Core Graduate Courses | Cross-Listed Courses | Topics Courses
CORE COURSES
MAT1000HF/MAT457H1F
REAL ANALYSIS I
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
F. Tong
(View Timetable)
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
I. Uriarte-Tuero
(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: Lawrence C. Evans, Partial Differential Equations, 2nd Edition, AMS GSM19, ISBN 978-1-4704-6942-9.
Prerequisites:
You need to fulfill at least one of the prerequisites listed with number 1, and at least one of the prerequisites listed with number 2.
1a MAT357H1
1b MAT457H1 (can be taken concurrently)
2a MAT244H1 with at least 80/100
2b MAT267H1 with at least 70/100
Highly recommended, but not a prerequisite: MAT351Y1
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: 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.
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.
MAT1850HS
LINEAR ALGEBRA AND OPTIMIZATION
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
MAT1016HS/MAT437H1S
TOPICS IN OPERATOR ALGEBRAS: 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 classication 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
Prerequisites:
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
S. Kudla
(View Timetable)
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
MAT1200HS/MAT415H1S
ALGEBRIC NUMBER THEORY
A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; Diophantine equations.
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
MAT1304HS/CSC2414HS
Topics in Combinatorics: Algebraic Gems in Theoretical Computer Science and Discrete Mathematics
In the last few decades, algebraic methods have proven to be extremely powerful in several areas of computer science and discrete mathematics. Many of the recent and important advances in these fields have relied on very simple properties of polynomials. In this course we will see many interesting and often surprising applications of linear algebra and polynomials to complexity theory, combinatorics, cryptography and algorithm design. We will develop all the algebraic tools that we need along the way.
The main prerequisite is mathematical maturity. It might be helpful to have some familiarity with discrete math/algorithms and linear algebra. Students with an interest in discrete mathematics and/or theoretical computer science are welcome.
A tentative (and partial) list of topics that we will cover:
- Applications of linear algebra
- Rank and dimension arguments, expander graphs
- Polynomials methods in combinatorics and discrete geometry
- Polynomials and error correcting codes
- Algebraic methods in complexity theory and cryptography
- Interactive proofs
- Polynomial identity testing
- Primality testing
- Circuit lower bounds
- Introduction to arithmetic circuits/arithmetic computation
MAT1306HF/CSC2429HF
The Discrete Mathematics Toolkit: Expanders and pseudorandom graphs
An advanced course on topics in theoretical computer science. Topics will change from one instance of the course to another, and can include algorithms, advanced data structures, complexity theory, cryptography, discrete mathematics, distributed computing, graph theory, privacy, pseudorandomness, social choice, quantum computation, as well as topics from other theoretical areas. The course is suitable for students with background in theoretical computer science, or with an interest in the course and motivation to do background reading.
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MAT1340HS/MAT425H1S
DIFFERENTIAL TOPOLOGY
Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.
Prerequisites:
MAT257Y1, MAT327H1
MAT1342HF/MAT464H1F
Riemannian Geometry
Manin
(View Timetable)
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
MAT1500HS/ CSC2427HS
Advanced Topics in Graph Theory - The Probabilistic Method
The probabilistic method, pioneered by Erdos in the 1950's, is now one of the most important and broadly used proof techniques in graph theory and combinatorics. It also has wide applications in other areas of math and theoretical computer science. Roughly speaking: In order to prove the existence of an object with certain properties (eg a colouring of a graph), we devise a way to generate that object randomly, and then prove that the desired properties hold with positive probability.
This course will cover some of the most important tools in this area including, the first moment method, the Lovasz Local Lemma, random graphs, and concentration inequalities.
Prerequisite:
You'll need a solid background in graph theory, for example a standard undergraduate course in the area. You will also need to be nimble with basic discrete probability, be comfortable computing expected values and have a good intuition about randomness.
MAT1507HF/APM441H1F
Asymptotic and Perturbation Methods
Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained co-ordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering)
Joint undergraduate/graduate course - APM441H1/MAT1507H
Prerequisite
APM346H1/ MAT351Y1, MAT334H1/ MAT354H1
For a more detailed course description, please visit the link below;
https://www.math.utoronto.ca/ivrii/APM-textbook/contents.html
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
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
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.
Prerequisites:
APM 346H1, STA 347H1
Corequisites:
STA457H1
TOPICS COURSES
MAT1045HS
Topics in Ergodic Theory: Randomness in Groups
There are several methods for choosing a generic element in a group. One can equip the group with the word metric associated to some finite generating set and then take an element at random in a ball of radius R with respect to this metric. Alternatively, one can choose an element using a random walk process on the group. That is, let µ be a probability measure on a Γ. Then µ defines a random walk process where a sample path is a sequence of element wn ∈ Γ with the property that in each step the probability of transition from wn to wn+1 = wnγ is µ(γ). We can then consider wn, for a large n, to be a randomly chosen element. It is often easier to prove properties of elements chosen using a random walk process because there is an explicit probabilistic description of wn. For example, a random mapping class chosen using a random walk process is known to be pseudo-Anosov but the same is not known if the mapping class is chosen using the first method. One way to answer such questions is to consider if the two notion of randomness can ever be asymptotically the same. Here, asymptotic means that we should take the limit of each process at infinity. Consider the following simple case where M is a compact manifold of negative curvature. The Gromov boundary at infinity of the universal cover ∂Mf of M is a topological sphere. This sphere carries two types of measures corresponding to two notions of randomness in the fundamental group of M. First, we can consider a weak limit of uniform average of orbit points contained in large balls in Mf. This limit coincides with the conformal measure built from the visual metric and is referred to as the Patterson-Sullivan measure. On the other hand, for almost every sample path wn and every point x ∈ Mf, the sequence wn(x) converges to a point on ∂Mf. Hence, we can consider the corresponding hitting measure on ∂Mf where the measure of a set E ⊂ ∂T is the probability that, for x ∈ Mf and a sample path {wn} of the random walk, wn(x) converges to a point of E. The above question translate to: Question 1. Can the Patterson-Sullivan measure be the hitting measure of a random walk? In the setting of lattices in semi-simple Lie groups acting on the associated symmetric spaces, Question 0.1 is answered by a celebrated theorem of Furstenberg. He showed that the Lebesgue measure on the Furstenberg boundary of a symmetric space is the hitting measure of some random walk with finite first moment. In this class, we will go over the proof of Furstenberg’s theorem. We then attemp to generalize it to other settings, most importantly, the setting of the action of Mapping class group on Teichm¨uller space. More precisely, let T be the Teichmüller space of a surface S, that is the space of marked Riemann surfaces homeomorphic to S, and let Map(S) be the mapping class group, the group of isotopy class orientation preserving homeomorphisms of S. Then Map(S) acts on T by changing the marking. In this setting, we can use the Thurston boundary of Teichmüller space which can be identified with the space of projectivized measured foliations, ∂T = PMF, and is equipped with a natural Lebesgue measure µTh known as the Thurston measure. It follows from the work of Athreya-Bufetov-Eskin-Mirzakhani that the PattersonSullivan measure is in the Lebesgue class and Kaimanovich-Masur showed that the Thurston boundary can be used at the Poisson boundary for the action of mapping class group on Teichmüller space. The goal of the class is to prove the following theorem
Theorem 2. There is a measure µ on the mapping class group with finite first-moment such that the corresponding hitting measure ν is absolutely continuous with respect to the Thurston measure µTh on ∂T (S).
MAT1062HF
Topics in PDE I: Introduction to Nonlinear Evolution equations
The goal of this course is to give an overview of some classical results in the theory of nonlinear evolution equations concerning their local and global wellposendess, and the asymptotic behavior of solutions.
We will start with a review of basic analysis tools (Fourier Transform, Sobolev spaces, Littewood-Paley decomposition and basic inequalities) and present several examples including nonlinear Schrödinger and Wave equations, Plasma and Fluid dynamics models, and talk about general properties of linear flows (e.g. dispersion and decay estimates).
We will then focus our attention on specific examples to highlight some of the main basic ideas and tools in the analysis of nonlinear equations. In particular, we plan to discuss:
- The local wellposendess of the Euler equations in 2 and 3 dimensions and the global regularity of solutions in 2d;
- The local and global wellposendess theory for some Nonlinear Schrödinger Equations;
- Some theorems on global wellposendess and asymptotics for small solutions of Nonlinear Schrödinger and Wave Equations.
If time permits we may also discuss the local and global wellposendess theory for the Water Waves equations of free surface hydrodynamics.
Suggested prerequisites: Some exposure to PDE (PDE I is more than sufficient) and to Real Analysis (Real Analysis I is more than sufficient).
MAT1103HF
Topics in Algebra I: Introduction to Algebraic D-modules
We are going to start by discussing a very elementary "analytic" question asked by I.M.Gelfand in the 50's: the question is whether certain integrals depending on a parameter have a meromorphic continuation with respect to that parameter. We shall then explain J. Bernstein's purely algebraic approach to this question - it is based on some deep properties of the algebra D of (linear) differential operators in several variables with polynomial coefficients and modules over it. We shall introduce a very important concept of holonomic D-module and use it to resolve Gelfand's question. After that we'll continue to study modules over this algebra (emphasizing the connection between D-modules and systems of linear differential equations with polynomial coefficients) .
In the second half of the course, we will generalize the above discussion to the case when D is replaced by the sheaf of differential operators on an arbitrary smooth complex algebraic variety. We will study such things as inverse and direct images of D-modules. This will require working with derived categories - we will devote several lectures to the definition and basic properties of this notion.
In the very end of the course, we will have a brief discussion of the notion of differential equations with regular singularities and the so-called Riemann-Hilbert correspondence.
Prerequisites:
basic undergraduate algebra (rings, modules etc.) for the first half of the course; very basic algebraic geometry (algebraic varieties and coherent sheaves) during the second half.
MAT1126HS
Lie Groups and Hamiltonian PDEs
I. Introduction and main notions.
- Lie groups and Lie algebras.
- Adjoint and coadjoint orbits.
- Central extensions.
- The Lie-Poisson (or Euler) equations for Lie groups.
- Bihamiltonian systems.
II. Geometry of infinite-dimensional Lie groups and their orbits.
- Affine Kac-Moody Lie algebras and groups.
- Definition of the affine Kac-Moody Lie algebras.
- Affine Lie groups.
- Their coadjoint orbits.
- The quotient (WZW) construction of the affine groups.
2. The Virasoro algebra and group. The KdV equation.
- The group of circle diffeomorphisms.
- The Virasoro group and coadjoint action.
- Virasoro coadjoint orbits.
- The Virasoro group and Korteweg-de Vries equation.
- Bihamiltonian structure of the KdV and Camassa-Holm equation.
3. Groups of (pseudo)differential operators. Integrable KP-KdV hierarchies.
- Pseudodifferential operators and cocycles on them.
- The Lie group of pseudodifferential operators of complex degree.
- Integrable KP-KdV hierarchies.
4. Groups of diffeomorphisms. The hydrodynamical Euler equation.
- The Lie group of volume-preserving di_eomorphisms and its Lie algebra.
- Coadjoint action and Casimirs.
III. Applications to PDEs in geometric fluid dynamics (time permitted).
- Ideal hydrodynamics and optimal mass transport. Otto's calculus.
- Compressible fluid dynamics.
- Dynamics of point vortices.
- Structures on and dynamics of vortex sheets.
References:
1. B. Khesin and R. Wendt \The geometry of infinite-dimensional groups,"
Ergebnisse der Mathematik und Grenzgebiete 3.Folge, 51, Springer-Verlag (2008),
xviii+304pp, see https://www.math.utoronto.ca/khesin/papers/Lecture_notes.pdf 2. A. Pressley and G. Segal: \Loop Groups," Clarendon Press, Oxford (1986)
Prerequisites:
A basic course (or familiarity with main notions) of symplectic geometry would be helpful.
MAT1191HF
Topics in Algebraic Geometry: Toric Geometry and Newton Polyhedra
A. Khovanskii
(View Timetable)
Newton polyhedron is a geometric generalization of the degree of a polynomial. Newton polyhedra connect the theory of convex bodies with algebraic geometry of toric varieties. For example, Dehn–Zommerville duality for simple polyhedra leads to the computation of the cohomology rings of smooth toric varieties. BKK (Bernstein– Kouchnirenko–Khovanskii) theorem computes the number of solutions of generic systems of algebraic equations via mixed volume of convex polyhedra. It suggested so-called Khovanskii–Tessier inequalities in algebraic geometry, which imply the famous Aleksandrov–Fenshel inequalities and their generalizations in the the theory of mixed volumes. Recently Join Hu won the Fields medal for his brilliant discoveries related to such inequalities. Toric geometry provides a visual version of algebraic geometry which is easier to learn. I will try to present the material in the most understandable way.
Grading: one final presentation or written report.
Prerequisites: MAT247H1/MAT347Y1, MAT401H1, MAT354H1
MAT1192HS
Advanced Topics in Algebraic Geometry: p-adic Motives
This is a graduate course that will revolve around Geisser-Levine's proof of the mod p versions of the Beilinson-Lichtenbaum conjectures. One concrete way of stating is that the mod-p K-theory of a characteristic p field is generated in degree 1 with relations only in degree 2. This has several important implications in the theory of algebraic cycles and motives. The proof also involves a host of modern characters in p-adic arithmetic geometry, which I will make heavy digressions into.
The class will roughly go as follows:
- background on motivic complexes, crystalline cohomology and Hodge-Witt cohomology,
- background on algebraic and Milnor K-theory,
- the language of motivic homotopy theory,
- the homotopy coniveau tower and effective motives,
- the Bloch-Kato-Gabber theorem and
- proof of the Geisser-Levine theorem.
Literature
1. J.Milnor. Topology from the Differentiable Viewpoint.
2. Yu. Burda, and A. Khovanskii. Degree of rational mapping, and the theorems of Sturm and Tarski. Journal of Fixed Point Theory and Applications. Vol. 3, No. 1, 2008, 79–93.
3. Topology and Intersection Theory of Divisors. Handout.
MAT1210HS
Topics in Number Theory: Shimura curves, geometry and arithmetic
Shimura curves CB are generalizations of modular curves. They are among the most basic examples of Shimura varieties and they have a rich and beautiful structure. Over the complex numbers, they arise as quotients of the upper half plane by discrete sub-groups of SL2(R) coming from the unit groups O_B of maximal orders OB in quaternion algebras B. In the case of quaternion algebras over Q, they are moduli spaces of abelian surfaces with an action of OB and this leads to models both over Q and to integral models over Z. The arithmetic surfaces over Spec Z arising in this way are fascinating
objects. In particular, (formal neighborhoods of) their _bers at primes of bad reduction have a p-adic uniformization. Aside from its intrinsic interest, in depth study of Shimura curves provides a valuable model for the study of more general Shimura varieties, a _eld of great current importance.
Topics may include:
complex theory: Quaternion algebras B over Q, maximal orders OB, type numbers, class numbers and strong approximation, embedded quadratic sub_elds, Fuchsian groups O_B, elliptic _xed points, genus formula
moduli theory/C: Abelian surfaces with OB action, CB as moduli space, special points/CM-points
moduli theory over Z: Integral model over Z (Drinfeld), primes of good reduction: p - D(B), primes of bad reduction: p j D(B)
p-adic uniformization: Drinfeld space b, p-adic uniformization, special cycles
Shimura curves over totally real _elds: Exotic canonical models
Prerequisites:
Basic algebraic geometry, algebraic number theory including ideles and
ad_eles, and complex analysis. Some knowledge of schemes and abelian varieties would
be useful for the later parts of the course.
References:
M. Eichler, Lectures on Modular Correspondences, Tata Institute of Fun-
damental Research, Bombay 1957.
M.-F. Vigneras, Arithm_etique des alg_ebras de quaternions, Lecture Notes in Math. 800,
Springer, 1980.
S. Kudla, M. Rapoport, and T. Yang, Modular Forms and Special Cycles on Shimura
Curves, Annals of Math. Studies 161, Princeton U. Press, 2006.
MAT1128HF
Topics in Probability: Gaussian random measures
Random measures based on Gaussian random variables appear in modern probability, statistical physics, and statistics. This course will provide an introduction.
MAT1210HF
Topics in Number Theory: Class Field Theory
This course will be a continuation of MAT415/1200. It will give an introduction to class field theory, the study of abelian extensions of number fields and p-adic fields, focusing on statements and examples such as the Kronecker-Weber Theorem. At the beginning, we will review inertia groups and decomposition groups, and we will use that foundation to introduce the Galois theory of local fields.
Main Textbook: Janusz - Algebraic Number Fields
MAT1304HF
Topics in Combinatorics: Extremal combinatorics
L. Gishboliner
(View Timetable)
Covering both basics and more advanced topics in Ramsey theory and extremal graph theory. The course "Introduction to combinatorics" would be a prerequisite.
A rough syllabus is:
- Ramsey's theorem and probabilistic lower bounds.
- The Erdos-Szekeres lemma, cups-caps theorem.
- Theorems of Hales-Jewett and Van der Waerden.
- Hypergraph Ramsey numbers, the stepping-up lemma.
- Turan's theorem and the Erdos-Stone theorem.
- The Kovari-Sos-Turan theorem and its generalization to hypergraphs, supersaturation.
- Extremal numbers of trees and cycles, algebraic constructions.
- The Ajtai-Komlos-Szemeredi theorem.
- Lower bounds for Ramsey numbers via the Lovasz local lemma.
- Dependent random choice.
- Ramsey numbers of bounded-degree graphs.
- The Szemeredi regularity lemma and the removal lemma (if time permitting).
MAT1305HF
Topics in Geometric Topology: Geometry, Arithmetic, and Dynamics of Discrete Groups
Modern research in the geometry, topology, and group theory often combines geometric, arithmetic and dynamical aspects of discrete groups. This course is mostly devoted to hyperbolic manifolds and orbifolds, but also will deal with the general theory of discrete subgroups of Lie groups and arithmetic groups. Vinberg’s theory of hyperbolic reflection groups will also be discussed, as it provides a lot of interesting examples and methods which turn out to be useful for different purposes. One of the goals of this course is to sketch the proof of the famous Mostow rigidity theorem via ergodic methods. In conclusion, I am going to talk about very recent results giving a geometric characterization of arithmetic hyperbolic manifolds through their totally geodesic subspaces.
MAT1351HF
Topics in Homotopy Theory: Rational homotopy theory with geometric applications
F. Manin
(View Timetable)
Rational homotopy theory is a set of algebraic tools used to study simply connected spaces -- in fact, it gives a complete and relatively straightforward algebraic description of such spaces if one is willing to disregard finite order information, or rather several such descriptions. I will start by introducing prerequisites not covered in the core algebraic topology course: fibrations, Postnikov systems, the Serre spectral sequence and obstruction theory. This will allow us to introduce Sullivan's model of rational homotopy theory and finally discuss applications in topology and geometry of manifolds. Classical applications include:
- All but a few Riemannian manifolds have infinitely many distinct closed geodesics (Vigué-Poirrier and Sullivan).
- The homotopy automorphism group of a compact simply connected space is an arithmetic group (Sullivan).
We may focus on different aspects depending on time and student interests, including recent results.
Prerequisites:
core courses in algebraic topology and Riemannian geometry
MAT1344HF
Introduction to Symplectic Geometry
This is an introductory course on symplectic geometry. The material that we plan to cover includes:
- Linear symplectic algebra.
- Symplectic manifolds
- Normal form theorems
- Lie groups, Hamiltonian group actions
- Moment maps, Symplectic reductions
- The Atiyah-Guillemin-Sternberg convexity theorem
- Equivariant cohomology and Duistermaat-Heckman theory
- Quasi-hamiltonian group actions
- Moduli spaces of flat connections
Prerequisites:
Knowledge of manifolds, vector fields and differential forms.
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: Microlocal sheaf theory and symplectic topology
Symplectic geometry is the natural geometry governing classical mechanics. Beginning in the 1980s, this subject was revolutionized by the introduction of new ideas arising from counting solutions of certain natural partial differential equations. This has led to a resolution of fundamental conjectures in the subject and to remarkable applications in low dimensional topology and algebraic geometry. More recently, a more topological approach has emerged from the study of derived categories of sheaves on manifolds. This course will be an introduction to the latter approach. The first part of the course will develop the basics of sheaf theory on manifolds and the second part will be devoted to applications of this theory to symplectic topology, such as the Gromov non-squeezing theorem and non-displaceablility results. Only basic knowledge of manifolds and familiarity with homological algebra will be assumed; in particular, we will not assume prior experience with sheaves or symplectic topology.
References:
Kashiwara and Schapira, Sheaves on Manifolds.
Tamarkin, Microlocal condition for non-displaceablility.
Zhang, Quantitative Tamarkin category.
Guillermou, Sheaves and symplectic geometry of cotangent bundles.
MAT1347HF
Topics in Symplectic Geometry and Topology: Morse homology, Floer homology, and Fukaya categories
The class will start by teaching Morse theory, using the Morse/Smale/Witten picture (the closest version to Floer homology). Other than finite dimensional manifold there are some nice applications to loop spaces (Milnor's book).
For Floer theory, the plan is to focus on Lagrangian Floer homology, though Hamiltonian Floer homology will also be covered. The focus would be on the proof of various Arnol'd conjectures, such as the Hamiltonian version on a closed manifold and the Lagrangian version in a cotangent bundle. We'll also distinguish Lagrangians from each other, for instance in CP^2.
Finally, we'll see how Fukaya categories come into the picture, by being the category whose Hom spaces are Floer homology groups. Here the focus moves to the products, higher structure maps, and algebra. The class might focus either on the compact version or the wrapped version: wrapped Fukaya categories require a bit more setup, but they're also hot at the moment and can prove nicer results. Fukaya-Seidel-Smith theorem would be an excellent stopping point, but if it ends up being too ambitious even the computation of some mirror symmetry/loop space examples would make for a satisfying conclusion.
MAT1435HS
Topics in Set Theory: Forcing and its Applications
This will course on the set-theoretic forcing techniques concentrated on its applications to other fields of mathematics following the trend of the currently running semester-long program at the Fields Institute.
Forcing and the corresponding forcing axioms will be introduced in great details.
When applying forcing axioms to other areas of mathematics in most cases we will introduce the corresponding combinatorial dichotomies that are more natural and easier to apply. We will also stress that many of these applications have deep counterparts that are equally interesting and require no additional set theoretic axioms, i.e., mathematical theorems that would have difficult to find without the detour though forcing.
The following textbooks will be useful:
- K. Kunen, Set theory: An Introduction To Independence Proofs, 1980 or 2011 edition.
- S. Todorcevic, Notes on Forcing Axiom, World Scientific, 2014.
While the first part will concentrate to the more basic part of the theory of Forcing accessible to the first year graduate students or bright undergraduates, the second part will present the side condition method in the realm of Forcing Axioms bringing us closer to the current research in this area.
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.
MAT1525HS
Topics in Inverse Problems and Image Analysis: Variational Methods in Imaging and Generative Neural Networks
This course aims to provide the analytic tools for understanding some spectacularly successful generative neural networks from a mathematical point of view. Examples will include Wasserstein GANs and score-based diffusion models. We will spend much of the time on the requisite background from convex analysis, paradigmatic variational problems from image analysis, basics of optimal transport theory, stochastic differential equations and gradient flows in metric spaces.
Prerequisites:
Basic real analysis and functional analysis will be helpful. Evaluation: 20% attendance, 40% oral presentation of your project, 40% written report of the project.
Recommended References:
“Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling” by F. Santambrogio, Birkhauser, 2015.
“Gradient flows in Metric Spaces and in the Space of Probability Measures”, by L. Ambrosio, N. Gigli and G. Savare, Birkhauser, 2008.
“Wasserstein GANs with Gradient Penalty Compute Congested Transport” by T. Milne and A. Nachman, PMLR, pp. 103–129, 2022.
“Diffusion Models: A Comprehensive Survey of Methods and Applications”, by L. Yang, Z. Zhang, Y. Song, S. Hong, R. Xu, Y. Zhao, Y. Shao, W. Zhang, B. Cui, and M.-H. Yang, ACM Computing Surveys (56) pp 1–39, (2023).
“Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance”, by H. Wu, J. Kohler, and F. Noe”, Advances in Neural Information Processing Systems, (2022).
MAT1800HF
Methods of Applied Mathematics
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.
MAT1855HF
Mathematical Problems in Economics
This course surveys a number of economic topics of current research interest in which mathematical developments have (and are expected to continue to) contribute crucial advances. These include the theory of matching and pricing, problems of asymmetric information, the principalagent framework, auction theory, mechanism (and information) design, portfolio optimization and hedging. These topics are partly unified through mathematical techniques such as linear programming (optimal transport and its emerging relevance figure prominently — think of trying to pair N workers with N firms so as to maximize the total surplus), nonsmooth analysis, the calculus of variations, and differential equations. We may also consider topics such as matching with unobservable heterogeneity and/or imperfectly transferable utility, and equilibria involving agents who respond nonlinearly to prices, which go beyond the variational framework.
The necessary mathematics (beyond measure theory and integration) will be developed in parallel with the applications, as well as any necessary background in economics.
Recommended texts:
Basov. Multidimensional Screening. Springer, 2005.
Chiappori. Matching with Transfers: The Economics of Love and Marriage. Princeton, 2017.
Galichon. Optimal Transport Methods in Economics. Princeton, 2016.
Henry-Labordère. Model-free Hedging: A Martingale Optimal Transport Viewpoint. CRC Press 2017.
Mas-Colell, Whinston and Green. Microeconomic Theory. Oxford, 1995.
Sotomayor and Roth. Two-Sided Matching: a Study in Game-Theoretic Modeling and Analysis. Cambridge Press, 1992.
Santambrogio Optimal Transport for Applied Mathematicians. Birkhauser 2015.
Vohra. Mechanism Design: A Linear Programming Approach. Cambridge 2011.
MAT1845HF
Dynamical Systems: Introduction to ergodic theory
The course will be a graduate-level introduction to Ergodic Theory. The aim is to gain a qualitative understanding of the statistical properties of some fairly elementary, usually discrete time, chaotic dynamical systems. We will bring some motivation from physics, tools, and a general perspective from probability/measure theory. I plan to cover a very brief history of ergodic theory; ergodicity, the ergodic theorem, (measure-theoretic) mixing; unique ergodic examples and Fustemberg (counter) example; metric entropy and
information; Markov chains.
Depending on time and interest, we may cover some of the following topics: decay of correlations and the central limit theorem; thermodynamic formalism; links with number theory; basic infinite ergodic theory.
MAT1845HS
Dynamical Systems: Topics in arithmetic dynamics
Various problems in diophantine geometry can be cast in a dynamical language and thereby more general conjectures may be formulated.
A key example that will feature in this course is the Manin-Mamford conjecture, generalized by Zhang in the dynamical setting. Though this conjecture is still open, special cases have been established. We will explore some such results and their proofs. This will lead us to studying arithmetic height functions and also complex dynamics.
We will also discuss relative versions of the Manin-Mumford conjecture in families of rational maps, with emphasis on work of Baker and DeMarco. We will learn about Silverman's specialization result and see how it relates to stability of a `dynamical pair'.
The course will be based on recent papers in the young and thriving field which has many open questions.
Based on the participants' choices, we may decide on additional topics to cover. I encourage you to meet me individually, so that I can learn more about your background and what you would like to learn.
Useful references and textbooks
John Milnor, Dynamics in One Complex Variable, Third Edition
Mark Hindry and Joseph Silverman, Diophantine geometry, an introduction
Joseph Silverman, The Arithmetic of Dynamical Systems
Enrico Bombieri and Walter Gubler, Heights in Diophantine Geometry