Moduli spaces of sheaves on K3 surfaces have been well-studied when defined over the complex numbers, because they are one of the known families of hyperkaehler varieties. However, many of their arithmetic properties when defined over an arbitrary field are still unknown. In this talk, I will tell you about a new result in this direction: two such moduli spaces of the same dimension, when defined over a finite field, have the same number of points defined over every finite field extension of the base field, which is surprising when the moduli spaces are not birational. The way to get at this result is to study the cohomology groups of the moduli spaces as Galois representations. Over an arbitrary field, we find that all of the cohomology groups are isomorphic as Galois representations.

I will describe some aspects of the geometry of K3 surfaces in positive characteristic, including derived-category replacements for the classical Torelli theorem, supersingular analogues of twistor spaces, and some consequences for the arithmetic of certain elliptic curves over function fields. Some of the work described is joint with Daniel Bragg, and some is joint with Martin Olsson.

We will give a survey of recent progress on constructing local system over punctured projective lines using techniques from automorphic forms and geometric Langlands. Applications include solutions of particular cases of the inverse Galois problem and existence of motives with exceptional Galois groups.

The perturbation theory of linear operators has a long history. Introduced by Rayleigh in the 1870's, it was used for the first time in quantum mechanics in an article by Schrödinger published in 1926. The mathematical study of the perturbation theory of self-adjoint operators was initiated by Rellich in 1937, and has been since then the matter of a large number of contributions in the mathematical literature.

Perturbation theory of nonlinear operators plays a key role in quantum physics and chemistry, where it is used in particular to compute the response properties of molecular systems to external electromagnetic fields (polarizability, hyperpolarizability, magnetic susceptibility, NMR shielding tensor, optical rotation, ...) within the framework of mean-field models.

In this talk, I will recall the basics of linear perturbation linear, present some recent theoretical results [1] on nonlinear perturbation theory, and show how this approach can be also used to speed-up numerical simulations [2,3] and compute effective a posteriori error bounds.

[1] E. Cancès and N. Mourad, A mathematical perspective on density functional perturbation theory, Nonlinearity 27 (2014) 1999-2034.
[2] E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralik, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, CRM 352 (2014) 941-946.
[3] E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralik, A perturbation-method-based post-processing for the planewave discretization of Kohn-Sham models, in preparation.

The fundamental group of a topological space is usually defined in terms of homotopy classes of based loops. The group structure is given by composition of loops. If X is a complex algebraic variety, one has an underlying topological space, and hence a fundamental group. Hain showed that the nilpotent completion of the group ring of this topological fundamental group carries a mixed Hodge structure. Grothendieck defined a fundamental group for schemes defined over any field. Applying this to a complex algebraic variety, one obtains the profinite completion of the topological fundamental group. This group comes with a natural action of the absolute Galois group of the field of definition. The above indicates that varieties over fields of characteristic zero have two notions of fundamental group, armed with either a Galois action or a mixed Hodge structure. This is similar to the situation with homology and cohomology groups, where one has both an etale and Betti version carrying Galois actions and Hodge structures. An important guiding principle is that both of these versions of homology and cohomology should come from an underlying ``motivic'' theory. This is a homology and cohomology theory for algebraic varieties over a field k taking values the abelian tensor category of mixed motives over k, denoted M_k. There should be functors from M_k to both the category of Galois modules and mixed Hodge structures. When applied to the motivic homology of a variety X, these functors should yield the etale homology or Betti homology of X. In this way, motives glue these two different theories together more strongly than just the comparison isomorphisms. Nori has recently given a definition of the category of mixed motives. In this talk, we will show how this category relates to the fundamental group. In fact, more generally there is a motivic version of paths between two different points x and y of X which is important for applications. We also show that the multiplication and comultiplication maps are motivic, and compare with Hain's theory.