\indent Given a Pfaffian system on a smooth manifold, we shall discuss the notion of reduced submanifold and how to find them. This was motivated from the problem of deciding the minimality of generic CR manifolds. As best known by the Noether's theorem conservation laws arise from the symmetry of differential equations. We approach the conservation laws from the viewpoint of the reduction of Pfaffian systems and discuss some possible applications.

\indent Counting curves on threefolds has been defined in several conjecturally equivalent instances, by integrating with respect to a virtual class on a moduli space (of stable maps for Gromov-Witten theory, ideal sheaves for Donaldson-Thomas theory, and stable pairs for Pandharipande-Thomas theory). The analogous picture for K3 surfaces is incomplete. The Gromov-Witten theory has been calculated by Maulik, Pandharipande, and Thomas, and was shown to give rise to modular forms. In joint work with Benjamin Bakker we define and compute an analog of DT/PT theory on K3 surfaces via stable pairs and show that it similarly gives rise to modular forms on $\Gamma(4)$.

\indent I will first describe a generalization of a result by Haagerup and Thorbjornsen on the asymptotic norm of non-commutative polynomials of random matrices, in the case of unitary matrices. Then I will show how such results help us refine our understanding of the outputs of random quantum channels. In particular one obtains optimal estimates for the minimum output entropy of a large class of typical quantum channels. The first part of this talk is based on joint work with Camille Male, and the second part is based on joint work with Serban Belinschi and Ion Nechita.

It is well-known from Weil that the isomorphism classes of rank n vector bundles on an algebraic curve can be written as the set of certain double cosets of GL(n,A), where A is the adeles of the curve. I will introduce such presentation in the realm of algebraic geometry and discuss two of its applications: the first is the Tamagawa number formula (proved by Gaitsgory-Lurie), and the second is the Verlinde formula in positive characteristic

\indent One-parameter convolution semigroups of probability measures on Euclidean space are related to limits of partial products of infinitesimal triangular systems of measures, in the sense that such limits are embeddable into one-parameter convolution semigroups. It is a long-standing problem related to the central limit theorem that on an arbitrary locally compact group such a result cannot be tackled unless the infinitesimal system is commutative and additional conditions on the underlying group and/or the limiting measure are satisfied. We shall develop the main steps towards the solution of the problem of embeddable limits and connect the problem with the embedding of infinitely divisible probability measures on the group. The problem, in full generality, is still open.

\indent We consider the question of global in time existence and uniqueness of solutions of the infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial data that is small in its kinetic energy and height, we show that the 2-D full water wave equation is uniquely solvable almost globally in time. And for any initial interface that is small in its steepness and velocity, we show that the 3-D full water wave equation is uniquely solvable globally in time.