Lecture Series

Mixed characteristic vanishing theorems and applications

A lecture series by Bhargav Bhatt, Shou Yoshikawa and Jakub Witaszek.

Abstract: The goal of this series of talks is to explain some recent work on vanishing theorems in mixed characteristic algebraic geometry and as well as the ensuing progress in the minimal model program (MMP).

In the first two talks, Bhatt will explain a variant of the Kodaira vanishing theorem in mixed characteristic as well as some immediate applications. In the third talk, Yoshikawa will give an introduction to the MMP in mixed characteristic and explain why a variant of Kodaira vanishing theorem is reasonable for the MMP in mixed characteristic. In the final talk, Witaszek will explain the strategy of the proof of the MMP for arithmetic threefolds of residue characteristics zero or greater than five, and explore further topics on mixed characteristic vanishing theorems.

The work presented in these talks relies on work of several mathematicians besides the speakers, including Lurie, Ma, Patakfalvi, Schwede, Takamatsu, Tucker, and Waldron.

Schedule: 12-1 pm Mountain Time (Salt Lake City Time) on May 17 (Bhatt), 20 (Bhatt), 24 (Yoshikawa @4pm), 27 (Witaszek)

Recent progress in K-stability of Fano varieties

A lecture series by Yuchen Liu and Ziquan Zhuang.

Abstract: K-stability was first introduced by Tian to characterize the solution of the Kähler-Einstein problem on Fano varieties. In the last decade, a purely algebraic geometric study of K-stability has prospered, based on the birational classification theory of varieties centered around the minimal model program. In the first part of this lecture series, we will discuss algebraic geometer's gradually evolving understanding of the K-stability through birational geometry and valuation theory. We will introduce the higher rank finite generation (HRFG) conjecture, and explain its connection to several foundational questions in K-stability. In the second part, we will discuss our recent joint work with Chenyang Xu, on settling the finite generation conjecture.

Schedule: 12-1 pm Mountain Time (Salt Lake City Time) on May 18 (Liu), 21 (Zhuang), 25 (Zhuang), 28 (Liu)