Time: March 15-16 2025
March 15
9:30-10:20
Speaker: Liding Huang, Xiamen University
Title: The form-type Calabi-Yau equation on a class of complex manifolds
Abstract:
The Calabi-Yau theorem says that given any smooth representative $\Phi$ of the first Chern class, there exists a unique K\”ahler metric $\omega$ cohomologous to $\alpha$ such that $Ricci(\omega)=\Phi$. It is natural to investigate whether similar results hold when the manifolds is non-K\”ahler. In this talk, we will introduce the form type Calabi-Yau equation which can used to prove a version of the Calabi conjecture for balanced metrics. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.
10:40-11:30
Speaker: Ling Wang, Peking University
Title: Flat level sets of Allen-Cahn equation in half-space
Abstract:
In this talk, I will present a half-space Bernstein theorem for Allen-Cahn equation. More precisely, I will show that every solution $u$ of the Allen-Cahn equation in the half-space $\overline{\mathbb{R}^{n}_{+}}$ with $|u|\leq 1$, boundary value given by the restriction of a one-dimensional solution on $\{x_1=0\}$ and monotone condition $\partial_{x_n}u>0$ as well as limiting condition $\lim_{x_n\to\pm\infty}u(x',x_n)=\pm 1$ must itself be one-dimensional. This talk is based on recent work joint with Wenkui Du and Yang Yang.
15:00-15:50
Speaker: Kewei Zhang, Beijing Normal University
Title: Quantization methods in Kahler geometry
Abstract:
I will first review the classical quantization methods for projective manifolds developed by Prof. Tian, which plays significant roles in the study of the YTD conjecture. Then I will introduce a quantization method for non-algebraic Kahler manifolds, which was proposed by R. Berman.I will also discuss its potential application in the cscK problem.
16:10-17:00
Speaker: Jianchun Chu, Peking University
Title: On Kahler manifolds with non-negative mixed curvature
Abstract:
The mixed curvature comes from a linear combination of the Ricci and holomorphic sectional curvature. In this talk, we will first survey recent progress. Then we discuss a structure theorem for compact Kahler manifolds with non-negative mixed curvature. This is a joint work with Man-Chun Lee and Jintian Zhu.
March 16
8:30-9:20
Speaker: Jian Song, Rutgers University
Title: Overview of the analytic MMP with Ricci flow
Abstract:
In this talk, I will give an overview of the analytic minimal model program with Ricci flow proposed by Tian and myself.
9:30-11:00
Speaker: Wangjian Jian, Institute of Mathematics, C.A.S.
Title: The curvature and diameter estimates for the finite-time Kahler-Ricci flow
Abstract:
First, we will recall the basic set-up of the finite-time Kahler-Ricci flow on the AMMP background. Then we will recall some previous results on the curvature estimates, especially Perelman's estimates in Fano case. Then we will talk about the recent Li-Yau type and Harnack type estimates for general cases, and talk about some details of the proofs. Finally, we will talk about remaining problems.
11:20-12:10
Speaker: Max Hallgren, Rutgers University
Title: Finite-Time Singularities of the Ricci Flow on Kähler Surfaces
Abstract:
By work of Song-Weinkove, it is understood that the Ricci flow on any Kähler surface can canonically be continued through singularities in a continuous way until its volume collapses.
This talk will discuss recent progress in understanding a more detailed picture of the singularity formation in this context.
