## Department Colloquium

### Friday, February 28th, 2020

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Anup Dixit (Queen's University)

**Title:** The generalized Brauer-Siegel conjecture.

**Abstract: ** One of the principal objects of study in number theory are number fields $K$, which are finite field extensions of $\mathbb{Q}$. The ring of integers of $K$ is analogous to integers $\mathbb{Z}$ in $\mathbb{Q}$. A natural question to investigate is if the ring of integers of $K$ is a unique factorization domain. The answer lies in the study of the invariant class number of $K$, which captures how far the ring of integers of $K$ is from having unique factorization. The origins of this problem can be traced back to Gauss, who conjectured that there are finitely many imaginary quadratic fields with this property. This was proved in the mid-twentieth century, independently by Baker, Heegner and Stark. A more intricate question is to understand how class number varies on varying number fields. In this context, the generalized Brauer-Siegel conjecture, formulated by M. Tsfasman and S. Vl\u{a}du\c{t} in 2002, predicts the behavior of class number times the regulator over certain families of number fields. In this talk, we will discuss recent progress towards this conjecture, in particular, establishing it in special cases.

**Anup Dixit** is a Coleman Postdoctoral Fellow at Queen's University under the supervision of M. Ram Murty. He obtained his Ph.D. in Mathematics from the University of Toronto in 2018. He is interested in analytic as well as algebraic number theory. He has worked on families of L-functions, behaviour of the class number on varying number fields, infinite extensions of number fields, universality of functions and Euler-Kronecker constants.