Let C be an elliptic or hyperelliptic curve over a p-adic field K. Then C is equipped with a Galois representation, given by the action of the absolute Galois group of K on the Tate module of C. The behaviour of this representation depends on the reduction type of C. We will focus on the case of C having bad reduction, and acquiring potentially good reduction over a wildly ramified extension of K. We will show that, if C is an elliptic curve, the Galois representation can be completely determined in this case, thus allowing one to fully classify Galois representations attached to elliptic curves. Furthermore, the same can be done for a special family of hyperelliptic curves, obtaining a result which is surprisingly similar to that for the corresponding elliptic curves case.
 

We will survey an analogy between random integers and random permutations, which goes back to works of Erdős and Kac and of Billingsley.
This analogy inspired results and proofs about permutations, originating in the setting of integers, and vice versa.
Extensions of this analogy will be described, involving the generalized Ewens measure on permutations, based on joint work with D. Elboim.
If time permits, an analogous analogy, this time between random polynomials over a finite field and random permutations, will be discussed and formalized, with some applications.
 

Chowla's conjecture from the 1960s is the assertion that the Möbius function does not correlate with its own shifts. I'll discuss some recent works where with collaborators we have made progress on this conjecture.

In this talk, I will talk about two seemingly disjoint topics - Vinogradov’s mean value theorem, a classically important topic of study in additive number theory concerning solutions to a specific system of diophantine equations, and Incidence geometry, a collection of combinatorial results which focus on estimating the number of incidences between an arbitrary set of points and curves. I will give a brief overview of these two topics along with some basic proofs and applications, and then point out how these subjects connect together.

Local-to-global principles are a key tool in arithmetic geometry. Through a theorem of Siegel on representations of totally positive numbers as sums of squares in number fields we give a concrete introduction to the Hasse principle, and briefly talk about other local-to-global principles. No prerequisites from algebraic number theory are assumed, although some familiarity is helpful for context.

In this session, Ben Fehrman and Markus Upmeier will give their thoughts on how to deliver a good talk for a conference or a seminar and tips for what to do and what to avoid. There will be a particular emphasis on how to give a good talk online. 

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The classical bulk reconstruction is provided by the sl(2)-orbit theorem, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us to prove that the number of self-dual flux vacua is finite. I will point out how the finiteness result for supersymmetric fluxes relates to the Hodge conjecture.