Berger conjecture, valuations and torsions

Abstract: Let $(R,\m_R,\k)$ 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 $\Omega_R$ 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 previous known results. This is  joint work with Craig Huneke
and Sarasij Maitra.