**Mathematics Department**

**Commutative algebra seminar**

**Date and time:** 15 December 2022, 11 am

**Venue:** Ramanujan Hall

**Title:** Berger conjecture, valuations, and torsions

**Abstract: **Let R,m,k be a one-dimensional complete local reduced k-algebra over a field of characteristic zero. Berger conjectured that R is regular if and only if the universally finite module of differentials O is torsion-free. We discuss methods that have been used in the past to prove cases where the conjecture holds. When R is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of R. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of R and orders of units of the integral closure of R. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of R. This also generalizes previously known results. This is joint work with Craig Huneke and Sarasij Maitra.