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.
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.
In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.
In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.
Motivated by an open conjecture in anabelian geometry, we investigate which arithmetic properties of the rationals are encoded in the absolute Galois group G_Q. We give a model-theoretic framework for studying absolute Galois groups and discuss in what respect orderings and valuations of the field are known to their first-order theory. Some questions regarding local-global principles and the transfer to elementary extensions of Q are raised.