Core Graduate Courses | Cross-Listed Courses | Topics Courses
Mathematics graduate students can take suitable graduate courses in other departments (CS, Physics, Chemistry, Eng. etc. ) to satisfy their course credit requirement. Two-thirds of the course requirements for each degree should be in the Mathematics Department.
Please contact the Graduate Office early on if you are looking for enrollment in popular courses, as they fill up fast.
CORE COURSES
MAT1000HF (MAT457H1F)
REAL ANALYSIS I
I. Uriarte-Tuero
(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^p-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
Elliott 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
Basic Functional Analysis: Banach spaces, Hilbert space, Hahn Banach theorem, open mapping theorem, closed graph theorem, uniform boundedness principle, Alaoglu's theorem, Frechet spaces.
Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, more topological vector spaces, Schwartz space, distributions.
Spectral theory: spectral theorem for bounded self-adjoint operators, specializations to compact operators and/or extensions to unbounded operators, as time permits.
Textbooks:
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley; 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
- Review of holomorphic and harmonic functions
- Topology of a space of holomorphic functions; Series and infinite products; Weierstrass p-function, gamma function; Weierstrass and Mittag-Leffler theorems; Normal families (compact subsets of H())
- Conformal mappings; Riemann mapping theorem; Schwarz-Christoffel formula
- Riemann surfaces; Riemann surface associated with an elliptic curve; Inversion of an elliptic integral, Abel’s theorem
- Analytic continuation; Sheaf of holomorphic functions; Monodromy theorem; Little Picard theorem
Recommended prerequisites:
A first course in complex analysis and a course in real analysis. Measure theory is not required.
Textbook:
L. Ahlfors: Complex Analysis, 3rd Edition, McGraw-Hill
Recommended References:
H. Cartan, Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover; T. Gamelin, Complex Analysis, Springer; O. Ivrii, The Bierstone Lectures on Complex Analysis, course notes online
MAT1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
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:
Lawrence C. Evans, Partial Differential Equations, 2nd Edition, AMS GSM19, ISBN 978-1-4704-6942-9.
MAT1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
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.
Prerequisites:
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:
Robert C. McOwen, Partial Differential Equations, 2nd edition, ISBN 0-13-009335-1.
MAT1100HF
ALGEBRA I
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.
MAT1101HS
ALGEBRA II
Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois Theory, solution of equations by radicals.
Commutative Rings:
oetherian 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.
MAT1300HF
DIFFERENTIAL TOPOLOGY
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
MAT1301HS
ALGEBRAIC TOPOLOGY
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
MAT1600HF
MATHEMATICAL PROBABILITY I
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.
Textbook:
Lecture notes and a list of recommended books will be provided.
Recommended prerequisite:
Real Analysis I.
MAT1601HS
MATHEMATICAL PROBABILITY II
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.
Recommended prerequisites:
Real Analysis I and Probability I.
MAT1850HF
LINEAR ALGEBRA AND OPTIMIZATION
A. Stinchcombe
(View Timetable)
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.
CROSS-LISTED
MAT1011HF/MAT436H1F
Introduction to Linear Operators
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).
Joint undergraduate/graduate course - MAT436H1/MAT1011H
Prerequisite:
5.0 MAT credits, including MAT224H1/ MAT247H1 and MAT237Y1/ MAT257Y1
MAT1202HF/MAT417H1F
Analytic Number Theory
A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; Diophantine approximation, modular forms.
Joint undergraduate/graduate course - MAT417H1/MAT1202H
Prerequisite:
MAT334H1/ MAT354H1
MAT1342HF/MAT464H1F
Riemannian Geometry
Riemannian metrics. Levi-Civita connection. Geodesics. Exponential map. Second fundamental form. Complete manifolds and Hopf-Rinow theorem. Curvature tensors. Ricci curvature and scalar curvature. Spaces of constant curvature.
Joint undergraduate/graduate course - MAT464H1/MAT1342H
Prerequisite:
MAT367H1
MAT1404HF/MAT409H1F
Set Theory
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.
Joint undergraduate/graduate course - MAT409H1/MAT1404H
Prerequisite:
MAT357H1
MAT1723HF/APM421H1F
Mathematical Foundations of Quantum Mechanics and Quantum Information Theory
Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.
Joint undergraduate/graduate course - APM421H1/MAT1723H
Prerequisite:
(MAT224H1/ MAT247H1, MAT337H1)/ MAT357H1
MAT1016HS/MAT437H1S
K-Theory and C* Algebras
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, 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 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.
Joint undergraduate/graduate course - MAT437H1/MAT1016H
Prerequisite:
5.0 MAT credits, including MAT224H1/ MAT247H1 and MAT237Y1/ MAT257Y1
Recommended Preparation:
Students are encouraged to execute basic research that answers the question, what is an abelian group?
MAT1194HS/MAT449H1S
Algebraic Curves
Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities. This course will be offered in alternating years.
Prerequisite:
MAT347Y1, MAT354H1
MAT1196HS/MAT445H1S
Representation Theory
A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.
Joint undergraduate/graduate - MAT445H1/MAT1196H
Prerequisite:
MAT347Y1
MAT1302HS/APM461H1S
Combinatorial Methods
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Joint undergraduate/graduate course - APM461H1/MAT1302H
Prerequisite:
MAT224H1/ MAT247H1, MAT137Y1/ MAT157Y1, MAT301H1/ MAT347Y1
Recommended Preparation:
MAT344H1
MAT1508HS/APM446H1S
Applied Nonlinear Equations
Partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. Nonlinear evolution equations. Existence and long-time behaviour of solutions. Existence of static, traveling wave, self-similar, topological and localized solutions. Stability. Formation of singularities and pattern formation. Fixed point theorems, spectral analysis, bifurcation theory. Equations considered in this course may include: Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics), Cahn-Hilliard (material science, biology), nonlinear Schroedinger equation (quantum and plasma physics, water waves, etc). mean curvature flow (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), and Chern-Simons equations (particle and condensed matter physics).
Joint undergraduate/graduate course - APM446H1/MAT1508H
Prerequisite:
APM346H1/ MAT351Y1
MAT1700HS/APM426H1S
General Relativity
Einstein's theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein's equations. Cosmological implications: big bang and inflationary universe. Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from black hole dynamics and gravitational waves. The Penrose singularity theorem.
Joint undergraduate/graduate course - APM426H1/MAT1700H
Prerequisite:
MAT363H1/ MAT367H1
MAT1856HS/APM466H1S
Mathematical Theory of Finance
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.
Joint undergraduate/graduate course - APM466H1/MAT1856H
Prerequisite:
APM346H1, STA347H1
Corequisite:
STA457H1
MAT1501HS/CSC2410HS
Graph Theory
This course covers many of the most important aspects of graph theory, including the development and analysis of algorithms for problems which arise in applications of graph theory. Topics include minimum spanning tree, shortest path problems, network flow with applications to bipartite matching, general matching, planarity testing, graph colourability, etc. Attention will be given to the mathematical theory that supports the algorithms presented.
TOPICS COURSES
MAT1013HF
Theory of Several Complex Variables II: Complex Geometry
Course Outline:
- Introduction to complex manifolds, and Kahler manifolds
- Hodge Theory, Cohomology of Kahler Manifolds
- Introduction to sheaf theory and sheaf cohomology
- Hormander's estimates for the d-bar equation
- The Kodaira vanishing/embedding theorem
- Multiplier ideal sheaves, vanishing theorems in algebraic geometry (eg. Kawamata-Viehweg), effective estimates
- Possible further topics:
- Ohsawa-Takegoshi L2 extension theorem
- Invariance of plurigenera
- Applications and analogies with convex geometry, including Brunn-Minkowksi theory, Berndtsson's positivity of direct image sheaves
- Yau's solution of the Calabi conjecture and applications
Pre-requisites: Familiarity with differential geometry and basic complex analysis.
MAT1062HF
Topics in Partial Differential Equations I: Geometric Fluid Dynamics
This course deals with various problems in geometry, Lie theory, and Hamiltonian systems, motivated by hydrodynamics and magnetohydrodynamics. We discuss the dynamics of an ideal fluid from the group-theoretic and Hamiltonian points of view. We cover geometry of conservation laws of the Euler equation, point vortex approximations, topology of steady flows and their nonlinear stability, relation of the energy and helicity of vector fields, geometry of diffeomorphism groups, relation to vortex sheets, as well descriptions of magnetohydrodynamics and of the Korteweg-de Vries equation in the Lie group framework.
MAT1105HF
Topics in Representation Theory: Representations of reductive p-adic groups
M. Gerbelli-Gauthier
(View Timetable)
Week 1 - review of representation of finite groups, representation theory of GL_2(F_q)
Week 2 - p-adic numbers and p-adic Lie groups; lattices and compact subgroups.
Week 3 - smooth and admissible representations.
Week 4 - Hecke algebras and Haar measure
Week 5 - induction: general results
Week 6 - Parabolic induction and Jacquet functors
Week 7 - The Satake isomorphism.
Week 8 - Supercuspidal representations
Week 9 - Representation theory of GL_2(Q_p)
Week 10 - Compact representations
Week 11 - Decompositions of parabolic inductions
Week 12 - Bernstein decomposition
Prerequisites: MAT1196HS/MAT445H1S
Textbook/references:
Renard, David. Représentations des groupes réductifs p-adiques. Paris: Société mathématique de France, 2010
Murnaghan, F. "Representations of reductive p-adic groups. Course Notes." 2009
Cartier, Pierre. "Representations of p-adic groups: a survey." Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part. Vol. 1. 1979.
Ngo, Bau Chau. "Harmonic analysis and representation theory of p-adic reductive groups", 2016
MAT1128HF
Gaussian distributions with applications
We will begin with an overview of multivariate Gaussian distributions and their fundamental properties. This includes tools such as Gaussian integration by parts, interpolation techniques, concentration inequalities, and comparison inequalities. Building on this, we will explore a range of applications and related topics, potentially including the Johnson-Lindenstrauss Lemma, Perceptron Capacity, structured Gaussian matrices, Dvoretzky’s Theorem, Gaussian isoperimetry, the Gaussian Correlation Inequality, Stein’s method for normal approximation, and the Sherrington-Kirkpatrick model.
MAT1190HF
Algebraic Geometry
This course will be an introduction to the theory of schemes, following chapter 2 in Hartshorne. The following topics will be covered:
Sheaves and locally ringed spaces
Schemes
Quasi-coherent sheaves
Open and closed immersions
Fibre products
Separated and proper morphisms
Valuative criterion
Projective morphisms
Kähler differentials
Prerequisites:
Algebra 1 & 2, in particular we expect familiarity with rings, ideals, modules, tensor products.
Textbook:
Artshorne, Algebraic geometry
Complementary reading of Vakil's book is highly encouraged.
MAT1210HF
Topics in Number Theory: Galois cohomology with applications to number
We will introduce Galois cohomology, develop its basic properties, and show how it can be used in several important areas of number theory including class field theory, Iwasawa theory, Stark's conjectures and the inverse problem of Galois theory. No prior knowledge of cohomology is required as we will develop the theory from scratch. However, graduate algebra and number theory will be useful.
MAT1312HF
Topics in Geometry: Geometry of Quantum Mechanics
A new course on the geometry of quantum mechanics, looking back a century to the beginning of the subject, and using modern geometric tools to express its core ideas. It is aimed at students with an understanding of the basics of manifold theory and complex analysis, and can be cross-listed.
In addition to revisiting all the main concepts of quantum mechanics with a modern toolkit, we will review a line of recent developments in the exact WKB method whereby the perturbative solutions to singular differential equations (such as the Schrodinger equation) are shown to be
Borel summable. This would bring us to the research level in the subject (Kontsevich-Soibelman,
Nikolaev).
Topics to be covered:
Groupoids and the Heisenberg approach to Quantum Mechanics (c.f. Connes’ Noncommutative Geometry)
The Schrodinger equation on Riemannian manifolds
Hilbert spaces, C_ algebras, and Von Neumann algebras (c.f. Witten’s recent work on an algebra of observables for de Sitter spacetime)
Representations of the Lorentz group, elementary particles (c.f. Wigner’s paper on irreducible representations)
The Dirac equation, Spin manifolds, generalized geometry.
Heisenberg spin chains, the vacuum state (c.f. Penrose Spin Chains)
The WKB method and its exact version (c.f. text of Kawai-Takei, recent papers of Nikolaev)
Quantum Field Theory: Atiyah-Segal axioms for topological and conformal field theories.
Evaluation will be by a series of assignments which are designed to teach the material through a series of elaborate worked examples.
MAT1435HF
Topics in Set Theory: Forcing
I will give a detailed forcing analysis of compact sets of Baire class one function that was initiated in my JAMS 1999 paper and will show how it leads to the solution of the separable quotient problem for dual Banach spaces.
MAT1502HF
Optimal Transportation and Its Applications
This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, and nonlinear partial differential equations. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass | think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.
Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Amp_ere), geometric inequalities with sharp constants, image processing, nonsmooth Riemannian and Lorentzian geometry, many-body quantum mechanics, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium ows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry.
This course aims to help prepare graduate students for the Fall 2026 Fields thematic program on Optimal Transport in Natural Sciences and Statistics.
Prerequisite(s):
Corequisite:
Measure theory, e.g. comparable to MAT 457F/1000F or equivalent
References:
F Santambrogio. `Optimal transport for applied mathematicians.' Birkhauser 2015.
Ambrosio, Gigli and Savare: `Gradient Flows in Metric Spaces and in the Space of Probability
Measures'. Birkhauser 2005 (and subsequent works).
D Burago, Y Burago and S Ivanov "A course in metric geometry" AMS 2001.
Figalli: `The Monge-Amp_ere equation and its applications.' EMS 2017.
McCann and Guillen `Five lectures on optimal transport' In Analysis and Geometry of Metric
Measure Spaces G. Dafni et al, eds. Providence: Amer. Math. Soc. (2013) 145-180
Villani: `Topics in Optimal Transportation', AMS GSM #58 2003
Villani `Optimal Transport: Old and New', Springer-Verlag 2009.
MAT1510HF
Deep Learning Theory & Data Science
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.
MAT1520HF
Wave propagation: Introduction to nonlinear dispersive equations
- Integrals depending on a parameter: The Laplace method, the stationary phase method.
- Applications to linear PDEs.
- Review of Sobolev spaces, Fourier Transform
- The linear Schrödinger equation: Basic properties, Strichartz estimates, Smoothing effects
- The nonlinear Schrödinger equation: Wellposedness theory, long time behaviour
- Formation of singularities
- Weakly nonlinear waves. Derivation of universal PDEs using the method of multiple scales.
MAT1739HF
Topics in Mathematical Physics: Introduction to PDE in physics and geometry - techniques and applications
This course will cover the general topics such as static solutions and solitons, stability and dynamics, emerging solutions and pattern formation, symmetries and conservation laws. Specific topics and PDEs will be chosen with input from the students and could range from topological solitons in physical systems to electrical impulses in neurons, from vortices and monopoles to evolving surfaces, from gauge field theory and superconductivity to the density functional theory (DFT) of matter.
MAT1120HS
Lie Groups and Lie Algebras I
Definition; maximal tori, Weyl group, tangent bundle, Lie algebra, exponential map, elementary representation theory.
Suggested text:
J.F. Adams, Lectures on Lie groups
T. Broecker and T. Tom Dieck, Representations of compact Lie groups
MAT1128HS
Topics in Probability: Stochastic differential equations
Brownian motion, martingales & stopping times, white noise, stochastic integral, stochastic ode's, Markov processes, Feynman-Kac formula, connection with PDE, discrete approximations, boundary conditions & local times, Cameron-Martin-Girsanov formula, Wiener chaos, stochastic partial differential equations.
Homework:
Around 8 problem sets
Exam:
Take home final exam
Prerequisite:
Graduate courses in real analysis and probability. Undergraduate courses in ODE, PDE.
MAT1190HS
Algebraic Geometry II
This is the sequel to the algebraic geometry course proposed by Michael Groechenig, which will cover Chapter II of Hartshorne.
This course will cover Chapter III of Hartshorne, including derived functors, sheaf cohomology, Cech cohomology, the cohomology of projective space and other examples, Ext groups, Serre duality, higher direct images, flat and smooth morphisms, formal GAGA, the semicontinuity theorem, and cohomology and base change. It is meant as the second-half of a year long course meant to provide graduate students with the necessary background to do research in algebraic geometry.
Prerequisites:
Commutative Algebra
Textbook:
Algebraic Geometry, Hartshorne
MAT1191HS
Topics in Algebraic Geometry: Prismatic F-gauges
Vologodsky
(View Timetable)
What is the linear algebra structure the de Rham cohomology of a p-adic variety carries? Drinfeld and Bhatt-Lurie (inspired by the Bhatt-Scholze theory of "prismatic cohomology") proposed an answer to this question. Their approach is based on an interpretation of the de Rham cohomology as the coherent cohomology of a certain algebraic stack, called 'the de Rham stack", functorially attached to every p-adic scheme and the study of deformations of the de Rham stack, called prismatization.
This will be an introductory course on the subject of prismatization.
Prerequisites:
Algebraic Geometry (Hartshorne, Chapter II).
Cohomology of sheaves. Familiarity with the language of derived categories will be useful.
MAT1192HS
Advanced Topics in Algebraic Geometry: Geometric representation theory
This course will be an introduction to geometric representation theory - the study of representations via the geometry and topology of naturally associated geometric objects, such as flag varieties. Some topics might include: category O, Beilinson-Bernstein localization theorem, and Kazhdan-Lusztig theory. Along the way, we will introduce the basic objects and build up the relevant theory, including some aspects of the theory of D-modules. Some knowledge of basic representation theory and algebraic geometry will be assumed.
References:
R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory
N. Chriss, V. Ginzburg, Representation theory and complex geometry
MAT1210HS
Topics in Number Theory: Elliptic Curves
We will cover the basics of arithmetic of elliptic curves, including proving Mordell's theorem, and defining the Selmer and Tate-Shafarevich groups. We will then move on to studying complex multiplication. Finally, if time permits, we will prove Elkies' landmark result, namely, that elliptic curves over Q have infinitely many supersingular primes.
MAT1304HS
Topics in Combinatorics - Advanced graph theory: structural, probabilistic and spectral
L. Gishboliner
(View Timetable)
The course consists of several topics in graph theory, including:
- Perfect graphs and the Lovasz theta function.
- The Erdos-Hajnal and Gyarfas-Sumner conjectures.
- The Nash-Williams and Gallai-Milgram theorems.
- Posa rotations.
- Random graphs: The giant component, appearance of small subgraphs, connectivity and Hamiltonicity.
- Chromatic thresholds.
- Spectral graph theory: Basics, the Friendship Theorem, strongly regular graphs, spectral inequalities, eigenvalue interlacing
- The Matrix-Tree theorem.
- De Brujin sequences and the BEST theorem.
MAT1314HS
Introduction to Noncommutative Geometry
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 classication 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. 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.
MAT1347HS
Topics in symplectic geometry and topology: mirror symmetry and Langlands duality
B. Gammage
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Broadly understood, "mirror symmetry" entails the question to find hidden algebraic structure in symplectic invariants. One formulation of this question is Kontsevich's homological mirror symmetry conjecture, which proposes that the Fukaya category of a symplectic manifold should admit a purely algebraic description. A more recent development is a "3d" version proposing a categorified version of this statement for a holomorphic symplectic manifold. Both are intimately related to geometric Langlands duality, which proposes, in representation-theoretic language, algebraic formulae for symplectic phenomena related to a complex reductive group. This course will survey some of the basic results in these theories, emphasizing the use of newly popular topological methods (constructible and microlocal sheaves) in symplectic geometry. The goal will be to understand the "symplectic" perspective on constructions which have arisen in other contexts in representation theory and homological algebra.
MAT1502HS
Topics in Geometric Analysis: Complex Monge-Ampere equations and Kahler geometry
This course will focus on a fundamental PDE in Kahler geometry: the complex Monge-Ampere equation, which describes the relationship between Kahler metrics and its Ricci curvature. The complex Monge-Ampere equation is the central tool for constructing and studying Kahler-Einstein metrics, which has far-reaching geometric consequences. We will begin with some classical results regarding the Calabi conjecture and Yau's subsequent proof, and move on to cover more modern developments.
Topics will include:
- The Calabi conjecture and Kahler-Einstein metrics
- Weak solutions and pluripotential theory
- A priori estimates and singular Kahler metrics
- Degenerate Monge-Ampere equations and geodesics in the space of Kahler metrics
Pre-requisite: Differential geometry, some exposure to PDEs (PDE I)
MAT1739HS
Topics in Mathematical Physics: Self-Similarity and the Einstein Vacuum Equations
Y. Shlapentokh-Rothman
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Self-Similarity and the Einstein Vacuum Equations:
This class will give on overview of self-similarity and naked singularities for the Einstein equations.
Topics to be covered may include Christodoulou’s naked singularities for the spherically symmetric Einstein-scalar field system, Singh’s study of the wave equation on Christodoulou’s naked singularity background, introduction to the double-null gauge and the characteristic initial value problem, Fefferman–Graham theory and the asymptotically self-similar regime for the Einstein vacuum equations, and twisted self-similarity and naked singularities for the Einstein vacuum equations.
MAT1800HS
Methods of Applied Mathematics
A. Stinchcombe
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This course will cover a variety of mathematical methods important to applied mathematics. The first part of the course will focus on numerical analysis and computation: how mathematical objects are represented by finite approximations in a way that permits efficient calculations. One of the goals of this part of the course will be to understand how to discretize and solve equations involving linear differential operators. The second part of the course will focus on nonlinear methods in applied mathematics. This can include topics from a variety of areas.
- Mathematical goals of numerical analysis: how a Banach space is approximated with a finitedimensional subset, numerical diameter, approximating operators between Banach spaces
- Floating point arithmetic: approximations of the real numbers, correctly rounded arithmetic, rounding error, cancellation error, condition number
- Numerical use of sequences, series, and asymptotic series
- Function approximation and interpolation: Weierstrass theorem, best approximating polynomial, Chebyshev approximation, Lagrange interpolation, Runge phenomenon, convergence of Fourier and Sturm-Liouville eigenfunction expansions
- Quadrature: Gaussian quadrature, adaptive integration, Euler-Maclaurin formula, Newton-Cotes formulas
- Bounded and compact operators: theory and examples
- Discretizing integral operators, solving integral equations
- Unbounded operators: theory, including densely-defined operators, self-adjointness, spectra, and examples
- Discretizing differential operators: ordinary differential operators and partial differential operators, finite differences, finite elements
- Numerical linear algebra: least squares, Gram-Schmidt procedure, singular value decomposition, solving linear systems, computing eigenvalues
- Topics in nonlinear methods, depending on instructor: sparse sensing, nonlinear optimization and simplex methods, inverse problems, wavelet methods, high-dimensional statistics, methods from data science, including kernel regression and deep neural networks, nonlinear dynamical systems and modeling.