testing, polynomial reconstruction and polynomial factoring.
MAT1340HS/MAT425H1S
DIFFERENTIAL TOPOLOGY
E. Meinrenken
Sard's theorem and transversality. Immersion and embedding theorems. Morse theory. Intersection theory. BorsukUlam theorem. Euler characteristic, PoincareHopf theorem and Hopf degree theorem. Additional topics may vary.
Prerequisites:
Introduction to Topology course (MAT327H) and Analysis (MAT257Y).
MAT1342HF/MAT464H1F
INTRODUCTION TO DIFFERENTIAL GEOMETRY: RIEMANNIAN GEOMETRY
R. Rotman
The topics include: Riemannian metrics, LeviCivita connection, geodesics, isometric embeddings and the Gauss formula, complete manifolds, variation of energy.
It will cover chapters 09 of the "Riemannian Geometry" book by Do Carmo.
MAT1404HF/MAT409H1F
INTRODUCTION TO MODEL THEORY AND SET THEORY
S. Todorcevic
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
Textbooks:
Set Theory by William Weiss
MAT1508HS/APM446H1S
TECH OF APPLIED MATH: APPLIED NONLINEAR EQUATIONS
A. Stinchcombe
In this course we study partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. We will touch upon questions of existence, longtime behaviour, formation of singularities, pattern formation. We will also address questions of existence of static, traveling wave, selfsimilar, topological and localized solutions and their stability.
Specifically we consider AllenCahn equation (material science), GinzburgLandau equation (condensed matter physics superfluidity and superconductivity ), CahnHilliard (material science, biology), Mean curvature flow and the equation for minimal and selfsimilar surfaces (geometry, material sciences), FisherKolmogorovPetrovskiiPiskunov (combustion theory, biology), KellerSegel equations (biology), GrossPitaevskii equation (BoseEinstein condensation) and ChernSimmons equations (particle physics and quantum Hall effect).
The course will be relatively selfcontained, but familiarity with elementary ordinary and partial differential equations and Fourier analysis will be assumed.
Prerequisites:
Elementary ordinary and partial differential equations, Fourier analysis, Elementary analysis and theory of functions or physics equivalent of these.
Textbook:
The instructor's notes
Recommended books:
R. McOwen, Partial Differential Equations, Prentice Hall, 2003
J. Ockedon, S. Howison, A. Lacey, A. Movchan, Applied Partial Differential Equations, Oxford University Press, 1999
Peter Grindrod Patterns and Waves: Theory and Applications of Reactiondiffusion Equations (Oxford Applied Mathematics & Computing Science) 1996
MAT1700HF/APM446H1F
GENERAL RELATIVITY
R. McCann
This course is an introduction to general relativity theory for students in mathematics and physics alike. Beginning from basic principles this course aims to discuss the modern theory of gravitation and the geometry of space and time, and will explore many of its consequences ranging from black holes to gravitational waves.
More specifically the course covers a discussion of the equivalence principle and its consequences, the geometry of curved spaces, and Einstein's field equations in the presence of matter; we will explore the geometry of the simplest spherically symmetric black hole spacetimes, and proceed to the dynamical formulation of general relativity, and the prediction of gravitational waves. If time permits we shall also discuss in some detail the postNewtonian approximation.
There will be student projects/presentations on selected topics, such as astrophysical applications, and Penrose's incompleteness theorem.
Prerequisites:
Some prior knowledge of special relativity, and elementary Riemannian geometry will be assumed, but are not strictly required, as relevant concepts will be introduced in the course.
References:
Robert Wald, General Relativity, UCP 1984 Norbert Straumann
General Relativity, Springer 2012 Stephen Hawking and George Ellis
The large scale structure of the spacetime, CUP 1973 Charles Misner, Kip Thorne, and John Wheeler, Gravitation, Freeman 1973
MAT1723HF/APM421H1F
FOUNDATIONS OF QUANTUM MECHANICS
M. Sigal
The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some topics which are at the forefront of active research. In particular we will present an introduction to quantum information theory, which has witnessed an explosion of research in the last decade and which involves some nice mathematics.
We will try to be as selfcontained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively timeconsuming we give an idea of the proof, if such exists, and/or explain the mathematics involved without providing all the details.
Prerequisites:
* Some familiarity with elementary ordinary and partial differential equations
* Knowledge of elementary theory of functions and operators would be helpful
References:
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2011
L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008
For material not contained in this book, e.g. quantum information theory, we will try to provide handouts and refer to online sources.
Useful, but optional, books on the subject are:
Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (Paperback  Sep 2000), Cambridge University Press, ISBN 0 521 63503 9 (paperback)
A. S. Holevo, Statistical Structure of Quantum Theory, Springer, 2001
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam, The Netherlands: North Holland
MAT1856HS (APM466H1S)
MATHEMATICAL THEORY OF FINANCE
L. Seco
Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, singleperiod finance, financial derivatives (treeapproximation and BlackScholes model for equity derivatives, American derivatives, numerical methods, lattice models for interestrate derivatives), value at risk, credit risk, portfolio theory.
Prerequisites:
APM 346H1, STA 347H1
TOPICS COURSES
MAT1103HF
TOPICS IN ALGEBRA I: HOMOLOGICAL ALGEBRA
M. Groechenig
This course is an introduction to the methods of homological algebra which became ubiquitous over the course of the 20th century. We will study abelian categories and the construction of derived functors as well as various examples thereof. Towards the end of the course, we will recast the classical theory within the framework of triangulated categories and take a glimpse at the modern approach via higher category theory.
Topics:

 exact sequences and diagram chasing
 basic category theory
 Tor and Ext
 abelian categories and derived functors
 Examples: sheaf cohomology and group (co)homology
 triangulated categories and derived categories
 outlook: stable infinitycategories
Prerequisites: rings and modules
MAT1128HS
TOPICS IN PROBABILITY: KPZ
J. Quastel
This course introduces students to nonstandard fluctuations arising in the KPZ universality class.
Prerequisites:
No explicit background is assumed but I would not suggest students take this course unless they have seen rigorous probability with measure theory roughly at the level of our core graduate course.
MAT1190HF
ALGEBRAIC GEOMETRY
A. Shankar
This is Part 1 of a yearlong course. Part 2 will be taught by Alexander Braverman in the winter semester.
In Part 1, we will follow Hartshorne's book starting from Chapter 2. The goal is to cover Sections 1 through 5. We will also use Ravi Vakil's notes for examples and exercises.
We will cover the following topics:
1. Sheaves
2. Schemes: definitions and examples.
3. Morphism between schemes.
4. Properties of schemes (connected, reduced, integral, noetherian, ect.) and morphisms (finite type, locally of finite type, finite, closed immersions, etc.)
5. Fiber products, separated morphisms, proper morphisms.
6. Quasi coherent and coherent sheaves of morphisms.
MAT1191HF
TOPICS IN ALGEBRA I: ALGEBRAIC GEOMETRY AND CONVEX GEOMETRY
A. Khovanskii
In the course I will discuss relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial.
Newton polyhedra connects the theory of convex polyhedra with algebraic geometry of toric varieties. For example, Dehn–Zommerville duality for simple polyhedra leads to the computation of the cohomology ring of smooth toric varieties. Riemann–Roch theorem for toric varieties provides valuable information on the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler–Maclaurin summation formula. Newton polyhedra allow to compute many discrete invariants of generic complete intersections. Newton–Okounkov bodies connect the theory of convex bodies (not necessary polyhedra) with algebraic geometry. These bodies provide a simple proof of the classical Alexandrov–Fenchel inequality (generalizing the isoperimetric inequality) and suggest analogues of these inequality in algebraic geometry.
Tropical geometry and the theory of Gröbner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry. All needed facts from algebraic geometry and convex geometry will be discussed in details during the course.
References
1. G.Kempf, F.Knudsen, D.Mamford, B.SaintDonat. Toroidal Embeddings, Springer Lecture Notes 339, 1973. 2
2. K.Kaveh, A.Khovanskii. Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Annals of Mathematics, V. 176, No 2, 925–978, 2012.
3. Some extra papers will be handout during the course.
Grading: one final presentation or written report.
MAT1192HF
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY: HODGE THEORY
D. Litt
We will develop Hodge theory, both in its analytic incarnation (for compact Kähler manifolds) and in its more algebraic incarnation (following work of Deligne and Illusie in positive characteristic). We will discuss many of the applications of Hodge theory to the topology and geometry of algebraic varieties. Time permitting, we will cover more advanced topics, such as the theory of weights and Hodge theory for varieties which are not necessarily smooth or proper, and nonabelian Hodge theory.
MAT1191HS
TOPICS IN ALGEBRAIC GEOMETRY
A. Braverman
This is the second half of the algebraic geometric course, following Arul's course in the fall. We will cover Representable functors, Quasicoherent sheaves, Line bundles and divisors, Morphisms to projective space, Grassmannians, Cohomology of quasicoherent sheaves, Applications to Curves, and other topics depending on interest of participants.
MAT1197HS
AUTOMORPHIC FORMS AND REPRESENTATION THEORY: AN INTRODUCTION TO THE LANGLANDS PROGRAM
J. Arthur
Automorphic representations are at the heart of the conjectures that make up the Langlands program. The trace formula provides the most powerful technique for attacking them. It is the analogue for automorphic representations of the Plancherel formula for ordinary representations.
We shall begin with some of the basic properties of automorphic representations. We shall then discuss some of the Langlands conjectures, especially the Principle of Functoriality. The rest of the course will be devoted to an introduction to the trace formula, and if time permits, some of its applications.
Prerequisites:
Core courses in Analysis and Algebra or their equivalents, and the basic theory of Lie (and/or) Algebraic Groups. This course will be a natural successor to this year's course MAT1196 on representations of real groups, but I will try to keep other prerequsites to a minimum. And in particular, readers without a background in Lie Groups could always think of the general linear group of nonsingular (N x N) matrices (under multiplication) in place of an arbitrary Lie/algebraic group G.
References:
S. Gelbart, An elementary introduction to the Langlands program, Bulletin of the American Mathematical Society, Vol. 10 (1984), p. 177 219.
J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, Vol. 4, 2005, p. 1 263.
MAT1210HS
TOPICS IN NUMBER THEORY: ELLIPTIC CURVES AND ELLIPTIC SURFACES
J. Desjardins
This course is divided into two parts: the first containing an introduction to the basic theory of elliptic curves, and the second with more advanced (but still accessible) topics of the general of elliptic surfaces.
Starting with the definition of elliptic curves, we will turn to studying their basic geometric properties, theory of reduction, Lfunction, MordellWeil Theorem. If time allows, we will define the Picard, Selmer and TateShafarevitch groups.
In the second half, we will give an overview of Shioda geometric theory of elliptic surfaces : KodairaNéron model, Tate's algorithm, base change and quadratic twists and NéronSeveri lattice. Elliptic surfaces are omnipresent in the theory of algebraic surfaces. We will see as well their relation with Del Pezzo surfaces in case of rational elliptic surfaces.
Prerequisites:
• An intuition of projective geometry
• Complex analysis, basic group theory, arithmetic in finite fields
References:
• The Arithmetic of Elliptic Curves, by Silverman.
• Elliptic Surfaces, survey paper by Schütt and Shioda.
MAT1210HF
TOPICS IN NUMBER THEORY: TRANSCENDENTAL NUMBER THEORY
H. Kim
We begin with the introduction to wellknown transcendental numbers such as e, pi, and move on to LindemannWeierstrass theorem, SchneiderLang theorem, Baker's theorem on linear independence of linear forms in logarithms of algebraic numbers, and transcendence of special values of Lfunctions such as Shimura's algebraicity result on critical values of Lfunctions associated to elliptic cusp forms and their RankinSelberg convolutions.
References:
R. Murty and P. Rath, Transcendental Numbers
A. Baker, Transcendental Number Theory
Prerequisites:
Algebraic Number Theory
Complex Analysis
MAT1305HS
TOPICS IN GEOMETRIC TOPOLOGY: DIFFEOMORPHISM GROUPS
A. Kupers
This course is about the dramatic advances made during the last decade in the study of diffeomorphism groups of manifolds. After a discussion of the relevant foundational results in differential topology, we survey classical results obtained for diffeomorphisms of lowdimensional manifolds using geometric techniques and for diffeomorphisms of highdimensional manifold using surgery theory. After this, we explain how the computation of the homotopy type of the cobordism category and the calculus of embeddings can be combined to understand diffeomorphisms far outside the ranges of classical techniques. The goal is to bring students to the forefront of research.
The topics we will discuss include but are not limited to: mapping class groups, the Smale conjecture, the hcobordism theorem, exotic spheres, surgery theory, embedding calculus, cobordism categories, the Madsen–Weiss theorem, homological stability, embedding calculus, configuration space integrals, smoothing theory.
MAT1309HF
GEOMETRIC INEQUALITIES
A. Nabutovsky
Isoperimetric inequality; Various generalizations of the isoperimetric inequality and related inequalities; BrunnMinkowski and AlexandrovFenchel inequalities; Sobolev inequalities and some of their applications; Besicovitch inequality; Introduction to systolic geometry; Gromov's systolic inequality; Complexity of optimal slicings and sweepouts; Geometric inequalities for the lengths of shortest periodic geodesics, shortest geodesic loops, etc.
Most of the course will be nontechnical and accessible even to advanced undergraduate students. I plan to discuss many easily stated open problems.
Prerequisites:
Familiarity with some basics of algebraic topology (in particular, the fundamental group) as well as a previous exposure to basics of Riemannian geometry is helpful, but not required.
Textbook:
Yu. Burago and V. Zalgaller "Geometric inequalities" + expository and research papers.
MAT1312HF
TOPICS IN GEOMETRY: SL(2,Z)
K. Rafi
The group SL(2,R) lies in the intersection of many different fields of mathematics including Hyperbolic Geometry, Homogeneous Dynamics, Algebraic Geometry and Number Theory. Many general phenomena have their simplest case appearing in the group SL(2,R). We examine several geometric and dynamical aspects of this group and attempt to build bridges between them. Possible topics to be covered include:
• SL(2,Z) as a braid group.
• Hyperbolic plane as the symmetric space associated to SL(2,R).
• Hyperbolic plane as the Teichmüller space of the torus.
• Continued Fractions and combinatorial properties of curves on a surface.
• Geodesic flow and counting problems in modular curve.
• Horocycle flow and the Prime Number Theorem.
MAT1312HS
TOPICS IN GEOMETRY: CLASSICAL MECHANICS
M. Gualtieri
This will be a new course on classical mechanics, focusing on the basic structure of the subject and how it is expressed with modern geometric tools.
MAT1314HS
INTRODUCTION TO NONCOMMUTATIVE GEOMETRY
G. Elliott
Some of the most basic objects of study in Connes's noncommutative geometryfor instance, the noncommutative toriwill 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:
M. Khalkhali, Basic Noncommutative Geometry (EMS Series of Lectures in Mathematics, 2010.)
A. Connes, Noncommutative Geometry, Academic Press, 1994.
J. GraciaBondia, 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 KTheory for C*Algebras, Cambridge University Press, 2000.
G.K. Pedersen, Analysis Now, Springer, 1989.
MAT1318HS
SEMINAR IN GEOMETRY AND TOPOLOGY: COMPARISON GEOMETRY
V. Kapovitch
This course would be a direct continuation of the MAT1342/MAT464 Differential geometry course. It would cover various comparison theorems (Rauch and Toponogov comparison) and their applications such as BishopGromov volume comparison, critical point theory of distance functions, diameter sphere theorem, negative and nonnegative curvature, GromollMeyer splitting theorem and CheegerGromoll soul theorem.
Zoom Invite; Passcode: 873664
MAT1347HF
TOPICS IN SYMPLECTIC GEOMETRY AND TOPOLOGY: SYMPLECTIC TOPOLOGY AND MORSE THEORY
B. Khesin
(The numbers below approximately correspond to the week numbers.)
1) Preliminaries/reminder: Symplectic manifolds, Hamiltonian fields, Darboux theorem, Lagrangian manifolds and foliations, integrable systems.
2) Symplectic properties of billiards and, time permitted, geodesics on an ellipsoid. 4
34) Symplectic fixed points theorems: the Poincare–Birkhoff theorem, Arnold’s conjecture, the Conley–Zehnder theorem.
46) Morse theory: Morse inequalities, LusternikSchnirelmann category, applications to geodesics, other ramifications (the Morse–Witten complex, Morse–Novikov theory); the end of proof for Conley–Zehnder.
78) A glimpse of generating functions for symplectomorphisms, nonsqueezing results, symplectic capacities, Floer homology.
910) The Hofer metric, geometry of and geodesics on symplectomorphism groups.
1112) Contact structures, Legendrian knots, their invariants and Bennequin inequality; a glimpse of contact homology of Legendrian knots.
References:
1. S. Tabachnikov, ”Introduction to symplectic topology” Lecture notes, (PennState U.): http://www.personal.psu.edu/sot2/courses/symplectic.pdf
2. D. McDuff and D. Salamon: ”Introduction to symplectic topology” (Oxford Math. Monographs, 1998)
3. V. Arnold and A. Givental ”Symplectic geometry” Dynamical systems, IV, 1–138, Encyclopaedia Math. Sci., vol. 4, (Springer 2001)
Prerequisite:
Familiarity with the main notions of symplectic geometry.
MAT1435HF
TOPICS IN SET THEORY: HOMOGENEOUS STRUCTURES, TOPOLOGICAL DYNAMICS OF THEIR AUTOMORPHISM GROUPS AND THE CORRESPONDING RAMSEY INDEX THEORY
S. Todorcevic
Fraisse theory of homogeneous structures. Topological dynamics of automorphism groups. Structural Ramsey theory. These subjects will be first introduced starting from their initial stages building gradually towards the modern developments. For example, Fraisse theory of metric structures is one such modern development as well as the categorical approach to Fraisse theory. Recent developments in the structural Ramsey theory will also be one of the focuses.
MAT1502HS
TOPICS IN GEOMETRIC ANALYSIS
S. Alexakis
This is topics course on questions of interest in general relativity and cosmology. The main topic we will cover is the behaviour of spacetime metrics upon approach to the big bang singularity, as well as the final crunch singularities and also black hole interiors. This is a subject on which very little is rigorously known. We will study some longstanding conjectures, as well as a set of rigorous results which are obtained for simplified models of the Einstein equations near singularities. In particular a conjecture that generically spacetimes exhibit a chaotic behaviour upon approach to these singularities will be studied.
Prerequisites:
Some background knowledge on general relativity or Riemannian geometry is desirable, but not strictly necessary. An indepth knowledge of PDEs is not necessary for this course.
Textbook:
"The Cosmological Singularity" by Belinskii and Henneaux (Cambridge University Press, 2018).
MAT1510HF
DEEP LEARNING: THEORY & DATA SCIENCE
V. Papyan
Deep learning systems have revolutionized field after another, leading to unprecedented empirical performance. Yet, their intricate structure led most practitioners and researchers to regard them as blackboxes, with little that could be understood. In this course, we will review experimental and theoretical works aiming to improve our understanding of modern deep learning systems.
Link to course website: https://sites.google.com/view/mat1510
Prerequisite:
Undergraduate Linear Algebra
MAT1525HS
TOPICS IN INVERSE PROBLEMS & IMAGE ANALYSIS: VARIATIONAL METHODS IN IMAGING AND NEURAL NETWORKS
A. Nachman
This course aims to provide the analytic tools for understanding the spectacularly successful Wasserstein Generative Neural Networks from a mathematical point of view. We will spend a fair amount of time on the requisite background from Convex Analysis, model variational problems from Image Analysis and basics of Optimal Transport Theory.
Brief outline of the course:
1. Introduction to Wasserstein Generative Neural Networks
2. The RudinOsherFatemi model for image restoration
3. Introduction to Congested Optimal Transport
4. Wasserstein GANs compute a congested transport cost
5. Image restoration with learned regularizers
6. Projected GANs
Prerequisite:
Basic Real Analysis and Functional analysis will be helpful.
Recommended References:
"An introduction to Total Variation for Image Analysis" by Antonin Chambolle, Vicent Caselles, Matteo Novaga, Daniel Cremers and Thomas Pock, archivesouvertes.fr, 2009.
"Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling" by Filippo Santambrogio, Birkhauser, 2015.
"Wasserstein GANs with Gradient Penalty Compute Congested Transport" by Tristan Milne and Adrian Nachman,
Conference on Learning Theory, PMLR, 2022, pp. 103–129.
MAT1846HS
TOPICS IN DYNAMICS: HOMOGENEOUS DYNAMICS & APPLICATIONS
W. Pan
After studying some basics of hyperbolic geometry, we will discuss the PattersonSullivan theory for Kleinian groups. We will also discuss how to generalize the PattersonSullivan theory to discrete subgroups of higherrank semisimple Lie groups.
References:
1. J. Ratcliffe. Foundations of hyperbolic manifolds. Graduate Texts in Mathematics, 149. Springer, New York, 2006.
2. K. Matsuzaki and M. Taniguchi. Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs.
3. J. F. Quint. An overview of PattersonSullivan theory https://www.math.ubordeaux.fr/~jquint/publications/courszurich.pdf
4. J. F. Quint. Mesures de PattersonSullivan en rang sup´erieur. Geom. Funct. Anal. 12 (2002), no. 4, 776–809.
MAT1901HF
TOPICS IN DYNAMICS: INTRODUCTION TO THE RIEMANNIAN CURVATURE DIMENSION CONDITION
N. Gigli
The aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.
We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of JordanKinderlehrerOtto we shall see that such heat flow can be equivalently characterized as gradient flow of the CheegerDirichlet energy on L2 and as gradient flow of the BoltzmannShannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the LottVillaniSturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of `infinitesimally Hilbertian' spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds.
We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.
We shall finally see how these are linked to the lower Ricci curvature bound  most notably we shall prove the Laplacian comparison theorem  and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth to notice that such statement gives new information  compared to those available through CheegerColding's theory of Riccilimit spaces  even about the structure of smooth Riemannian manifolds.
Prerequisites:
Some familiarity with Riemannian geometry and optimal transport theory in the case cost=distance2 is preferred, but not required: I shall provide the necessary background when needed.
Students requiring individual instruction in mathematical topics should consult with the Mathematics Graduate Office.