2021-2022 Graduate Courses Descriptions

 

Core Graduate Courses | Cross-Listed and Topics Courses



Core Courses

MAT1000HF (MAT 457H1F)
REAL ANALYSIS
I
Y. Shlapentokh-Rothman 
(View Timetable)

Measure Theory: Lebesgue 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 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.

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MAT1001HS (MAT 458H1S)
REAL ANALYSIS II
A. Burchard
(View Timetable)

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 self-adjoint 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 (MAT 454H1S)
COMPLEX
ANALYSIS
M. Goldstein
(View Timetable)

 

  1. Review of holomorphic and harmonic functions (Chapters 1-4 in Ahlfors).
  2. Topology of a space of holomorphic functions: Series and infinite products, Weierstrass p-function, Weierstrass and Mittag-Leffler theorems.
  3. Normal families: Normal families and equicontinuity, theorems of Montel and Picard.
  4. Conformal mappings: Riemann mapping theorem, Schwarz-Christoffel formula.
  5. Riemann surfaces: Riemann surface associated with an elliptic curve, inversion of an elliptic integral, Abel’s theorem.
  6. 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, McGraw-Hill

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
R. McCann
(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.

Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743

References:

  • R. McOwen, Partial Differential Equations, (2nd ed),
    Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,
    Paperback: 2002 Pearson ISBN-13 978-0130093356
  • Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2

 

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MAT1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
I. M. Sigal
(View Timetable)

This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, and fixed point theorems.  One important theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.

The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.

Other topics in PDE will also be discussed.

Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743

Reference:

  • R. McOwen, Partial Differential Equations, (2nd ed),
    Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,
    Paperback: 2002 Pearson ISBN-13 978-0130093356
  • Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2

 

MAT1100HF
ALGEBRA I
I. Varma
(View Timetable)

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

Group Theory: Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.

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

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

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

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

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

 

MAT1101HS
ALGEBRA II
F. Herzig
(View Timetable)

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

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

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 
Y. Liokumovich
(View Timetable)

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

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

Prerequisites:  linear algebra; vector calculus; point set topology

Textbook:  
Differential Topology, Victor Guillemin and Alan Pollack,
American Mathematical Society ISBN-10: 0821851934, ISBN-13: 978-0821851937

MAT1301HS
ALGEBRAIC TOPOLOGY 
V. Kapovitch
(View Timetable)

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

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

Textbook:  
Allen Hatcher, Algebraic Topology

Recommended Textbooks:
Munkres, Topology
Munkres, Algebraic Topology

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MAT1600HF
MATHEMATICAL PROBABILITY I
D. Dauvergne
(View Timetable)

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

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MAT1601HS
MATHEMATICAL PROBABILITY II
B. Landon
(View Timetable)

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
(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 COURSES 

 

MAT1011HF/MAT436H1F
INTRODUCTION TO LINEAR OPERATORS
G. A. Elliott
(View Timetable)

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

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

Textbook:
Gert K. Pedersen, Analysis Now

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

 

 

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MAT1016HS/MAT437H1S
TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS
G. A. Elliott
(View Timetable)

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

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

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

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

Textbook:
Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras

Recommended References:
Edward G. Effros, Dimensions and C*-algebras
Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras
Kenneth R. Davidson, C*-Algebras by Example

 

MAT1155HS/MAT448H1S
COMMUTATIVE ALGEBRA
E. Bierstone
(View Timetable) 

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. 

 

 

MAT1202HF/MAT417H1F
ANALYTIC NUMBER THEORY 
H. Kim
(View Timetable)   

1. Introduction to some of the famous problems and theorems of the subject
2. Simple tools from elementary number theory, algebra and analysis
3. Dirichlet’s theorem on primes in arithmetic progressions
4. Prime Number Theorem
5. Prime number theorem for arithmetic progressions
6. An introduction to sieve methods
7. A selection, if time permits, of some subset of the following topics:
a. further zeta-function theory
b. L-functions and character sums
c. exponential sums and uniform distribution
d. Hardy-Littlewood-Ramanujan method
e. further theory of prime distribution
Prerequisites:
1. A half-year course in complex variables such as MAT 334. (This is the most important prerequisite.)
2. A course in groups, rings, fields, such as MAT 347.
3. A half year course in introductory number theory such as MAT 315.
4. A commitment to attend all lectures.

References:
A) General Analytic Number Theory:

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

B) More specialized texts:

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

 

 

MAT1302HS/APM461H1S
COMBINATORIAL THEORY
S. Kopparty
(View Timetable)

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: ALGEBRAIC GEMS IN THEORETICAL COMPUTER SCIENCE AND DISCRETE MATHEMATICS
S. Saraf
(View Timetable)

In the last few decades, algebraic methods have proven to be extremely powerful in several areas of computer science. In this course we will see some of the  important advances in theoretical computer science that have relied on very simple properties of polynomials. In particular, we will see some interesting and  surprising applications of linear algebra and polynomials to complexity theory, cryptography, algorithm design and combinatorics. We will develop all the algebraic tools that we need along the way.

A tentative list of topics that will be covered includes design of algebraic algorithms, applications of polynomials in cryptography and complexity theory, design and analysis of error correcting codes and other pseudorandom objects, and algebraic circuit complexity.

 

 

 

MAT1342HF/MAT464H1F
INTRODUCTION TO DIFFERENTIAL GEOMETRY: RIEMANNIAN GEOMETRY
R. Rotman
(View Timetable)

The topics include:
Riemannian metrics, Levi-Civita connection, geodesics, isometric embeddings and the Gauss formula, complete manifolds, variation of energy.

It will cover chapters 0-9 of the "Riemannian Geometry" book by Do Carmo.

 

 

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

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

Prerequisite: 
MAT357H1

Textbooks:
http://www.math.toronto.edu/weiss/set_theory.html

 

 

MAT1507HF/APM441H1F
ASYMPTOTIC METHODS AND PERTURBATION
C. Sulem
(View Timetable)

1. Local Methods
Classification of regular/ singular points of linear ODEs Approximate solutions near regular, regular-singular irregular singular points, irregular point at infinity Asymptotic series. Some examples of nonlinear differential equations.
2. Asymptotic expansion of integrals
Laplace method Method of stationary phase Steepest descent
3. Perturbation methods
Regular and singular perturbation theory
4. Global Analysis
Boundary layer theory
WKB theory : Formal expansion, conditions for validity, geometrical optics
Multiple scale analysis for ODEs: Resonance and secular behavior, damped oscillator
Multiple scale analysis for PDEs

Prerequisites:
2nd year calculus
ODEs and PDE courses
Complex variables

 

 

MAT1508HS/APM446H1S
TECH OF APPLIED MATH: APPLIED NONLINEAR EQUATIONS
J. Arbunich
(View Timetable)

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

Specifically we consider Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics -superfluidity and superconductivity ), Cahn-Hilliard (material science, biology), Mean curvature flow and the equation for minimal and self-similar surfaces (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), Gross-Pitaevskii equation (Bose-Einstein condensation) and Chern-Simmons equations (particle physics and quantum Hall effect).

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

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 Reaction-diffusion Equations (Oxford Applied Mathematics & Computing Science) 1996

 

MAT1723HF/APM421H1F
FOUNDATIONS OF QUANTUM MECHANICS
M. Sigal
(View Timetable)

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 self-contained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively time-consuming 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 on-line 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

 

 

 

 

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MAT1840HF/MAT482H1F
CONTROL THEORY

B. Khesin
(View Timetable)

The course focuses on the key notions of Calculus of Variations and Optimal Control Theory: key examples of variational problems, (un)constrained optimization, first- and second-order conditions, Euler-Lagrange equation, variational problems with constraints, examples of control systems, the maximum principle, the Hamilton-Jacobi-Bellman equation (time permitting), holonomic and nonholonomic constraints, Frobenius theorem, Riemannian and sub-Riemannian geodesics.

Prerequisites:
Courses of multivariable calculus, ordinary differential equations,
and linear algebra.

Textbooks:
1) D. Liberzon "Calculus of Variations and Optimal Control Theory: A Concise Introduction'' 2012, Princeton Univ. Press (chapters 1,2,3 and partially chapters 4,5,7)
2) A. Agrachev, D. Barilari, and U. Boscain "Introduction to Riemannian and Sub-Riemannian geometry'' 2018, e-preprint (chapter 2 and partially chapters 1,3)

Detailed Course Outline:
(The following is a tentative outline of the material to be covered.)

I. Introduction: examples, (un)constrained optimization, Lagrange
multipliers first and second variations.

II. Calculus of variations: examples (Dido problem, catenary, brachistochrone), weak and strong extrema, Euler-Lagrange equation, introduction to Hamiltonian formalism, integral and non-integral constraints.

III. From calculus of variations to optimal control: control system, cost functional, target set.

IV. The maximum principle: statement, ideas of proof, weak form, examples.

V. The Hamilton-Jacobi-Bellman equation: discrete problems, the principle of optimality.

VI. Lie brackets of vector fields, Frobenius theorem, nonholonomic constraints, examples (ball rolling, car parking), Chow-Rashevskii theorem, sub-Riemannian metrics.

 

 

 

MAT1856HS (APM466H1S)
MATHEMATICAL THEORY OF FINANCE
L. Seco
(View Timetable)

Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.

Prerequisites
APM 346H1, STA 347H1

 


 

TOPICS COURSES

 

 

MAT1005HF
FOURIER ANALYSIS 
I. Uriarte-Tuero
(View Timetable)  

 

Fourier series and integrals: pointwise convergence, convergence in norm. Maximal function: approximations of the identity, Marcinkiewicz interpolation, weak (1,1) inequality for the maximal function. Hilbert transform: Lp bounds and weak type bounds, truncated integrals and pointwise convergence, multipliers. Calder'on-Zygmund operators: operators with bounded Fourier transform of the kernel, the method of rotations, singular integrals with even kernel, the Calder’on-Zygmund theorem, truncated integrals and the principal value.

 

Time permitting, H^1 and BMO: the spaces H^1 and BMO, interpolation, the John-Nirenberg lemma.

 

 

  

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MAT1045HF
TOPICS IN ERGODIC THEORY 
G. Tiozzo
(View Timetable)  

 

This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups.

Overview.

We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(FN ).

Prerequisites:

An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichm¨uller theory is needed.

 

MAT1062HF
TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS I: INTRODUCTION TO NONLINEAR EVOLUTION PDES
F. Pusateri
(View Timetable)

 

  • Review/Basics (Week 1-2) (T)

ODE material

Basics of Fourier analysis and Functional Analysis

Basics of Littlewood-Paley theory

  • Linear equations (Week 3-5) (T+N)

Schrodinger, Airy, Wave, Water Waves

Decay estimates, Stationary phase

Strichartz estimates

  • Nonlinear Schrodinger (NLS) equation (Week 6-8) (T+C+N)

Local existence

Global existence for subcritical NLS

Cubic NLS in 2d (small data)

Cubic NLS in 1d (small data, long time asymptotics)

  • Nonlinear wave equations (Weeks 9-11) (S)

Local existence and blowup for low power nonlinearities

Some Theorems on global existence (3d energy critical small data, quadratic with null condition)

  • Additional topics:
    Nonlinear Resonances
    Normal forms for PDEs
    Multilinear Harmonic analysis with applications.

 

MAT1064HS
ELLIPTIC BOUNDARY VALUE PROBLEMS ON NONSMOOTH DOMAINS
K. Serkh
(View Timetable)

The behavior of the solutions of elliptic boundary value problems on domains with corners, edges, and conical points is a subject of major practical importance, with applications to aerodynamics, hydrodynamics, electromagnetics, elasticity, fracture mechanics, etc. The purpose of this course is to provide an introduction to the theory of linear elliptic boundary value problems on nonsmooth domains, both the classical approach, and the various contemporary approaches using function spaces. More concretely, we will be concerned with the following problem.

The shift theorem states that, when the Dirichlet problem for Poisson's equation \nabla^2 u = f is solved on a two-dimensional region with a smooth boundary, and f belongs to the Sobolev space H^{s}, the solution u belongs to H^{s+2}. When the region has a corner, this shift theorem no longer holds in general, and only holds for certain ranges of s. It turns out that the presence of a corner results in singularities in the solution u, making it much less regular than it would be otherwise. For the result to be in H^{s+2} for general values of s, a certain set of singular functions must be subtracted from u. This course will answer the following questions, for a general elliptic operator L: H^{s+2} \to H^{s} on a nonsmooth domain. For which values of s is the shift theorem true? When is L Fredholm? When is it semi-Fredholm? What are the singular functions that need to be subtracted for the shift theorem to hold in general? Under which assumptions on f does the solution belong to H^{s+2}? Under which assumptions on the boundary values does the solution belong to H^{s+2}? What happens near corners and edges in higher dimensions? What are the asymptotic expansions of the solutions near corners and edges?

References:

Grisvard, P. Elliptic Problems in Nonsmooth Domains. Pittman Publishing Inc., 1985.
Dauge, M. Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions. Springer-Verlag, 1980.

 

 

 

MAT1104HF
TOPICS IN ALGEBRA II: TOPOLOGICAL GALOIS THOERY SOLVABILITY AND NONSOLVABILITY OF EQUATIONS IN FINITE TERMS
A. Khovanskii
(View Timetable)

Numerous unsuccessful attempts to solve a series of algebraic and differential equations “in explicit form” led mathematicians to belief that explicit solutions for these equations simple do not exist. Galois theory explains why algebraic equation is usually not solvable by means of radicals It belongs to algebra. It is very understandable and has a lot of other applications. Liouville theory explains why integral of elementary function usually is not an elementary function. Liouville theory is very elegant and understandable.

About 50 years ago, I started to develop a topological version of Galois theory for functions in one complex variable. According to it, there are topological restrictions on the way the Riemann surface of a function representable by radicals or by quadratures covers the complex plane. If the function does not satisfy these restrictions, then it is not representable by radicals or by quadratures. Beside its geometric clarity the topological results on nonsolvability are stronger than the algebraic results. 

In this course, I plan to present Liouville Theory and Galois Theory in detail and to discuss topological results on nonsolvability by radical and by quadratures. I expect some basic knowledge in complex analysis and in elementary algebra.

References:

1) Askold Khovanskii. Topological Galois Theory.
2) A few original papers will be suggested as additional material.

 

 

MAT1191HF 
TOPICS IN ALGEBRAIC GEOMETRY: ALGEBRAIC GROUPS
S. Kudla
(View Timetable)

Algebraic groups are group objects in the category of algebraic varieties or, in more down to earth language, groups define by polynomial equations. They arise naturally and play an important role in many areas of mathematics. This course will provide a basic introduction to the theory of linear algebraic groups. Of particular interest will be the reductive algebraic groups, as these are an essential part of the foundations of the theory of automorphic representations and the Langlands program.

Topics may include:

  • Background form algebraic geometry, a review
  • Linear algebraic groups: basic structures, Lie algebras
  • Borel's fixed point theorem, parabolic subgroups
  • Weyl group, roots and root datum
  • Classiffication, presentation of G, isogenies, existence theorem
  • Galois cohomology, F-groups,
  • The root datum of an F-reductive group
  • The Langlands dual group
  • Examples, classical groups, G2, and others

Prerequisites:

Basic algebraic geometry and basic scheme theory, e.g. first 2 chapters of Hartshorne.

Reference:

T. A. Springer, Linear Algebraic Groups, 2nd edition.

 

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MAT1191HS 
TOPICS IN ALGEBRAIC GEOMETRY: SEMI-SIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
A. Braverman
(View Timetable)

Introduction to Lie algebras, nilpotent, solvable and semi-simple Lie algebras over complex numbers. Classification of simple Lie algebras and their finite-dimensional representations.

Verma modules and Weyl character formula. If time permits (which is unlikely) we might include some discussion of semi-simple Lie algebras over fields of positive characteristic.

 

MAT1192HS
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY: DIOPHANTINE GEOMETRY
V. Dimitrov
(View Timetable) 

Weil height functions. Roth's and Schmidt's theorems on Diophantine Approximations in projective space. The unit equation in two and several variables. Small points and the Lehmer problem. Abelian varieties and the theorem of Mordell-Weil. Faltings's theorem and some more recent developments. The abc conjecture and Vojta's conjectures.

Textbook:
E. Bombieri and W. Gubler: Heights in Diophantine Geometry (Cambridge New Monographs in Mathematics, 2006).

Course prerequisite:
A graduate-level understanding of abstract algebra and of the rudiments of algebraic number theory.



 

MAT1210HS
TOPICS IN NUMBER THEORY: INTRODUCTION TO L-FUNCTIONS
H. Kim
(View Timetable)

 

An L-function is a type of generating function formed out of local data associated with either an arithmetic-geometric object (such as elliptic curves and Shimura varieties) or with an automorphic form. When an L-function is associated with an automorphic form, we call it automorphic L-function. L-functions play a central role in modern number theory. In this introductory course, we will study various L-functions and their properties, and their applications in number theory. Students will be expected to do final projects and give presentation.

Topics to be covered:
1. Riemann zeta function and prime number theorem
2. Dirichlet L-function and Dirichlet's theorem on arithmetic progression
3. Dedekind zeta function and Hecke's L-function
4. Tate's thesis (adelic treatment of Hecke's L-function)
5. Modular L-functions
6. Low-lying zeros
7. Chebotarev density theorem

Prerequisite:

Complex analysis, some algebraic number theory


 

 

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MAT1306HS 
THE DISCRETE MATHEMATICS TOOLKIT
S. Kopparty
(View Timetable)

We will learn tools to study basic topics in discrete mathematics, including enumeration, symmetry, extremal combinatorics, set systems, Ramsey theory, discrepancy, additive combinatorics and quasirandomness. There will be emphasis on general techniques, including probabilistic methods, linear-algebra methods, analytic methods, topological methods and geometric methods. Applications to theoretical computer science will be discussed throughout.

 

MAT1309HF
GEOMETRIC INEQUALITIES
A. Nabutovsky
(View Timetable)

Isoperimetric inequality; Various generalizations of the isoperimetric inequality and related inequalities; Brunn-Minkowski and Alexandrov-Fenchel inequalities; Sobolev inequalities and some of their applications; Besicovitch inequality;  Introduction to systolic geometry; Gromov's systolic inequality; Complexity of optimal slicings and sweep-outs; Geometric inequalities for the lengths of shortest periodic geodesics, shortest geodesic loops, etc. 

 

Most of the course will be non-technical 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" and supplement it by a few expository and research papers.

 

 

 

MAT1314HS
INTRODUCTION TO NONCOMMUTATIVE GEOMETRY
G. Elliott
(View Timetable)

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

Prerequisite:
The Spectral theorem.

References:
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.

 

MAT1341HS
TOPICS IN DIFFERENTIAL GEOMETRY: POISSON GEOMETRY AND LIE ALGEBROIDS
E. Meinrenken
(View Timetable)

 

This course is an introduction to the theory of Lie algebroids and their applications to Poisson geometry. One focus of the course will be the problem of integration of Poisson manifolds to symplectic groupoids.

Topics covered:

Poisson brackets and Poisson bivector fields
Symplectic foliation
Weinstein splitting theorem
Lie algebroids and Lie groupoids
Symplectic groupoids
Poisson Lie groups and Poisson actions

Prerequisites:

Topology I (Manifolds) and basic knowledge of symplectic geometry. Knowledge of Lie theory is helpful but not required.

 

MAT1350HF
TOPICS IN ALGEBRAIC TOPOLOGY I: ALGEBRAIC KNOT THEORY AND COMPUTATION
D. Bar-Natan
(View Timetable)

 

 

The destination will be "a poly-time computable knot invariant with good algebraic properties". But you will be taking the course for the journey, not for the destination: What are knots and what are some of the problems around them? Why care about "invariants with good algebraic properties"? What is the "Yang-Baxter equation"? What are "virtual tangles"? What are "Hopf algebras"? Why would a topologist care about computations in Heisenberg algebras more than most physicists? How does Gaussian integration, and how do Feynman diagrams, arise in pure algebra? What is the "Drinfel'd Double Procedure"? Are we there yet?

The professor for this class does not believe anything that he does unless it is coded and the code runs. A useful life skill you will learn here is that even the incredibly abstract can become a computer program, often with no loss to its beauty.

 

 

 

MAT1351HF
TOPICS IN HOMOTOPY: HIGHER CATEGORIES WITH APPLICATIONS
A. Kupers
(View Timetable)

 

In this course we will learn the foundations of higher category theory, or more precisely ∞-categories. This is a technical subject that lies at the heart of recent results in homotopy, algebraic geometry, and number theory. Our approach will be through Joyal–Lurie’s quasicategories, though we will also discuss some alternative models. The end-goal of this course will be to develop the tools required to construct and study the stable ∞-category of spectra. Spectra are the objects of study in stable homotopy theory, and classical approaches to spectra have to contend with difficult coherence issues that are avoided in the higher-categorical approach. The topics we will discuss include: quasicategories as ∞-categories, complete Segal spaces as ∞-categories, comma ∞-categories, (co)limits, (co)cartesian fibrations, adjunctions, stable ∞-categories, t-structures, symmetric monoidal ∞-categories, the stable homotopy category, the smash product, commutative ring spectra, the Dold–Kan correspondence.

 

MAT1435HF
TOPICS IN SET THEORY: GEOMETRICAL PARADOXES AND RAMSEY THEORY
S. Unger
(View Timetable)

The goal of the course is to present measurable solutions to problems in classical geometry and Ramsey theory and their connection with current research.

From classical geometry, we consider problems about breaking subsets of Euclidean space into congruent pieces.  We will start by giving a proof of the Banach-Tarski-Hausdorff paradox and explaining the role of the axiom of choice and the sense in which the pieces of the paradox are non-measurable.  We will then prove two different measurable versions of the paradox.

From Ramsey theory, we will focus on colorings of subsets of the natural numbers.  After proving the basic finite and infinite Ramsey theorems, we will show that even a weak version of the axiom of choice implies the failure of a natural 'infinite dimensional' Ramsey statement.  We will then characterize the sets for which this infinite dimensional Ramsey statement holds using an appropriate notion of measurability.

In many cases, necessary background will be developed during the course. In particular, we will present some material from both set theory and descriptive set theory.  We will assume some knowledge of combinatorics, analysis/topology and algebra.

 

 

 

MAT1435HS
TOPICS IN SET THEORY 
S. Todorcevic
(View Timetable)

 

Axiomatic approach to higher dimensional Ramsey theory with and without the availability of a pigeonhole principle in dimension one. The corresponding theory of Ramsey space will be presented with a special attention to its applications to functional analysis and topological dynamics.

 

MAT1502HF
TOPICS IN GEOMETRIC ANALYSIS: SCHRAMM LOEWNER EVOLUTION AND LATTICE MODELS
I. Binder
(View Timetable)
(View Fields Program Details)

In recent years, significant progress has been obtained in the rigorous understanding of the scaling limits of the various lattice models of statistical physics. One of the instrumental tools in this development is the Schramm Loewner Evolution (SLE), invented by Oded Schramm in 1998.

The course will introduce the students to these developments. The topics will include the definition and geometric properties of SLE, including the necessary background in Geometric Function Theory; basic properties of the lattice models, such as Percolation, Ising, Potts, and Self Avoiding Random Walk; proofs of the existence of scaling limits and their relations to Schramm Loewner Evolution; the rate of convergence of critical interfaces to SLE curves and obtaining Schramm Loewner Evolution by welding.

References:

1. "Conformal Maps and Geometry", by D. Beliaev

2. "Conformally Invariant Processes in the Plane", by Gregory F. Lawler

3. "Schramm-Loewner evolution," by Anti Kemppainen

 

 

MAT1502HS
TOPICS IN GEOMETRIC ANALYSIS: OPTIMAL TRANSPORTATION, GEOMETRY AND DYNAMICS
R. McCann
(View Timetable)

 

This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. 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-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, 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 flows). 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. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.

This course aims to help prepare graduate students for the Fall 2022 Fields thematic semester on Nonsmooth Riemannian and Lorentzian Geometry. 


Prerequisites:

(Corequisite: measure theory, e.g. comparable to MAT 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-Ampère 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.

 

 

MAT1525HS
TOPICS IN INVERSE PROBLEMS & IMAGE ANALYSIS: COMPUTATIONAL INVERSE PROBLEMS

S. Alexakis
(View Timetable)

 

This course will consider a selection of inverse problems, all arising from concrete real-world problems. We will attempt to present both the theoretical aspects of the problems involved, but also the computational aspects of reconstruction, related to stability and regularization. Some of the problems discussed will involve X-Ray tomography, Gravimetry, Integral geometry and Tomography, and some more PDE inverse problems. The course will use a variety of sources, but will primarily rely on “Inverse Problems for Partial Differential Equations” by V. Isakov and “Regularization of Inverse Problems” by Engl, Hanke and Neubauer. I will try to make the course accessible to most students. Familiarity with graduate-level real analysis (Banach spaces, some Fourier Series) as well as some PDEs (at the undergraduate level at least) would be desirable. Most of all it is open to curious students with an open mind. Any missing background knowledge can also be acquired during the course, for those willing to put in the effort.

 

 

 

MAT1839HS
INTEGRAL EQUATION METHODS FOR THE NUMERICAL SOLUTION OF PDES
J. Bremer
(View Timetable)

 

Integral equation methods are widely used in both the analysis and numerical solution of elliptic boundary value problems. Although we will cover elementary results in the theory of linear integral equations, our focus will be on numerical solution of elliptic boundary value problems via integral equation methods. In fact, we will develop a state-of-the-art integral equation solver for a certain elliptic boundary value problem step-by-step. Approximately one-third of the course will be spent discussing theory, about one-third will concern state-of-the-art methods for the discretization of singular integral operators and the remaining portion will focus on fast direct solvers for integral equations. The course is expected to be of use to students interested in numerical analysis, regardless of whether their focus is on integral equations or not. Indeed, the process we will use to develop our solver can be viewed as a template for writing numerical code which exploit nontrivial results in mathematical analysis to solve a mathematical problem.

 

 

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MAT1841HF
MATHEMATICS OF MASSIVE DATA ANALYSIS: FUNDAMENTALS AND APPLICATIONS
W. Yu
(View Timetable)

 

This course covers the formulation and solution of applied problems. Sources of these problems are the Fields of engineering, physics, computer science, chemistry, biology, medicine, economics, statistics, and the social sciences. In this course, we will focus on the mathematical underpinnings of modern data science. The core of the class will revolve around understanding high-dimensional space, matrix factorization, and probabilistic techniques. Topics may include Markov chains, streaming/sketching algorithms, clustering, spectral graph theory, random graphs, wavelets, graphical models, and computational topology. The precise topics will vary with the instructor and the interests of the class. This course forms the foundation for research in applied mathematics.

 

MAT1846HS
TOPICS IN DYNAMICAL SYSTEMS: LENGTH AND LAPLACE RIGIDITY 
J. De Simoi
(View Timetable)

The problem of spectral rigidity was popularized by M. Kac in 1976 as the question “Can one hear the shape of a drum?” and has been a very active research topic ever since. The problem amounts to reconstruct a manifold, or a domain in ℝⁿ, from the knowledge of the spectrum of its Laplace–Beltrami operator. The corresponding dynamical problem consists in reconstructing a manifold, or a domain in ℝⁿ, from the knowledge of the length of all periodic geodesics.

In this course we will introduce in some detail the most important known results from both the Laplace and the dynamical point of view, among such results:

– Wave Trace formula (Andersson–Melrose, Guillemin–Duistermaat 1976)
– Spectral rigidity of surfaces of negative curvature (Guillemin–Kazhdan 1978)
– Marked Length Spectrum Determination of surfaces of negative curvature (Otal 1990)
– Counterexamples by Vignerás, Sunada and Gordon–Webb–Wolpert (1980–1992)
– Laplace Determination of symmetric convex analytic domains (Zelditch 2008)
– Length Spectral Rigidity of convex domains (—, Kaloshin, Wei 2016)
– Local determination of smooth nonpositively curved Anosov Manifolds (Guillarmou–Lefeuvre 2018)

Prerequisites:

The course will be accessible to graduate students with a basic knowledge of PDEs, differential geometry and real analysis. Prior knowledge in Dynamical Systems is optional.

 

MAT1901HS
READING IN PURE MATHEMATICS: TRANSSERIES, MODEL THEORY, AND HARDY FIELDS
L. van den Dries
(View Timetable)

The course is based on joint work with Matthias Aschenbrenner and Joris van der Hoeven. The focus of our book "Asymptotic Differential Algebra and Model Theory of Transseries" (Annals of Mathematics Studies 195, 2017, Princeton University Press) was on understanding the differential field of transseries. In the process we developed a wide ranging theory of rather general valued differential fields.

The purpose of the course is to apply this material to the study of Hardy fields, which are intriguing mathematical structures with a history going back to Du Bois-Reymond, Veronese, Borel, Hardy, Hausdorff, and Hahn, to mention some pioneers of over a century ago. Many formerly open problems about Hardy fields can now be answered, and interesting new problems can be posed. I will discuss also potential connections to o-minimality.

Hardy fields are algebraic as well as analytic in nature, so we are going to use also some analysis (real, complex, functional, ODE), plus a modest amount of model theory. I will try to make this course as self-contained as possible, but sometimes it will be inevitable to refer to the literature, like the book already mentioned, for proofs of auxiliary results. My two Fields workshop talks (January 11, 13) are meant as an introduction to the course.



 

 

 

 

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