Graduate » Current Students » Courses Descriptions (2022-23)

REAL ANALYSIS

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.

REAL ANALYSIS II

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).

COMPLEX

E. Bierstone

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- Review of holomorphic and harmonic functions (Chapters 1-4 in Ahlfors).
- Topology of a space of holomorphic functions: Series and infinite products, Weierstrass p-function, Weierstrass and Mittag-Leffler theorems.
- Normal families: Normal families and equicontinuity, theorems of Montel and Picard.
- 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.
- 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

PARTIAL DIFFERENTIAL EQUATIONS I

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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

PARTIAL DIFFERENTIAL EQUATIONS II

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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

ALGEBRA I

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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.

ALGEBRA II

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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

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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

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

MATHEMATICAL PROBABILITY I

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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

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.**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

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.

G. A. Elliott

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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

** **

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.**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

S. Kudla

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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.

J. Tsimerman

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Dedekind domains, ideal class group, splitting of prime ideals, finiteness of class number, Dirichlet unit theorem, further topics such as counting number fields as time allows.

**Prerequisites:**

Solid knowledge of abstract algebra is essential (e.g. Dummit and Foote, MAT347, MAT1100-1101)

**Reference(s):**

The main reference will be Problems in Algebraic Number Theory by J. Esmonde and Ram Murty Milne’s course notes: http://www.jmilne.org/math/CourseNotes/ant.html

COMBINATORIAL THEORY

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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.

TOPICS IN COMBINATORICS: ERROR CORRECTING CODES

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This course is about the theoretical computer science aspects of error-correcting codes. After an introduction to the classical results, we will see a number of modern topics — local testing and decoding, codes from expander graphs, Fourier analytic methods, list decoding, and connections to pseudorandomness and complexity theory.

TOPICS IN COMBINATORICS: ALGEBRAIC COMPLEXITY THEORY

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The goal of this course will be to understand to power and limitations of algebraic computation. Arithmetic circuits are a very natural model of computation for many natural algebraic algorithms such as matrix multiplication, computing fast fourier transforms, computing the determinant etc. The problem of proving lower bounds for arithmetic circuits is one of the most interesting and challenging problems in complexity theory. This course will discuss lower bounds for arithmetic circuits, both recent and classical. It will also discuss the very related problems of derandomizing polynomial identity testing, polynomial reconstruction and polynomial factoring.

E. Meinrenken

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Sard's theorem and transversality. Immersion and embedding theorems. Morse theory. Intersection theory. Borsuk-Ulam theorem. Euler characteristic, Poincare-Hopf theorem and Hopf degree theorem. Additional topics may vary.

**Prerequisites:**

Introduction to Topology course (MAT327H) and Analysis (MAT257Y).

** **

R. Rotman

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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.

INTRODUCTION TO MODEL THEORY AND 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.**Prerequisite:**

MAT357H1**Textbooks:**

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

TECH OF APPLIED MATH: APPLIED NONLINEAR EQUATIONS

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

R. McCann

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This course is an introduction to general relativity theory for students in mathematics and physics alike. Beginning from basic principles this course aims to discuss the modern theory of gravitation and the geometry of space and time, and will explore many of its consequences ranging from black holes to gravitational waves.

More specifically the course covers a discussion of the equivalence principle and its consequences, the geometry of curved spaces, and Einstein's field equations in the presence of matter; we will explore the geometry of the simplest spherically symmetric black hole spacetimes, and proceed to the dynamical formulation of general relativity, and the prediction of gravitational waves. If time permits we shall also discuss in some detail the post-Newtonian approximation.

There will be student projects/presentations on selected topics, such as astrophysical applications, and Penrose's incompleteness theorem.

**Prerequisites:**

Some prior knowledge of special relativity, and elementary Riemannian geometry will be assumed, but are not strictly required, as relevant concepts will be introduced in the course.

**References:**

Robert Wald, General Relativity, UCP 1984 Norbert Straumann

General Relativity, Springer 2012 Stephen Hawking and George Ellis

The large scale structure of the space-time, CUP 1973 Charles Misner, Kip Thorne, and John Wheeler, Gravitation, Freeman 1973

M. Sigal

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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

MATHEMATICAL THEORY OF FINANCE

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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 IN ALGEBRA I: HOMOLOGICAL ALGEBRA

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This course is an introduction to the methods of homological algebra which became ubiquitous over the course of the 20th century. We will study abelian categories and the construction of derived functors as well as various examples thereof. Towards the end of the course, we will recast the classical theory within the framework of triangulated categories and take a glimpse at the modern approach via higher category theory.

Topics:

- exact sequences and diagram chasing
- basic category theory
- Tor and Ext
- abelian categories and derived functors
- Examples: sheaf cohomology and group (co)homology
- triangulated categories and derived categories
- outlook: stable infinity-categories

**Prerequisites:** rings and modules

TOPICS IN PROBABILITY: KPZ

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This course introduces students to non-standard fluctuations arising in the KPZ universality class. **Prerequisites:**

No explicit background is assumed but I would not suggest students take this course unless they have seen rigorous probability with measure theory roughly at the level of our core graduate course.

ALGEBRAIC GEOMETRY

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This is Part 1 of a yearlong course. Part 2 will be taught by Alexander Braverman in the winter semester.

In Part 1, we will follow Hartshorne's book starting from Chapter 2. The goal is to cover Sections 1 through 5. We will also use Ravi Vakil's notes for examples and exercises.

We will cover the following topics:

1. Sheaves

2. Schemes: definitions and examples.

3. Morphism between schemes.

4. Properties of schemes (connected, reduced, integral, noetherian, ect.) and morphisms (finite type, locally of finite type, finite, closed immersions, etc.)

5. Fiber products, separated morphisms, proper morphisms.

6. Quasi coherent and coherent sheaves of morphisms.

TOPICS IN ALGEBRA I: ALGEBRAIC GEOMETRY AND CONVEX GEOMETRY

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In the course I will discuss relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial.

Newton polyhedra connects the theory of convex polyhedra with algebraic geometry of toric varieties. For example, Dehn–Zommerville duality for simple polyhedra leads to the computation of the cohomology ring of smooth toric varieties. Riemann–Roch theorem for toric varieties provides valuable information on the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler–Maclaurin summation formula. Newton polyhedra allow to compute many discrete invariants of generic complete intersections. Newton–Okounkov bodies connect the theory of convex bodies (not necessary polyhedra) with algebraic geometry. These bodies provide a simple proof of the classical Alexandrov–Fenchel inequality (generalizing the isoperimetric inequality) and suggest analogues of these inequality in algebraic geometry.

Tropical geometry and the theory of Gröbner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry. All needed facts from algebraic geometry and convex geometry will be discussed in details during the course.

**References**

1. G.Kempf, F.Knudsen, D.Mamford, B.Saint-Donat. Toroidal Embeddings, Springer Lecture Notes 339, 1973. 2

2. K.Kaveh, A.Khovanskii. Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Annals of Mathematics, V. 176, No 2, 925–978, 2012.

3. Some extra papers will be handout during the course.

Grading: one final presentation or written report.

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY: HODGE THEORY

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We will develop Hodge theory, both in its analytic incarnation (for compact Kähler manifolds) and in its more algebraic incarnation (following work of Deligne and Illusie in positive characteristic). We will discuss many of the applications of Hodge theory to the topology and geometry of algebraic varieties. Time permitting, we will cover more advanced topics, such as the theory of weights and Hodge theory for varieties which are not necessarily smooth or proper, and non-abelian Hodge theory.

TOPICS IN ALGEBRAIC GEOMETRY

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This is the second half of the algebraic geometric course, following Arul's course in the fall. We will cover Representable functors, Quasi-coherent sheaves, Line bundles and divisors, Morphisms to projective space, Grassmannians, Cohomology of quasi-coherent sheaves, Applications to Curves, and other topics depending on interest of participants.

AUTOMORPHIC FORMS AND REPRESENTATION THEORY:

J. Arthur

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Automorphic representations are at the heart of the conjectures that make up the Langlands program. The trace formula provides the most powerful technique for attacking them. It is the analogue for automorphic representations of the Plancherel formula for ordinary representations.

We shall begin with some of the basic properties of automorphic representations. We shall then discuss some of the Langlands conjectures, especially the Principle of Functoriality. The rest of the course will be devoted to an introduction to the trace formula, and if time permits, some of its applications. **Prerequisites:**

Core courses in Analysis and Algebra or their equivalents, and the basic theory of Lie (and/or) Algebraic Groups. This course will be a natural successor to this year's course MAT1196 on representations of real groups, but I will try to keep other prerequsites to a minimum. And in particular, readers without a background in Lie Groups could always think of the general linear group of nonsingular (N x N) matrices (under multiplication) in place of an arbitrary Lie/algebraic group G.**References:**

S. Gelbart, An elementary introduction to the Langlands program, Bulletin of the American Mathematical Society, Vol. 10 (1984), p. 177- 219.

J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, Vol. 4, 2005, p. 1- 263.

TOPICS IN NUMBER THEORY: ELLIPTIC CURVES AND ELLIPTIC SURFACES

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This course is divided into two parts: the first containing an introduction to the basic theory of elliptic curves, and the second with more advanced (but still accessible) topics of the general of elliptic surfaces.

Starting with the definition of elliptic curves, we will turn to studying their basic geometric properties, theory of reduction, L-function, Mordell-Weil Theorem. If time allows, we will define the Picard, Selmer and Tate-Shafarevitch groups.

In the second half, we will give an overview of Shioda geometric theory of elliptic surfaces : KodairaNéron model, Tate's algorithm, base change and quadratic twists and Néron-Severi lattice. Elliptic surfaces are omnipresent in the theory of algebraic surfaces. We will see as well their relation with Del Pezzo surfaces in case of rational elliptic surfaces.

**Prerequisites:**

• An intuition of projective geometry

• Complex analysis, basic group theory, arithmetic in finite fields

**References:**

• The Arithmetic of Elliptic Curves, by Silverman.

• Elliptic Surfaces, survey paper by Schütt and Shioda.

TOPICS IN NUMBER THEORY: TRANSCENDENTAL NUMBER THEORY

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We begin with the introduction to well-known transcendental numbers such as e, pi, and move on to Lindemann-Weierstrass theorem, Schneider-Lang theorem, Baker's theorem on linear independence of linear forms in logarithms of algebraic numbers, and transcendence of special values of L-functions such as Shimura's algebraicity result on critical values of L-functions associated to elliptic cusp forms and their Rankin-Selberg convolutions.

**References:**

R. Murty and P. Rath, Transcendental Numbers

A. Baker, Transcendental Number Theory

**Prerequisites:**

Algebraic Number Theory

Complex Analysis

TOPICS IN GEOMETRIC TOPOLOGY: DIFFEOMORPHISM GROUPS

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This course is about the dramatic advances made during the last decade in the study of diffeomorphism groups of manifolds. After a discussion of the relevant foundational results in differential topology, we survey classical results obtained for diffeomorphisms of low-dimensional manifolds using geometric techniques and for diffeomorphisms of high-dimensional manifold using surgery theory. After this, we explain how the computation of the homotopy type of the cobordism category and the calculus of embeddings can be combined to understand diffeomorphisms far outside the ranges of classical techniques. The goal is to bring students to the forefront of research.

The topics we will discuss include but are not limited to: mapping class groups, the Smale conjecture, the h-cobordism theorem, exotic spheres, surgery theory, embedding calculus, cobordism categories, the Madsen–Weiss theorem, homological stability, embedding calculus, configuration space integrals, smoothing theory.

GEOMETRIC INEQUALITIES

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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" + expository and research papers.

TOPICS IN GEOMETRY: SL(2,Z)

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The group SL(2,R) lies in the intersection of many different fields of mathematics including Hyperbolic Geometry, Homogeneous Dynamics, Algebraic Geometry and Number Theory. Many general phenomena have their simplest case appearing in the group SL(2,R). We examine several geometric and dynamical aspects of this group and attempt to build bridges between them. Possible topics to be covered include:

• SL(2,Z) as a braid group.

• Hyperbolic plane as the symmetric space associated to SL(2,R).

• Hyperbolic plane as the Teichmüller space of the torus.

• Continued Fractions and combinatorial properties of curves on a surface.

• Geodesic flow and counting problems in modular curve.

• Horocycle flow and the Prime Number Theorem.

TOPICS IN GEOMETRY: CLASSICAL MECHANICS

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This will be a new course on classical mechanics, focusing on the basic structure of the subject and how it is expressed with modern geometric tools.

INTRODUCTION TO NONCOMMUTATIVE GEOMETRY

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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.

SEMINAR IN GEOMETRY AND TOPOLOGY: COMPARISON GEOMETRY

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This course would be a direct continuation of the MAT1342/MAT464 Differential geometry course. It would cover various comparison theorems (Rauch and Toponogov comparison) and their applications such as Bishop-Gromov volume comparison, critical point theory of distance functions, diameter sphere theorem, negative and nonnegative curvature, Gromoll-Meyer splitting theorem and Cheeger-Gromoll soul theorem.

https://utoronto.zoom.us/j/87352812967 Passcode: 231588

The course Quercus page is below and is accessible to anyone at UofT even if they are not registered for the course:

The link is https://q.utoronto.ca/courses/296667

TOPICS IN SYMPLECTIC GEOMETRY AND TOPOLOGY: SYMPLECTIC TOPOLOGY AND MORSE THEORY

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(The numbers below approximately correspond to the week numbers.)

1) Preliminaries/reminder: Symplectic manifolds, Hamiltonian fields, Darboux theorem, Lagrangian manifolds and foliations, integrable systems.

2) Symplectic properties of billiards and, time permitted, geodesics on an ellipsoid. 4

3-4) Symplectic fixed points theorems: the Poincare–Birkhoff theorem, Arnold’s conjecture, the Conley–Zehnder theorem.

4-6) Morse theory: Morse inequalities, Lusternik-Schnirelmann category, applications to geodesics, other ramifications (the Morse–Witten complex, Morse–Novikov theory); the end of proof for Conley–Zehnder.

7-8) A glimpse of generating functions for symplectomorphisms, non-squeezing results, symplectic capacities, Floer homology.

9-10) The Hofer metric, geometry of and geodesics on symplectomorphism groups.

11-12) Contact structures, Legendrian knots, their invariants and Bennequin inequality; a glimpse of contact homology of Legendrian knots.

**References:**

1. S. Tabachnikov, ”Introduction to symplectic topology” Lecture notes, (PennState U.): http://www.personal.psu.edu/sot2/courses/symplectic.pdf

2. D. McDuff and D. Salamon: ”Introduction to symplectic topology” (Oxford Math. Monographs, 1998)

3. V. Arnold and A. Givental ”Symplectic geometry” Dynamical systems, IV, 1–138, Encyclopaedia Math. Sci., vol. 4, (Springer 2001)

**Prerequisite: **

Familiarity with the main notions of symplectic geometry.

TOPICS IN SET THEORY: HOMOGENEOUS STRUCTURES, TOPOLOGICAL DYNAMICS OF THEIR AUTOMORPHISM GROUPS AND THE CORRESPONDING RAMSEY INDEX THEORY

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Fraisse theory of homogeneous structures. Topological dynamics of automorphism groups. Structural Ramsey theory. These subjects will be first introduced starting from their initial stages building gradually towards the modern developments. For example, Fraisse theory of metric structures is one such modern development as well as the categorical approach to Fraisse theory. Recent developments in the structural Ramsey theory will also be one of the focuses.

S. Alexakis

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This is topics course on questions of interest in general relativity and cosmology. The main topic we will cover is the behaviour of space-time metrics upon approach to the big bang singularity, as well as the final crunch singularities and also black hole interiors. This is a subject on which very little is rigorously known. We will study some long-standing conjectures, as well as a set of rigorous results which are obtained for simplified models of the Einstein equations near singularities. In particular a conjecture that generically space-times exhibit a chaotic behaviour upon approach to these singularities will be studied.

**Prerequisites:**

Some background knowledge on general relativity or Riemannian geometry is desirable, but not strictly necessary. An in-depth knowledge of PDEs is not necessary for this course.

**Textbook:**

"The Cosmological Singularity" by Belinskii and Henneaux (Cambridge University Press, 2018).

DEEP LEARNING: THEORY & DATA SCIENCE

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Deep learning systems have revolutionized field after another, leading to unprecedented empirical performance. Yet, their intricate structure led most practitioners and researchers to regard them as blackboxes, with little that could be understood. In this course, we will review experimental and theoretical works aiming to improve our understanding of modern deep learning systems.

Link to course website: https://sites.google.com/view/mat1510

**Prerequisite:**

Undergraduate Linear Algebra

TOPICS IN INVERSE PROBLEMS & IMAGE ANALYSIS: VARIATIONAL METHODS IN IMAGING AND NEURAL NETWORKS

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This course aims to provide the analytic tools for understanding the spectacularly successful Wasserstein Generative Neural Networks from a mathematical point of view. We will spend a fair amount of time on the requisite background from Convex Analysis, model variational problems from Image Analysis and basics of Optimal Transport Theory.

Brief outline of the course:

1. Introduction to Wasserstein Generative Neural Networks

2. The Rudin-Osher-Fatemi model for image restoration

3. Introduction to Congested Optimal Transport

4. Wasserstein GANs compute a congested transport cost

5. Image restoration with learned regularizers

6. Projected GANs

**Prerequisite:**

Basic Real Analysis and Functional analysis will be helpful.

**Recommended References:**

"An introduction to Total Variation for Image Analysis" by Antonin Chambolle, Vicent Caselles, Matteo Novaga, Daniel Cremers and Thomas Pock, archives-ouvertes.fr, 2009.

"Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling" by Filippo Santambrogio, Birkhauser, 2015.

"Wasserstein GANs with Gradient Penalty Compute Congested Transport" by Tristan Milne and Adrian Nachman,

Conference on Learning Theory, PMLR, 2022, pp. 103–129.

TOPICS IN DYNAMICS: HOMOGENEOUS DYNAMICS & APPLICATIONS

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After studying some basics of hyperbolic geometry, we will discuss the Patterson-Sullivan theory for Kleinian groups. We will also discuss how to generalize the Patterson-Sullivan theory to discrete subgroups of higher-rank semisimple Lie groups.

**References:**

1. J. Ratcliffe. Foundations of hyperbolic manifolds. Graduate Texts in Mathematics, 149. Springer, New York, 2006.

2. K. Matsuzaki and M. Taniguchi. Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs.

3. J. F. Quint. An overview of Patterson-Sullivan theory https://www.math.u-bordeaux.fr/~jquint/publications/courszurich.pdf

4. J. F. Quint. Mesures de Patterson-Sullivan en rang sup´erieur. Geom. Funct. Anal. 12 (2002), no. 4, 776–809.

TOPICS IN DYNAMICS: INTRODUCTION TO THE RIEMANNIAN CURVATURE DIMENSION CONDITION

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The aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L2 and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of `infinitesimally Hilbertian' spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds.

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound - most notably we shall prove the Laplacian comparison theorem - and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth to notice that such statement gives new information - compared to those available through Cheeger-Colding's theory of Ricci-limit spaces - even about the structure of smooth Riemannian manifolds.

**Prerequisites**:

Some familiarity with Riemannian geometry and optimal transport theory in the case cost=distance2 is preferred, but not required: I shall provide the necessary background when needed.

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