# JAE School of Mathematics 2013: programme

Algebraic Geometry with Analytic Techniques: an Introduction to Complex Geometry

Complex Geometry is a rich and highly attractive field in modern mathematics. Its interaction with other fields such as algebraic geometry and differential geometry is obvious, but it also plays a role in the Conformal Field Theory and String Theory, so one may say that it is a multi-faceted area of geometry. In this course we provide a brief introduction that will range from the most basic concepts, such as complex manifolds, to the most common tools employed in the area, such as the Cech Cohomology, holomorphic bundles and their connections, and finishing off with Chern classes.

Geometry of Dynamics: Deterministic and Stochastic (This course will be given in english)

“Basic dynamical systems” is a course that every mathematician must take, nowadays, at some point or other in their education, especially if their emphasis is applied mathematics. The framework for such courses is the geometrical, phase space point of view of Poincaré, and dynamics is described in terms of autonomous vector fields (which generate flows) and maps (which could arise from time-­periodic vector fields).

However, many models of physical phenomena lead to nonautonomous and stochastic vector fields. Does the same geometrical approach to dynamical systems apply in this setting? Moreover, in what sense to nonautonomous and stochastic vector fields define dynamical systems?

In these lectures I will develop the basic building blocks of geometrical dynamical systems for autonomous, nonautonomous, and stochastic vector fields “side-­ŌĆÉby-­ŌĆÉside”. The idea is that students will find it easy to grasp the “traditional” approach to dynamical systems theory and this will serve as a foundation for the more difficult ideas associated with nonautonomous and stochastic dynamical systems. The emphasis in the lectures will be on concrete examples, rather than proofs of theorems (although a guide to the key theorems and their proofs will be given).

Introduction to partial differential equations of evolution

This course will provide an introduction to some of the main techniques and ideas that arise in the study of evolution equations.

Fluid Mechanics

The aims of the course will be to introduce the Euler and Navier-Stokes equations: the physical motivation, the most relevant characteristics, the open problems, etc., as well as to present some of the main research lines currently open for studying them.

Realization Problems in Foliation Theory

The aim is to introduce students to Foliation Theory, specifically to the problems of realization of sub-manifolds as leaves of foliation. Special emphasis will be placed on some interesting examples of the theory.

Number Theory

This course consists of several topics in elemental theory and elemental theory and analytic number theory. The first is an introduction to Number Theory using examples. The second explains how the trigonometric sums and some techniques with an analytic flavor may yield quite spectacular results on prime numbers. The third topic falls within what is known as additive number theory; we will use tools such as discrete Fourier analysis to demonstrate results in the sense that subsets of integers that avoid arithmetic progressions have low densities.

Symplectic Topology

The aim is to introduce Symplectic Topology to those attending the course. This will be done with the exposition of two results that are central to the history of this subject. The first result is the non-squeezing theorem or the symplectic camel, for the demonstration of which we will use pseudo-holomorphic curves, which explain the ground-breaking work by Gromov. We will then concentrate on Lagrangian sub-manifolds of a symplectic manifold. In particular, we address Arnold’s Conjecture; from the infinitesimal version coming from Sturm’s theorem, the finite version from the Morse theorem and the general case by Floer Homology.