13:00-14:00 Dimitar Jetchev (EPFL)

Explicit Isogenies and the Discrete Logarithm Problem for Curves of Genus 2 Over Finite Fields

The discrete logarithm problem (DLP) on groups of points on elliptic curves and Jacobians of curves of genus 2 over finite fields is one of the fundamental computational problems in mathematical cryptology. A proper understanding (both theoretical and practical) of the difficulty of solving this problem will allow for a better security assessment of multiple cryptographic schemes. The mathematics behind this problem is rich and involves ideas from both number theory and arithmetic algebraic geometry. In this talk, I will focus on how one can compare the hardness of the problem within certain classes of curves. More precisely, I will explain recent results proving that DLP is equally within an isogeny class of elliptic curves (a random self-reducibility property). Establishing such a result in genus 2 looks significantly more subtle. I will provide a new algorithm for explicitly computing cyclic isogenies between Jacobians of curves of genus 2 based on Shimura's theory of polarized abelian varieties with complex multiplication and how this algorithm can be used for proving random self-reducibility in genus 2. This is an ongoing work with Alina Dudeanu.

14:15-15:15 Hugo Duminil (Geneva)

Self-Avoiding Walk against Simple Random Walk

In this talk, we will discuss the Self-Avoiding Walk model, focusing on the comparison between this model and the Simple Random Walk model. We will describe both combinatorial and geometric aspects of the model, and in particular the use of probabilistic and geometric techniques to solve combinatorial questions.

15:30-16:30 Balazs Szegedy (Toronto)

Graph limit theory for algebraic geometers

The so-called graph limit space is the completion of the set of finite graphs in the topology generated by subgraph densities. This space comes together with a commutative algebra of functions and basic concepts from algebraic geometry can be defined on. The purpose of the talk is to present a connection between combinatorics and geometry. We show applications in extremal combinatorics and various open problems that are interesting outside combinatorics. (Joint work with Laszlo Lovasz)