Abstract. We will introduce Koopman operator and its adjoint, transfer (Perron-Frobenius) operator. We will discuss their properties and their use in proving properties such as ergodicity and mixing. No prerequisites assumed.

Abstract. Hodge theory gives a transcendental method for studying algebraic geometry by building ladders between algebraic geometry, topology and analysis. This trichotomy is well-known for compact Kähler manifolds (or smooth projective varieties). Then, we will study the generalization of this trichotomy by considering singular manifolds and relative situations.

Abstract. A toric variety is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. We will start the minicourse from scratch, so the only background required would be familiarity with the basics of algebraic geometry. The plan is to cover (a subset of) the following topics: affine toric varieties, general toric varieties, singularities, divisors, line bundles, cohomology, Kahler differentials on toric varieties. We will mostly be following Mircea's lecture notes (http://www-personal.umich.edu/~mmustata/toric_var.html).

Abstract. In this course, we will begin by discussing some basic theory and motivations for congruences of modular forms, and then we will turn to constructing the eigencurve, a rigid analytic space parametrizing systems of eigenvalues of modular forms. The eigencurve was originally constructed by Coleman and Mazur, but we will use techniques of Andreatta-Iovita which give a more conceptually satisfying picture. Prerequisites include the classical theory of modular forms, and some algebraic geometry although I will fill in any gaps that arise as needed. Familiarity with formal schemes and rigid analytic spaces couldn't hurt, but it's not expected.