Soluble groups, and other classes of groups that can be built from simpler groups, are useful test cases for studying group properties. I will talk about techniques for building profinite groups from simpler ones, and how  to use these to investigate the cohomology of such groups and recover information about the group structure.

I will explain the basics of deformation quantization of Poisson
algebras (an important tool in mathematical physics). Roughly, it is a
family of associative algebras deforming the original commutative
algebra. Following Fedosov, I will describe a classification of
quantizations of (algebraic) symplectic manifolds.
 

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

Many numerical solvers of ordinary differential equations require problems to be posed as a system of first order differential equations. This means that if one wishes to solve higher order problems, the system have to be rewritten, which is a cumbersome and error-prone process. This talk presents a technique for automatically doing such reformulations.

Choreographies are periodic solutions of the n-body problem in which all of the bodies have unit masses, share a common orbit and are uniformly spread along it. In this talk, I will present an algorithm for numerical computation and stability analysis of choreographies.  It is based on approximations by trigonometric polynomials, minimization of the action functional using a closed-form expression of the gradient, quasi-Newton methods, automatic differentiation and Floquet stability analysis.

In this talk we will introduce the (bounded) derived category of coherent sheaves on a smooth projective variety X, and explain how the geometry of X endows this category with a very rigid structure. In particular we will give an overview of a theorem of Orlov which states that any sufficiently ‘nice’ functor between such categories must be Fourier-Mukai.

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also discuss a related version of Heegaard Floer homology, which is more computable.

In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of metric properties, provided that the parabolic subgroups have these properties. I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.