Speaker: 帖经智（University of Georgia）
Time: 9:30-10:30, Dec 22，2023
Venue: 数学中心研讨室（至善楼602）
Online: Zoom: 811 5638 1768 (Password: msrc)
Title: The exact degree of space of harmonic polynomials on the Heisenberg group
Abstract:
I will first talk about the classical result about the dimension of linear space of harmonic polynomials on the Euclidean space $R^n$. Then I will talk about the analogous results on the Heisenberg group.
Cirriculum Vitae:
美国 University of Georgia 教授，1995年于University of Toronto获博士学位，1985年于兰州大学数学系获学士学位。 帖经智教授主要从事Harmonic Analysis、Several Complex Variable, Geometric Analysis、Mathematical Finance 等研究工作。其研究领域包括：Analysis and Geometry on Heisenberg Group, Pseudo-Hermitian manifolds。目前已在Canadian Journal of Mathematics，Journal of Geometric Analysis, Journal of Differential Geometry, Communication in PDEs, Journal of Optimization and Application 等国际学术期刊上发表SCI收录论文40余篇。
Speaker:Prof. Yong Luo (MSRC, CQUT)
Time: 14:00-15:00, Dec 11, 2023
Venue: 数学中心研讨室（至善楼602）
Title: 子流行几何1
Abstract:
在本讲中我们介绍黎曼和子流形几何中的基本概念和定理，包括黎曼度量，联络和曲率，以及子流形几何中的几个基本定义。
Speaker: Prof. Shu-Cheng Chang (MSRC, CQUT)
Time: 14:00-15:00, Nov 27, 2023
Venue: 数学中心研讨室（至善楼602）
Title: From Thurston Geometrization Conjecture to Mori Minimal Model Program (II)
Abstract:
A central problem of differential geometry is the geometrization problem on manifolds. In particular, it is to determine which manifolds admit certain geometric structures. One of methods is to understand and classify the singularity models of the corresponding nonlinear geometric evolution equation, and to connect it to the existence problem of geometric structures on manifolds. In 1982, R. Hamilton introduced the Ricci flow and then by studying the singularity models of Ricci flow, G. Perelman completely solved Thurston geometrization conjecture and Poincare conjecture for a closed 3-manifold in 2002-2003. On the other hand, Mori minimal model program in birational geometry can be viewed as the complex analogue of Thurston's geometrization conjecture. In 1985, H.-D. Cao introduced the Kähler-Ricci flow and then recaptured the Calabi-Yau Conjecture. Moreover, there is a conjecture picture by Song-Tian that the Kähler-Ricci flow should carry out the analytic minimal model program with scaling on projective varieties. Recently, Song-Weinkove established the above conjecture on a projective algebraic surface. Furthermore, via the Sasaki-Ricci flow, Chang-Lin-Wu proved the Sasaki analogue of minimal model program on closed quasi-regular Sasakian 5-manifolds. In this talk, I will address the above issues in some detail.
Speaker: Prof. Shu-Cheng Chang (MSRC, CQUT)
Time: 14:00-15:00, Nov 20, 2023
Venue: 数学中心研讨室（至善楼602）
Title: From Thurston Geometrization Conjecture to Mori Minimal Model Program (I)
Abstract:
A central problem of differential geometry is the geometrization problem on manifolds. In particular, it is to determine which manifolds admit certain geometric structures. One of methods is to understand and classify the singularity models of the corresponding nonlinear geometric evolution equation, and to connect it to the existence problem of geometric structures on manifolds. In 1982, R. Hamilton introduced the Ricci flow and then by studying the singularity models of Ricci flow, G. Perelman completely solved Thurston geometrization conjecture and Poincare conjecture for a closed 3-manifold in 2002-2003. On the other hand, Mori minimal model program in birational geometry can be viewed as the complex analogue of Thurston's geometrization conjecture. In 1985, H.-D. Cao introduced the Kähler-Ricci flow and then recaptured the Calabi-Yau Conjecture. Moreover, there is a conjecture picture by Song-Tian that the Kähler-Ricci flow should carry out the analytic minimal model program with scaling on projective varieties. Recently, Song-Weinkove established the above conjecture on a projective algebraic surface. Furthermore, via the Sasaki-Ricci flow, Chang-Lin-Wu proved the Sasaki analogue of minimal model program on closed quasi-regular Sasakian 5-manifolds. In this talk, I will address the above issues in some detail.
Speaker: Prof. Shu-Cheng Chang (MSRC, CQUT)
Time: 11:00-12:00, Nov 15, 2023
Venue: 数学中心研讨室（至善楼602）
Title: The Special Legendrian Graph
Abstract:
I will derive the n-dimensional special Legendrian equation in the (2n+1)-dimensional Euclidean space with the standard Sasakian structure. It is related to the special Lagrangian equation over the associated cone. In particular, the 2-dimensional special Legendrian equation is related to a 3-dimensional real Monge-Ampere equation in the cone over 2-dimensional Euclidean space. At last, I will address several open problems on the Berstein-type theorem and existence theorem.
Speaker: 谢君明 (罗格斯大学)
Time: 15:30-16:30, July 20, 2023
Venue: 数学中心研讨室（至善楼602）
Online: Zoom: 87060973893 (Password: msrc)
Title: Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature
Abstract:
In this talk, we will discuss the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. We prove a certain form of curvature estimates for such Ricci shrinkers, including a quadratic curvature lower bound estimate for noncompact ones with half PIC. As a consequence, we classify 4-dimensional complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature, except the half PIC case. We also treat the half PIC case under an additional assumption that the Ricci tensor has an eigenvalue with multiplicity 3. This talk is based on the joint work with Huai-Dong Cao.
Speaker: 李宇 (中国科学技术大学几何与物理研究中心)
Time: 10:00-11:30, June 6, 2023
Venue: Zoom Meeting
Title: On Kähler Ricci shrinker surfaces
Abstract:
We use convergence theories for Ricci shrinkers to show that non-compact Kähler Ricci shrinker surfaces have two distinct canonical neighborhoods outside a compact set. Therefore, we prove that Kähler Ricci shrinker surfaces have uniformly bounded sectional curvature. By combining this curvature estimate with previous research by multiple authors, we achieve a comprehensive classification of all Kähler Ricci shrinker surfaces. This work was done in collaboration with Bing Wang.
Speaker: 陈柏扬 (University of California, San Diego)
Time: 10:00-11:30, May 25, 2023
Venue: Zoom Meeting
Title: Curvature and gap theorems of gradient Ricci solitons
Abstract:
Ricci flow deforms the Riemannian structure of a manifold in the direction of its Ricci curvature and tends to regularize the metric. This provides useful information about the underlying space. Ricci solitons are special solutions to the Ricci flow and arise naturally in the singularity analysis of the flow. We shall discuss some curvature and entropy gap theorems of gradient Ricci solitons. This talk is based on joint works with Zilu Ma and Yongjia Zhang, Eric Chen and Man-Chun Lee.
Speaker: 张科伟 (北京师范大学)
Time: 10:00-11:30, May 18, 2023
Venue: 数学中心研讨室（至善楼602）& Zoom
Title: A pluripotential approach to the Kähler -Einstein problem
Abstract:
Searching for Kähler-Einstein metrics on Fano manifolds is an important problem in geometric analysis. By Tian’s solution of the YTD conjecture, it is known that a K-stable Fano manifold always admits a KE metric. In this talk I will present a pluripotential approach to this problem. More precisely, we show that a Fano manifold admits a unique KE metric if and only if its delta invariant is bigger than 1. This is a joint work with T. Darvas.
Speaker: Chien Lin (MSRC, CQUT)
Time: 10:00-11:30, May 11, 2023
Venue: 数学中心研讨室（至善楼602）& Zoom
Title: On complete gradient shrinking Ricci solitons
Abstract:
Recently, Yu Li and Bing Wang gave the classification of Kähler-Ricci shrinker surfaces by confirming the boundedness of their sectional curvatures. In the course of the proof, the asymptotic behavior of the potential function of a shrinking soliton plays an essential role so that today I want to present the relation between the potential function and the distance function in the Cao-Zhou’s paper “On complete gradient shrinking Ricci solitons (JDG, 2010)”.
Speaker: Prof. Yuhua Sun (Nankai University)
Time: 10:00-11:30, April 27, 2023
Venue: 数学中心研讨室（至善楼602）& Zoom
Title: Quasilinear Laplace Equations and Inequalities with Fractional Orders
Abstract:
We report our recent work on Laplace equations, and inequalities with fractional order, where we establish the existence of solution, fundamental solution, and Liouville results.
Speaker: Prof. Yong Luo (MSRC, CQUT)
Time: 10:00-11:30, April 20, 2023
Venue: 数学中心研讨室（至善楼602）& Zoom
Title: Volume gap for minimal submanifolds in spheres
Abstract:
In this report, we will talk about the paper “Volume gap for minimal submanifolds in spheres” (arXiv:2210.04654v1) by Jianquan Ge and Fagui Li, which concerns the lower bound of volume of minimal submanifolds in spheres.
Speaker: Prof. Shu-Cheng Chang (MSRC, CQUT)
Time: 10:00-11:30, April 13, 2023
Venue: 数学中心研讨室（至善楼602）& Zoom
Title: The Legendrian Mean Curvature Flow and Its Related Problems
Abstract:
In this talk, I will introduce the fundamentals of the Legendrian mean curvature flow, its blow-up analysis, Legendrian self-shrinkers, and applications to existence problems of minimal Legendrian surfaces in a Sasakian five-manifold.
Speaker: 程新跃 （重庆师范大学）
Time: 16:00-17:00, April 6, 2023
Venue: 数学中心研讨室（至善楼602）
Title: Volume comparison theorems and their applications in Finsler geometry
Abstract:
In this talk, we will mainly introduce the important research progress of volume comparison theorems in Finsler geometry. As the applications, we will introduce some important inequalities on Finsler manifolds with the weighted Ricci curvature bounded from below.
Speaker: Shu-Cheng Chang （MSRC, CQUT）
Time: 10:00-11:30, March 30, 2023
Venue: 数学中心研讨室（至善楼602）
Title: Introduction to the Sasaki-Ricci flows and the related topics
Abstract:
In this talk, I will introduce the fundamentals of the Sasaki-Ricci flow and its existence problems of canonical metrics in Sasakian manifolds.
