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

The course will cover various topics in Symplectic Geometry and Topology; topics will differ from year to year.

This course will cover various topics in Algebraic Topology; topics will differ from year to year. Consult the departmental website for more information.

This course will cover various topics in Homotopy Theory; topics will differ from year to year. Consult the departmental website for more information.

Course in Singularities of Differentiable Mappings that will also include topics in Ideals of Differentiable Functions. Topics will vary from year to year.

This course will provide a basic introduction to Abelian Varieties theory followed by a sketch of more advanced aspects. Topics will vary every year.

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.

This is an introductory course to the theory of large cardinals (higher infinity axioms) and their applications.

The course will cover various topics in Set Theory; topics will differ from year to year.

The goal of the course is to address the need for professional development activities within the MAT program. We currently offer, on a yearly basis, a career alumni panel and academic jobs application panel. We plan on offering around 3 to 4 sessions per year with some compulsory sessions (e.g., professional ethics and etiquette, sexual violence and harassment policy, milestones for completion of the graduate program and advice on navigating the program), and some voluntary sessions alternating the various topics as needed: career alumni panel, academic jobs applications, short descriptions aimed at a broad audience about research ("elevator pitch"), grant proposal and research paper writing, and submission strategies, etc.

The course will cover various ways to deliver mathematical contents to a general audience; topics will differ from year to year.

This course helps train students to become effective lecturers. It is not for degree credit and is not to be offered every year.

The course will cover an advanced topic in graph theory. The topics will differ from year to year.

This course will cover various topics in Geometric Analysis; topics will differ from year to year. Consult the departmental website for more information.

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

Asymptotic expansion of integrals Laplace method Method of stationary phase Steepest descent Perturbation methods Regular and singular perturbation theory.

Global Analysis Boundary layer theory WKB theory: Formal expansion, conditions for validity, geometrical optics Multiple scale analysis for ODEs: Resonance and secular behavior, damped oscillator Multiple scale analysis for PDEs.

Partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. Nonlinear evolution equations. Existence and long-time behaviour of solutions. Existence of static, traveling wave, self-similar, topological and localized solutions. Stability. Formation of singularities and pattern formation. Fixed point theorems, spectral analysis, bifurcation theory. Equations considered in this course may include: Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics), Cahn-Hilliard (material science, biology), nonlinear Schroedinger equation (quantum and plasma physics, water waves, etc). mean curvature flow (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), and Chern-Simons equations (particle and condensed matter physics).

Mathematical approaches to the study of natural languages, ranging from formal languages, to the study of syntactic structures via geometric and topological methods.

The course will review experimental and theoretical works aiming to improve our understanding of modern deep learning systems.

This course will aim to present some of the most striking recent developments in singularity formation in the general theory of relativity. Topics will vary every year.

This course will cover various topics in Inverse Problems and Image Analysis; topics will differ from year to year. Consult the departmental website for more information.

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.

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.

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.

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.

The goal of this course is to provide an introduction to the mathematical theory of General Relativity through the lens of the study of wave propagation on black holes. Topics will vary every year.

The course will cover various topics in Computational Mathematics; topics will differ from year to year.

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).

This course will cover various topics in Quantum Information Theory; topics will differ from year to year. Consult the departmental website for more information.

This course will cover a variety of mathematical methods important to applied mathematics. The first part of the course will focus on numerical analysis and computation: how mathematical objects are represented by finite approximations in a way that permits efficient calculations. One of the goals of this part of the course will be to understand how to discretize and solve equations involving linear differential operators. The second part of the course will focus on nonlinear methods in applied mathematics. This can include topics from a variety of areas. Consult departmental website for more details specific to annual offering.

This course covers the formulation and solution of applied problems. Topics will vary from year to year.