MenuEvents for 05/06/2024 from all calendars
Number Theory Seminar
Time: 11:00AM - 12:00PM
Location: BLOC 302
Speaker: John Sung Min Lee, University of Illinois at Chicago
Title: On the distribution in arithmetic progressions of primes of various properties related to elliptic curves
Abstract: Given an elliptic curve $E/\mathbb{Q}$ and a prime $p$ of good reduction for $E$, let $\widetilde{E}_p(\mathbb{F}_p)$ denote the group of $\mathbb{F}_p$-rational points of the reduction of $E$ modulo $p$. One can define primes with various properties associated with $E$ based on the structure of $\widetilde{E}_p(\mathbb{F}_p)$. For instance, we call $p$ a prime of cyclic reduction, $m$-divisibility, and Koblitz reduction for $E$ if $\widetilde{E}_p(\mathbb{F}_p)$ is cyclic, has an order divisible by $m$, and has a prime order, respectively. In this talk, we study how the aforementioned primes, for an individual elliptic curve or on average, are distributed across arithmetic progressions. Furthermore, we analyze whether these primes are equally distributed or biased over congruence classes modulo $n$. This work is a partial collaboration with Nathan Jones, Jacob Mayle, and Tian Wang.
Geometry Seminar
Time: 3:00PM - 4:00PM
Location: BLOC 302
Speaker: F. Gesmundo, U. Toulouse
Title: Collineation varieties of tensors
Abstract: A linear space of matrices gives rise naturally to a family of algebraic varieties: the k-th collineation variety arises from the parametrization induced by the minors of size k. We propose the study of collineation varieties to obtain `equations' for interesting varieties in the space of tensors. In recent work with Hanieh Keneshlou, we classify these varieties in the case of pencils of matrices and nets of matrices of small dimension. In this seminar, I will introduce this construction and show to what extent the collineation varieties, and their geometric invariants (e.g. their degree), separate orbit-closures and allow us to recover varieties of interest in the study of border rank and border subrank of tensors.
