Counting congruences between modular forms
For a given eigenform, how many other eigenforms of the same weight and level are congruent to it modulo a prime \(l\)? We will explain how modular representation theory can be used to help answer this question. We focus especially on the case when the level is the square of a prime \(p\) such that \(p = -1\) mod \(l\). This is ongoing joint work with Robert Pollack and Preston Wake.
On the Telhcirid problem
We consider the digital reverse of integers, in particular those of primes. A palindromic prime number is a popular example of a prime whose reverse is also a prime. The infinitude of such primes is one among the open conjectures in the area. We will discuss available partial results on almost primes and build on these to present reversed primes in arithmetic progression. The new results are from my joint work with Yuta Suzuki.
Equivariant localization theorem
The Atiyah-Bott localization theorem says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed-point subspace. After discussing this classical theorem in algebraic topology, we discuss a categorification of this result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology) in algebraic geometry. This is based on a joint work with Adeel Khan.
Difference graphs of étale algebras over finite fields
In 1992, Winnie Li studied properties of Cayley graphs arising from finite field extensions \(\mathbb{F}_{q^n}\) over \(\mathbb{F}_q\). The vertices of the graph are elements of \(\mathbb{F}_{q^n}\) and two vertices are adjacent if they differ by an element of norm one. In her paper, properties such as connectedness and girth are studied. Moreover, the spectrum of the graphs in this family are shown to be generalized Kloosterman sums. In this talk, we investigate these properties in a related class of graphs, where \(\mathbb{F}_{q^n}\) is replaced by an étale algebra of degree \(n\).
Iwasawa main conjectures for \(p\)-adic Lie extensions of graphs
If you would like to attend this event, please register here. Registration is free but it would help us with the preparation of logistics. The deadline to register is April 23rd.
Venue: LH-1, Department of Mathematics