In this talk we will give an introduction to the theory of p-adic (or rigid) cohomology. We will first define the theory for smooth affine varieties, then sketch the definition in general, next compute a simple example, and finally discuss some applications.
The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'.
I will discuss the structure of the Selberg class - in which certain expected properties of Dirichlet series and L-functions are axiomatised - along with the numerous interesting conjectures concerning the Dirichlet series in the Selberg class. Furthermore, results regarding the degree of the elements in the Selberg class shall be explored, culminating in the recent work of Kaczorowski and Perelli in which they prove the absence of elements with degree between one and two.
We consider the prime k-tuples conjecture, which predicts that a system of linear forms are simultaneously prime infinitely often, provided that there are no obvious obstructions. We discuss some motivations for this and some progress towards proving weakened forms of the conjecture.
I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others
I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others
Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the
first talk.
Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction.