Graduate » Current Students » Tentative Courses Descriptions (2018-19)

REAL ANALYSIS

J. De Simoi

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 ANALYSIS

E. Bierstone

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- Review of elementary properties of holomorphic functions. Cauchy’s integral formula, Taylor and Laurent series, residue calculus.
- Harmonic functions. Poisson's integral formula and Dirichlet's problem.
- Conformal mapping, Riemann mapping theorem.
- Analytic continuation, Monodromy Theorem, Riemann surfaces.
- Modular functions and the Picard Theorems.
- Other topics are possible, like product theorems, elliptic functions, and non-isolated removability theorems.

**Recommended prerequisites: **A first course in complex analysis and a course in real analysis. Measure theory is not required.

**Main References: **

L. Ahlfors: Complex Analysis, 3rd Edition

Stein and Shakarchi: Complex Analysis

**Additional References:**

T. Gamelin: Complex Analysis

W. Rudin: Real and Complex Analysis, 2nd or 3rd edition

D. Sarason: Complex Function Theory

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

F. Pusateri

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

TBA

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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

J. Arthur

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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.

**Textbooks:**

Dummit and Foote: Abstract Algebra, 3rd Edition

**Other References:**

Jacobson: Basic Algebra, Volumes I and II.

Cohn: Basic Algebra

Lang: Algebra 3rd Edition

M. Artin: Algebra.

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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:**

John M. Lee: Introduction to Smooth Manifolds

TOPOLOGY II

V. Kapovitch

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

B. Virag

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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

B. Virag

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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

INTRODUCTION TO LINEAR OPERATORS

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

TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS

G. A. Elliott

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a 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

TOPICS IN ERGODIC THEORY

G. Tiozzo

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An introduction to classical topics in ergodic theory, with applications to dynamical systems.

Part 1. The basics

Measure preserving transformations. The ergodic theorems: Birkhoff, Von Neumann, Kingman.

Ergodicity and mixing. Martingales and convergence. Entropy: Kolmogorov-Sinai entropy, topological entropy, the variational principle.

Part 2. Applications.

The geodesic flow on manifolds of negative curvature. Mixing and ergodic properties.

Applications to the Teichmuller flow: ergodicity and mixing.

**Prerequisite:**

Graduate real analysis and probability.

**Textbook(s):**A Simple Introduction to Ergodic Theory, K. Dajani and S. Dirskin, available at

http://www.staff.science.uu.nl/~kraai101/lecturenotes2009.pdf

Introduction to the Modern Theory of Dynamical Systems, A. Katok and B. Hasselblatt

**References:**

M. Einsledler and T. Ward, Ergodic theory with a view towards Number Theory

P. Walters, An introduction to Ergodic Theory

TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS I: HYPERBOLIC PDEs AND APPLICATIONS IN GENERAL RELATIVITY

S. Aretakis

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The course will present major results in hyperbolic PDEs with applications in general relativity. Topics will include the following: stability of Minkowski spacetime, memory effect, formation of trapped surfaces and stability of black holes.

TOPICS IN REPRESENTATION THEORY: REPRESENTATIONS OF OF REDUCTIVE P-ADIC GROUPRS

F. Murnaghan

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Introductory course on theory of complex representations of reductive p-adic groups: emphasis on general linear groups (in order to avoid possible issues with students' lack of background in the structure of reductive algebraic groups). Admissible representations, parabolic induction, Jacquet modules, Jacquet's Subrepresentation Theorem, supercuspidal representations, discrete series, types, distributions and characters, Bernstein Decomposition, p-adic symmetric spaces and representations distinguished by involutions.

**Prerequisite(s):**

Core algebra and analysis courses.

**Reference(s):**

My online course notes;

D. Renard: Representations des groupes reductifs p-adiques (SMF volume 17);

Casselman's notes;

various other online notes.

TOPICS IN PROBABILITY: SPIN GLASS

D. Panchenko

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The main subject of this course is the mathematical theory of the celebrated Parisi solution of the Sherrington-Kirkpatrick model of spin glasses. The methods involved also serve as an illustration of such classical topics in probability as the Gaussian interpolation and concentration of measure, Poisson processes, and representation results for exchangeable arrays.

**Textbook:**

D. Panchenko “The Sherrington-Kirkpatrick Model” (available electronically through U of T library).

**Prerequisites:**

MAT 1600, 1601.

INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY

P. Milman

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We will study the geometry of algebraic varieties. The course will cover affine and projective varieties, Zariski topology, dimension, smoothness, elimination (via resultants) etc. Topics will include classic results of Nullstellensatz, Bezout Theorem, Weierstrass Preparation, Puiseux Factorization, Chow Theorem and (time permitting) Chow Ideal Theorem and its application to Hironaka Valuation Criterion with proofs from the recommended books below (placed on reserve in the Math. Department's library) and/or printed lecture notes. All necessary results and concepts used will be introduced and explained.

**Prerequisite:**

Familiarity with the basic undergraduate program of the first 3 years in math.

**References:**

1. David Mumford "Algebraic Geometry I . Complex Projective Varieties"

2. M.F. Atiyah and I.G. Macdonald "Introduction to Commutative Algebra"

ALGEBRAIC GEOMETRY: ARITHMETIC TECHNIQUES

M. Groechenig

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This course is devoted to studying topological properties of complex algebraic varieties using arithmetic methods. At first we will review the (proven) Weil conjectures which relate geometry over finite fields and the complex numbers. Subsequently we will turn to p-adic integration and prove Batyrev's theorem: Birational Calabi-Yau varieties have equal Betti numbers. If time permits we will briefly encounter crepant resultions and the McKay correspondence and conclude the course by discussing purely complex geometric alternatives to those arithmetic techniques: mixed Hodge structures and motivic integration. **Prerequisites:**

- familiarity with the theory of manifolds or algebraic varieties

- feeling for (singular or de Rham) cohomology **Reference(s):**

- Etale cohomology and the Weil conjectures, by Freitag and Kiehl

- Lecture notes by Popa, Modern aspects of the cohomological study of algebraic varieties, http://www.math.northwestern.edu/~mpopa/571/

- An introduction to the theory of local zeta functions, by Igusa

TOPICS IN ALGEBRAIC GEOMETRY: HODGE THEORY

A. Braverman

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1. Hodge decomposition for smooth compact manifolds.

2. Hodge theory on compact Kahler manifolds. Hodge theorem - analytic proof.

3. Hodge theorem for smooth complete varieties. Algebraic proof due to Deligne-Illusie.

4. Introduction to mixed Hodge structures and Hodge theory on non-compact manifolds.

**Prerequisite(s):**Manifolds, differential forms, very basic algebraic geometry.

TOPICS IN ALGEBRAIC GEOMETRY: ALGEBRAIC CURVES AND TOPOLOGICAL GALOIS THEORY

A. Khovanskii

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Algebraic curves and compact Riemann surfaces comprise the most developed and arguably the most beautiful portion of algebraic geometry. Elegant tools and ideas of Galois Theory are heavily used in algebraic geometry.

We will cover basic results of algebraic curves theory including Abel, Riemann-Roch and Jacobi theorems, Weil's reciprocity law, group structure on cubic curve and theory of elliptic function. I will also provide an introduction to the theory of real algebraic curves including Harnack inequality and Petrovskii theorem on real algebraic curve of degree 6.

We will cover basic results of Galois Theory. I will explain topological criteriums for representability of algebraic functions by radicals and for solvability of some differential equations by quadratures. We will study Lioville's theorem on algebraic functions integrable in finite terms.

**Prerequisite:**

Basic knowledge in complex analysis and in elementary algebra.

**References:**

1. Phillip A. Griffiths. \Introduction to Algebraic Curves". AMS Translations of Mathematical Monographs. V. 76.

2. A.Khovanskii. \Topological Galois Theory. Solvability and Unsolvability of Equations in Finite Terms", Springer Monographs in Mathematics, 305 pp., 2014.

AUTOMORPHIC FORMS AND REPRESENTATION THEORY II: METHODS BEYOND ENDOSCOPY

J. Arthur

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Our goal will be to understand methods that have evolved in the past ten years for the study of Beyond Endoscopy. The course is aimed at students who are working on problems in Beyond Endoscopy, of more generally, any students with interests in automorphic forms and number theory.

Beyond Endoscopy is a proposal/program advanced by Langlands around 2000 for using the trace formula to study the analytic behaviour of automorphic L-functions. Its ultimate goal would be to establish the principle of functoriality. Functoriality is the fundamental pillar of the Langlands program, with enormous implications for number theory, arithmetic geometry and representation theory. We cannot hope for a general proof in the near term future, but the ideas in Beyond Endoscopy are suggestive and quite striking. Many of them lead to accessible problems that seem to be of great interest in their own right.

We shall review the general premises of Langlands proposal for Beyond Endoscopy, with some discussion of the background, as needed. We shall then study some of the papers since 2000 that have lead to progress. **Prerequisites:**

Core courses in analysis and algebra or their equivalents, basic theory of Lie (and/or algebraic groups), and perhaps also a course in algebraic or analytic number theory could be regarded as more essential background. **References:**

R. Langlands, Beyond Endoscopy, in Contributions to Automorphic Forms, Geometry and Number Theory, Johns Hopkins University Press, 2004, 611-697.

J. Arthur, An introduction to the trace formula, Clay Math Proceedings, Vol. 4, 2005, 1-263.

E. Frenkel, R.Langlands and B. C. Ngo, Formula des traces et functorialite: le debut d'un programme, Annales Sci. Math. Quebec 34 (2010), 199-243.

A. Altug, Beyond Endoscopy via the trace formula I: Poisson summation and isolation of special representations, Compositio Math. 151 (2015), 1791-1820.

J. Arthur, Problems Beyond Endoscopy, to appear in Proceedings of Conference in Honor of the 70th Birthday of Roger Howe.

ALGEBRAIC NUMBER THEORY

F. Herzig

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Review of some commutative algebra, Dedekind domains, ideal class group, splitting of prime ideals, discrete valuation rings and completions, p-adic fields, different and discriminant Galois extensions, Frobenius elements, finiteness of class number, unit theorem, ideles and adeles, further topics 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 Milne's course notes http://www.jmilne.org/math/CourseNotes/ant.html

COMBINATORIAL THEORY

K. Rafi

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: CIRCUIT COMPLEXITY

B. Rossman

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This course presents an introduction to Circuit Complexity, an area of Theoretical Computer Science that aims to prove unconditional lower bounds in combinatorial models of computation (for instance, showing that the k-clique problem requires monotone circuits of large size). We will begin by reviewing the classic lower bounds for bounded-depth and monotone circuits and formulas (switching lemmas, the polynomial method, gate elimination, matrix rank). We will then focus on a new set of techniques that pin down the average-case complexity of subgraph isomorphism problems on Erdős-Rényi random graphs. Along the way, we will encounter useful tools from the probabilistic method and highlight various open problems.

No prerequisites are required beyond basic mathematical maturity.

TOPICS IN ALGEBRAIC TOPOLOGY:GENERALIZED COMPLEX AND KAEHLER GEOMETRY

M. Gualtieri

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This will be an introductory course in generalized geometry, with a special emphasis on Dirac, generalized complex and Kahler geometry. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror

symmetry. The main references for this class are the published papers on generalized complex and Kahler geometry, but we will also draw from more recent developments in the physics literature.

A very basic familiarity with complex and symplectic manifolds will be assumed; here is a list of topics which will be covered in the lecture course:

- Gerbes, B-fields, and exact Courant algebroids;

- Relation to sigma models in physics;

- Linear algebra of a split-signature real bilinear form; pure spinors;

- Generalized Riemannian structures and the generalized Hodge star;

- Integrability, Dirac structures, Lie algebroids and bialgebroids;

- Generalized complex structures; examples of such;

- Kodaira-Spencer-Kuranishi deformation theory for generalized complex structures;

- Local structure theory for generalized complex structures;

- Generalized K\"ahler geometry;

- Hodge decomposition theorem for Generalized K\"ahler structures

- Hermitian geometry; the Gray-Hervella classification

- Equivalence theorem Generalized K\"ahler=Bihermitian

- Reduction of Courant algebroids and generalized complex structures.

- Generalized Calabi-Yau structures and the Hitchin functional

- Ramond-Ramond versus Neveu-Schwarz fluxes; D-branes.

**Prerequisite:**

A basic familiarity with smooth manifolds, complex structures, and ideally symplectic structures.

**References:**

The main texts are all drawn from the literature in generalized geometry over the past 10 years. This includes the main papers by Hitchin, Gualtieri, Cavalcanti, Goto et al. in the math literature and by Gates, Hull, Rocek, Lindstrom, Zabzine et al. in the physics literature.

SEMINAR IN GEOMETRY: COMPARISON GEOMETRY

V. Kapovitch

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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, diameter sphere theorem and the 1/4-piching sphere theorem, manifolds of nonpositive, negative and nonnegative curvature, Gromoll-Meyer splitting theorem and Cheeger-Gromoll soul theorem.

INTRODUCTION TO NONCOMMUTATIVE GEOMETRY

M. Marcolli

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This course will give an overview of the main techniques in Noncommutative Geometry, with special emphasis on its application to physics, especially to geometric models for particle physics and cosmology, as well as to the interplay between quantum statistical mechanics and number theory.

SEMINAR IN GEOMETRY AND TOPOLOGY: GEOMETRY AND DYNAMICS IN TEICHMULLER SPACE

K. Rafi

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We start with covering certain parts of Thurston’s paper “Minimal Stretch Maps Between Hyperbolic Surfaces” where the Thurston metric is first defined and developed. We then review some of the more recent progress in the area. One goal of this course is to develop notes on Thurston’s paper adding details for some of the arguments in the paper and making the area more accessible to graduate students.

DIFFERENTIAL TOPLOGY

R. McCann

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

o Manifolds: Embedded and intrinsic definitions, equivalence.

o Charts and coordinates, tangent space.

o Smooth maps, immersions, submersions, diffeomorphisms, smooth embeddings.

o Regular values and Sard's lemma, transversality. Homotopy and stability.

o Manifolds with boundary, oriented manifolds. Intersection index, Lefschetz number.

o Whitney's immersion and embedding theorems.

o Borsuk-Ulam and Brouwer theorems, the Euler characteristic and Hopf-Poincare theorem.

o Additional topics: differential forms and integration, linking numbers and Hopf invariant, elements of Morse theory, Frobenius integrability theorem, topological dimension.

**Prerequisites: **

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

INTRODUCTION TO DIFFERENTIAL GEOMETRY: RIEMANNIAN GEOMETRY

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 SYMPLECTIC GEOMERY

Y. Karshon

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This is an introductory course in symplectic geometry and topology. We will discuss a variety of concepts, examples, and theorems, which may include, but are not restricted to, these topics: Moser's method and Darboux's theorem; Hamiltonian group actions and momentum maps; almost complex structures and holomorphic curves; Gromov's nonsqueezing theorem.

**Prerequisite(s):**

Manifolds and differential forms; homology.

**Reference(s):**

Cannas Da Silva's "Lectures on Symplectic Geometry".

Additional reference will be posted on the course website.

SET THEORY: FORCING

S. Todorcevic

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This will be a Forcing course specialized to the side condition method and its variations. While we will present some of the very basic forcing constructions we will concentrate on forcing that can be iterated. For example, we will present the proof of the consistency of the Proper Forcing Axiom.

**Prerequisite:**

MAT 409

**Textbook(s):**

K. Kunen, Set Theory, 1980 or 2011 edition.

T.Jech, Set Theory, 2003 edition

S. Shelah, Proper and Improper Forcing, first or second edition.

TEACHING LARGE MATHEMATICS CLASSES

J. Repka

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The goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems.

Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.

The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later (private) assessment.

APPLIED ANALYSIS: INTRODUCTION TO SPECTRAL THEORY

M. Sigal

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Spectral theory permeates the entire mathematics and plays a fundamental role in physical sciences, engineering and beyond. It originated with an attempt to give a quantative description of the motion of a string, then migrated into the theories of oscillations and waves, dynamical systems and quantum mechanics (giving criteria of stability and values of allowed energy levels of quantum systems, respectively). In this introductory course, we give basic concepts of spectral analysis, starting with the matrix theory and then proceeding to differential equations including the motion of the string and oscillations of an electron in its orbit around the nucleus. Finally we will relate it to the theory of stability of stationary solutions and travelling waves, aka solitons. **Prerequisites:**

Linear algebra and multivariable calculus, elementary differential equations and elementary real analysis (vector spaces, norms and inner products, etc). **References:**

The instructors on-line lecture notes which are drawn partly from I.M. Sigal and P. Hislop, Introduction to Spectral Theory, Springer but with more background and more applications.

TOPICS IN GEOMETRIC ANALYSIS: BROWNIAN MOTION ON MANIFOLDS

R. Haslhofer

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There has been a long-standing interaction between probability and geometry, witnessed most strikingly by random motion on manifolds. Classical examples for this include the probabilistic proof of the Atiyah-Singer index theorem and Driver’s integration by parts formula on path space. More recently, it has been discovered that Brownian motion can be used to characterize solutions of the Einstein’s equations and of Hamilton’s Ricci flow via certain sharp estimates on path space.

In this course we will provide a general introduction to Brownian motion on manifolds and some of its applications. In the first half of the course we will develop the classical theory of Brownian motion in Euclidean space and on manifolds. In the second half of the course, we will consider applications, including spectral gap estimates via coupling, a probabilistic proof of the Chern-Gauss-Bonnet theorem, integration by parts on path space, and the recent characterizations of solutions of the Einstein equations.

**Prerequisite(s):**

Some basic background in probability and geometry is recommended, i.e. it is helpful to be (somewhat) familiar with fundamental notions such as Markov processes, martingales, Riemannian manifolds, and curvature.

**Reference(s):**

B. Driver: A Primer in Riemannian Geometry and Stochastic Analysis on Path Spaces, FIM notes, 1995

P. Morters, Y. Peres: Brownian Motion, CUP, 2010

R. Haslhofer, A. Naber: Ricci Curvature and Bochner Formulas for Martingales, CPAM, 2018

E. Hsu: Stochastic Analysis on Manifolds, AMS, 2002

L. Rogers, D. Williams: Diffusions, Markov Processes and Martingales, CUP, 1994

D. Stroock: An Introduction to the Analysis on Paths on a Riemannian Manifold, AMS, 2000

ASYMPTOTIC AND PERTURBATION METHODS

C. Sulem

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- Local Methods.
- Classification of regular and singular points of linear ODEs. Approximate solutions near regular, regular-singular and irregular singular points; irregular point at infinity.
- Asymptotic series.
- Some examples of nonlinear differential equations.

- Asymptotic expansion of integrals.
- Laplace method.
- Method of stationary phase.
- Steepest descent.

- Perturbation methods.
- Regular perturbation theory.
- Singular perturbation theory.

- Global Analysis.
- Boundary layer theory.
- WKB theory : Formal expansion, conditions for validity, shortwave asymptotics: geometrical optics and semi-classics.
- Bohr-Sommerfeld approximation.
- 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

**Textbooks:**

Advanced Mathematical Methods for Scientists and Engineers; Asymptotic Methods and Perturbation Theory, by Carl M. Bender and Steven A. Orszag, 1st edition, McGraw-Hill, 1978, 2nd ed., Springer 1999.

Applied Asymptotic Analysis by Peter Miller, AMS, Grad. Studies in Math, Vol 75.

T

M. Sigal

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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

MATHEMATICAL AND COMPUTATIONAL LINGUISTICS

M. Marcolli

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In this course I plan to discuss mathematical approaches to the study of natural languages, ranging from formal languages, to the study of syntactic structures via geometric and topological methods.

TOPICS IN INVERSE PROBLEMS AND IMAGE ANALYS

A. Nachman

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This course will present some of the highly developed Variational, PDE and Differential Geometric techniques for treating problems in Image Analysis. We will also explore the recent amazingly successful Deep Learning approaches to some of these problems.

The first part of the course will cover the theory and algorithms of Total Variation Regularization for Image Denoising and Restauration.

The second part of the course will be on the infinite dimensional Riemannian geometry of diffeomorphisms for Image Registration.

The third part of the course will be an introduction to Deep Learning and Geometric Deep Learning.

**Prerequisite(s):**

Some prior familiarity with real analysis and functional analysis will be helpful.

**Reference(s):**

Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, by Yves Meyer, ISBN: 0821829203

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

Shapes and Diffeomorphisms by Laurent Younes. ISBN 978-3-642-12055-8

GENERAL RELATIVITY

S. Alexakis

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General Relativity is a geometric theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It describes the evolution of gravitational physical systems such as planetary systems, galaxies, black holes and ultimately the universe as a whole.

This course will strive in a very intuitive, but still rigorous, way to uncover the subtle connections of Einstein's theory with geometry and analysis. We will prove purely mathematical theorems which have direct consequences for fundamental phenomena of our natural world. Moreover, this course will provide students with the background needed for studying recent research papers in the field and for initiating their own research projects.

The following topics will be covered:

A. Geometry: Introduction to Riemannian, Lorentzian and Null geometry,

B. Einstein equations and fundamental predictions: gravitational waves, black holes, trapped surfaces,

Penrose incompleteness theorem

C. Black hole dynamics: Schwarzschild and Kerr families

**Prerequisites:**

Some familiarity with basic differential or Riemannian geometry is desirable but not required. All students should be familiar with advanced multivariable calculus.

**References:**

We will mainly follow the lecturer's ''Lecture notes on general relativity'' (available online).

Robert Wald's "General relativity",

Hawking and Ellis' ''The large scale structure of spacetime''.

(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

Syllabus (not all topics would be covered):

* Schroedinger equation

* Quantum observables

* Spectrum and evolution

* Atoms and molecules

* Density matrix and open systems

* Quasiclassical asymptotics

* Perturbation theory

* Adiabatic theory and Berry phase

* Self-consistent approximations

* Bose-Einstein condensation

* Open systems and Lindblad evolution

* Quantum entropy

* Quantum channels and information processing

* Quantum Shannon theoroms

**References:**

S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2011

For material not contained in this book, e.g. quantum information theory, we will try to provide handouts and refer to on-line sources.

**Additional reference:**

L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008.

COMPUTATIONAL MATHEMATICS: NUMERICAL METHODS

A. Stinchcombe

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Computer simulation is a viable approach to gain insight in nearly all scientific fields. In this course, we will study the essential methods of numerical linear algebra upon which most computer simulations depend. We will then consider finite difference methods for ordinary and partial differential equations and conclude with some modern methods for partial differential equations. Topics: stability and accuracy of numerical methods in floating point arithmetic; matrix factorizations (QR, SVD, Cholesky); direct and iterative methods for solving linear systems; methods for eigenvalue problems; Newton's method for non-linear systems of equations; Taylor series method, Runge-Kutta methods, multistep methods, and Richardson extrapolation for ordinary differential equations; finite difference and finite volume methods for linear partial differential equations; consistency, stability, and convergence; introduction to spectral methods and boundary integral equation methods **Prerequisites:**

Ordinary differential equations and linear algebra, some exposure to partial differential equations, some programming experience. **Textbooks:**

Numerical Linear Algebra by Trefethen and Bau;

Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations by Ascher and Petzold

QUANTUM COMPUTING, FOUNDATIONS TO FRONTIER

H. Yuen

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This course will give a broad overview of the field of quantum computing. We will start with a crash course in the fundamentals of quantum computing (qubits, quantum circuits, basic quantum algorithms such as Grover’s search algorithm and Shor’s factoring algorithm). Armed with the basics, we will then explore topics at the frontier of quantum computing: quantum complexity theory, device-independent quantum cryptography, quantum machine learning algorithms, and quantum supremacy. Students will make project presentations at the end of the course. This is a theoretical course that requires a strong background in linear algebra, probability theory and mathematical maturity. **Prerequisite(s):**

Familiarity with analysis of algorithms and complexity is a major plus but not required.

CONTROL THEORY

B. Khesin

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The course focuses on the key notions of control theory: holonomic and nonholonomic constraints, Frobenius theorem, examples of brachistochrone, parallel parking, sliding skate, and rolling sphere. It gives an introduction to calculus of variations, optimal control, maximum principle, Riemannian and subriemannian geodesics, notions of symmetry and stability.

**Prerequisites:**

MAT235Y1/ MAT237Y1/ MAT257Y1 (multivariable calculus) MAT244H1/ MAT267H1 (differential equations), MAT223 (linear algebra); recommended: MAT363/ MAT367 (differential geometry)

**References:**

1) D. Liberzon “Calculus of Variations and Optimal Control Theory” 2012, Princeton Univ. Press

2) A. Agrachev, D. Barilari, and U. Boscain “Introduction to Riemannian and Sub-Riemannian geometry“ 2018

INTRODUCTION TO MODERN HOLOMORPHIC DYNAMICS

M. Yampolsky

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The course will introduce the subject from a modern viewpoint, starting from basic concepts and touching on problems of current interest. Some of the topics covered will include; dynamics on the Riemann sphere, Fatou and Julia sets, quasiconformal mappings, almost complex structures and Measurable Riemann Mapping theorem, Fatou-Sullivan structure theory, dynamics of quadratic polynomials, parabolic implosion, the Mandelbrot set, quadratic-like maps and Douady-Hubbard straightening theorem.

**Prerequisite:**

A graduate-level course in Complex Analysis.

MATHEMATICAL PROBLEMS IN ECONOMICS

R. McCann

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This course surveys a number of economic topics of current research interest in which mathematical developments have (and are expected to continue to) contribute crucial advances. These include the theory of matching and pricing, problems of asymmetric information, the principle-agent framework, auction theory, mechanism (and information) design, portfolio optimization and hedging. These topics are unified through a linear program approach in which the mathematical theory of optimal transport and its emerging relevance figure prominently.

**References:**

Basov. Multidimensional Screening. Springer, 2005.

Chiappori. Matching with Transfers: The Economics of Love and Marriage. Princeton, 2017.

Galichon. Optimal Transport Methods in Economics. Princeton, 2016.

Henry-Labord\`ere. Model-free Hedging: A Martingale Optimal Transport Viewpoint. CRC Press 2017.

Mas-Colell, Whinston and Green. Microeconomic Theory. Oxford, 1995.

Sotomayor and Roth. Two-Sided Matching: a Study in Game-Theoretic Modeling and Analysis. Cambridge Press, 1992.

Santambrogio. Optimal Transport for Applied Mathematicians. Birkhauser 2015.

Vohra. Mechanism Design: A Linear Programming Approach. Cambridge 2011.

MATHEMATICAL THEORY OF FINANCE

L. Seco

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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

READING IN PURE MATHEMATICS: TOPOLOGY AND GEOMETRY OF AUTOMORPHISM GROUPS OF FREE GROUPS

M. Bestvina

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Outer automorphism group $Out(F_n)$ of the free group $F_n$ of rank $n$ is part of the third great family of groups much studied in geometric group theory, the other two being arithmetic groups and mapping class groups. The goal of the course is to introduce the spaces on which $Out(F_n)$ acts and the basic techniques for studying this group. Analogies with mapping class groups guide the subject and some familiarity with them will be helpful, but not required.

J. Chaika

This course will begin by introducing and developing classical ergodic theory and topological dynamics. From here it will talk about how ergodic theory and dynamics have provided results and families of questions about surfaces and Teichmuller space.

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