Abstract:
Gauss-Bhargava's higher composition laws: a new perespective

Abstract:
In this talk I will discuss some extension problems for finite flat group schemes and p-divisible groups.

Abstract:
The number of isomorphism classes of various types of elliptic curves over finite fields has attracted interest. We look at this problem over the family of elliptic curves over a finite field containing a point of small order.

Abstract:
In this talk, we present a few results on torsion subgroups of Jacobian varieties. We display explicit constructions of abelian varieties with rational torsion points of prescribed order over the rational field. We also explore some arithmetic questions on Jacobian varieties of low dimension.

Abstract:
In this talk, we will give an overview of the history of the proof of the modularity of elliptic curves over number fields, beginning with Wiles' proof of the semistable case over $\mathbb{Q}$, which in particular implies Fermat's Last Theorem. We will then explain how modularity was established for all elliptic curves over $\mathbb{Q}$, and outline the developments that led to the results in the totally real and CM cases. We will also briefly mention the recent works of Boxer-Calegari-Gee-Pilloni on two-dimensional abelian varieties. If time permits, we will outline some aspects of the method of proof, especially how a modularity lifting theorem - proved using an $R=\mathbb{T}$ theorem - implies modularity.