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

The course will cover various topics in Algebraic Groups; topics will differ from year to year.

The theory of Lie groups and Lie algebras is a classical and well-established subject of mathematics. The plan for this course is to give an introduction to the foundations of this theory, with emphasis on compact Lie groups and semi-simple Lie algebras. Yearly topics will be provided on the departmental website's course offering page when the course is offered.

The course will cover various topics in Lie Groups and Fluid Dynamics; topics will differ from year to year.

This course will serve as an introduction to the universality theory of random matrices. No prior exposure to random matrix theory is required or assumed. Yearly topics will be provided on the departmental website's course offering page when the course is offered.

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. This course will be offered in alternating years.

The course will cover various topics in Algebraic Geometry; topics will differ from year to year.

The course will cover various topics in Algebraic Geometry; topics will differ from year to year.

The course will cover various topics in Algebraic Geometry; topics will differ from year to year.

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.

Basic properties of automorphic representations.

Study of methods that have evolved in the past ten years for the study of Beyond Endoscopy.

This course is an introduction to Siegel modular forms.

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.

A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; Diophantine approximation, modular forms.

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

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 neighbourhoods, 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.

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.

A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.

This course will cover some important classical and modern themes in the study of finite fields. These will include: solutions of equations; pseudorandomness; exponential sums and Fourier techniques; algebraic curves over finite fields, the Weil theorems; additive combinatorics and the sum-product phenomenon; applications to combinatorics, theoretical computer science and number theory.

The course will cover various topics in Geometric Topology; topics will differ from year to year.

Study basic topics in discrete mathematics, including enumeration, symmetry, extremal combinatorics, set systems, Ramsey theory, discrepancy, additive combinatorics and quasirandomness.

The course will focus on fundamental geometric insights in Geometric Measure Theory and Geometric Analysis. Topics will vary every year.

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

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

The course is intended to provide a short introduction to several important areas of geometry and topology.

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.

The course will cover various topics in Differential Geometry; topics will differ from year to year.

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