Speaker：Dingxin Zhang (Tsinghua University)

Time：2025-3-12, 9:00 & 2025-3-13, 14:30

Location：Conference Room S206 & C610 

Abstract:

I: We present the theory of regular holonomic D-modules over a one-dimensional base, with a focus on the V-filtration and vanishing cycle constructions. The study of the one-dimensional case provides insights that may extend to analogous constructions in higher dimensions.

II: Let f be a convergent power series in n+1 variables with an isolated critical point at 0. Consider the finite-dimensional C-algebra R = C{x₀,...,xₙ}/(∂f/∂x₀,...,∂f/∂xₙ). There exists an N such that fᴺ= 0 in R. The Brionçon–Skoda theorem states that we can take N = n+1. This remarkable result remained without an algebraic proof for many years. We will present Varchenko's interpretation of this theorem using D-modules and Hodge theory: if the Jordan blocks of the local monodromy operator acting on the Milnor fiber of f have sizes ≤ r, then f^r= 0 in R.