Oxford

Recently, as part of a project to find CY manifolds for which both the Hodge numbers (h^{11}, h^{21}) are small, manifolds have been found with Hodge numbers (4,1) and (1,1). The one-dimensional special geometries of their complex structures are more complicated than those previously studied. I will review these, emphasising the role of the fundamental period and Picard-Fuchs equation. Two arithmetic aspects arise: the first is the role of \zeta(3) in the monodromy matrices and the second is the fact, perhaps natural to a number theorist, that through a study of the CY manifolds over finite fields, modular functions can be associated to the singular manifolds of the family. This is a report on joint work with Volker Braun, Xenia de la Ossa and Duco van Straten.

I will introduce the general idea of p-adic Hodge theory from the view point of a beginner. Also, I will give a sketch of the proof of the crystalline comparison theorem in the case of good reduction using 'almost mathematics'.

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p[t])$

We pose the question of how to characterize "generic" elements of finitely generated groups. We set the scene by discussing recent results for linear groups in characteristic zero. To conclude we describe some new work in positive characteristic.

 This is joint work with Angus Macintyre. We prove a general model-completeness theorem for Henselian valued fields
stating that a Henselian valued field of characteristic zero with value group a Z-group and with finite ramification is model-complete in the language of rings provided that its residue field is model-complete. We apply this to extensions of p-adic fields showing that any finite or infinite extension of p-adics with finite ramification is model-complete in the language of rings.

A meromorphic connection on a complex curve can be interpreted as a representation of a simple Lie algebroid.  By integrating this Lie algebroid to a Lie groupoid, one obtains a complex surface on which the parallel transport of the connection is globally well-defined and holomorphic, despite the apparent singularities of the corresponding differential equations.  I will describe these groupoids and explain how they can be used to illuminate various aspects of the classical theory of singular ODEs, such as the resummation of divergent series solutions.  (This talk is based on joint work with Marco Gualtieri and Songhao Li.)

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.