Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.
I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.
 

I will review the notion of classifying topos of a first-order (geometric) theory and explain the central role enjoyed by theories of presheaf type (i.e. classified by a presheaf topos) in the context of the topos-theoretic investigation of the model theory of geometric theories. After presenting a few main results and characterizations for theories of presheaf type, I will illustrate the generality of the point of view provided by this class of theories by discussing a topos-theoretic framework unifying and generalizing Fraissé’s construction in model theory and topological Galois theory and leading to an approach to the problem of the independence from l of l-adic cohomology.

The Constantin-Lax-Majda model is a 1d system which shares certain features (related to vortex stretching) with the 3d Euler equation. The model is explicitly solvable and exhibits finite-time blow-up for an open subset of smooth initial data. In 1990s De Gregorio suggested adding a transport term to the system, which is analogous to the transport term in the Euler equation. It turns out the transport term has some regularizing effects, which we will discuss in the lecture.

I will  address the study of decay rates of solutions to dissipative equations. The characterization of these rates will first be given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. The understanding of the behavior of the linear  combined with the decay character and the Fourier Splitting method is then used to obtain some  upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case. 

I will discuss the specific long-time behaviour of kinetic solutions to stochastic conservation laws with non-linear multiplicative noises.

We consider the Euler–Voigt equations in a bounded domain as an approximation for the 3D Euler equations. We adopt suitable physical conditions and show that the solutions of the Voigt equations are global, do not smooth out the solutions and converge to the solutions of the Euler equations, hence they represent a good model.

We describe a general program for the classification of flat connections on topological manifolds. This is motivated by the classification of locally homogeneous geometric structures on manifolds, in the spirit of Ehresmann and Thurston.  This leads to interesting dynamical systems arising from mapping class group actions on character varieties. The mapping class group action is a discrete version of a continuous object, namely the extension of the Teichmueller flow to a  unversal character variety over over the tangent bundle of Teichmuller space. We give several examples of this construction
and discuss joint work with Giovanni Forni on a mixing property of this suspended flow.

I shall discuss Zauner's conjecture about the existence of n^2 mutually equidistant points in complex projective space CP^{n-1} with its standard Fubini-Study metric. This is the so-called SIC-POVM problem, and is related to properties of the moment mapping that embeds CP^{n-1} into the Lie algebra su(n). In the case n=3, there is an obvious 1-parameter family of such sets of 9 points under the action of SU(3) and we shall sketch a proof that there are no others. This is joint work with Lane Hughston.

The moduli space M(G) of Higgs bundles for a complex reductive group G on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G.  In this talk, I will focus on the real groups SU^*(2m), SO^*(4m) and Sp(m,m). The image under the Hitchin map of Higgs bundles for these groups is completely contained in the discriminant locus of the base and our analysis is carried out by describing the whole
(singular) fibres they intersect. These turn out to be certain subvarieties of the moduli space of rank 1 torsion-free sheaves on a non-reduced curve. If time permits we will also discuss another class of complex Lagrangians in M(G) which can be constructed from symplectic representations of G.

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real
Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane. In this talk, I will explain how to compute the mod 2 Betti numbers of M_R using Morse theory. A key role in this calculation is played by the Abel-Jacobi map from symmetric products of X to the Jacobian of X.