**Roberto Emparan** (Universitat de Barcelona)

**Title:** Black holes in the limit of very many dimensions.

**Abstract:** One-hundred years after Einstein formulated General Relativity, the pivotal role of its most fundamental and fascinating objects – the black holes – is nowadays recognized in many areas of physics, even beyond astrophysics and cosmology. Still, solving the theory that governs their dynamics remains a formidable challenge that continues to demand new ideas. I will argue that, from many points of view, it is natural to consider the number of spacetime dimensions, D, as an adjustable parameter in the theory. Then we can use it for a perturbative expansion of the theory around the limit of very many dimensions, that is, considering 1/D as a small number. We will see that in this limit the gravitational field of a black hole simplifies greatly and its equations often turn out to be analytically tractable. A simple picture emerges in which, among other things, the shape of the black hole is determined by the same equations that describe soap bubbles.

**Istvan Gyongy** (The University of Edinburgh)

**Title:** On stochastic partial differential equations of parabolic type.

**Abstract:** A brief introduction to the theory of stochastic partial differential equations will be presented. Applications to nonlinear filtering problems will be discussed. In particular, new results in the innovation problem will be given. This part of the talk is based on joint work with Nick Krylov.

**Carles Simó** (Universitat de Barcelona)

**Title:** Splitting, return maps and confined motion in the planar RTBP: Theory and praxis.

**Abstract:** Return maps to a given domain close to a broken separatrix are useful to understand many dynamical properties. The talk will concentrate on symplectic 2D maps and related problems.

The main ingredients are the return time to the domain and the splitting of separatrices. Different models are obtained, depending on the problem at hand. They allow to produce realistic quantitative estimates on the boundaries of confined motion.

The methodology is applied to the planar restricted three-body problem. Comparing the theoretical predictions with a careful numerical study one has deeper understanding and a source of new problems.

This is a joint work with Regina Martínez.

**Enric Ventura** (Universitat Politècnica Catalunya)

**Title:** The Conjugacy Problem and other algorithmically related questions

**Abstract:** A 1911 paper by Max Dehn iniciated a very fruitful line of research, which is in our days extremely dynamic and active: Algorithmic Group Theory. He introduced the famous Word, Conjugacy and Isomorphism Problems for discrete groups; against the original intuition, during the 1950’s, the three of them were showed to be unsolvable in general.

In this talk I will present a 2006 solution to the Conjugacy Problem for free-by-cyclic groups (an intriguing open question at that moment), and the further interesting development started when trying to extend this same result to the wilder class of free-by-free groups. This project is still going on, has proved to be very fruitful, and connects with a more primitive notion in algorithmic algebra: the so-called orbit decidability.

We will end the talk proving that a very easy-to-estate linear algebra problem about matrices over the integers, happens to be algorithmically unsolvable.

**Jim Wright** (The University of Edinburgh)

**Title:** Lebesgue Constants: connections with pointwise ergodic theorems.

**Abstract:** The classical Lebesgue constant for continuous periodic functions is useful in the study of pointwise and uniformly convergent fourier series. We examine variants for functions with a sparse spectrum and in particular we look at extensions to functions of several variables. Interestingly there are some connections with extensions and generalisations of Bourgain’s work on pointwise ergodic theorems along sparse subsets of integers.