Upcoming Talks:
Title: Noether inequality for irregular threefolds of general type
Speaker: 张通，华东师范大学
Time: 16:00-17:30 on Friday, June 7th, 2024
Venue: B627, Shuangqing Complex Building
Abstract: Finding the Noether inequality for varieties of general type is a natural problem in the study of algebraic geometry, which dates back to the work of M. Noether for surfaces. Later, Debarre proved that irregular surfaces of general type satisfy a stronger Noether inequality. Recently, J. Chen, M. Chen and C. Jiang established the optimal Noether inequality in dimension three. In this talk, I will introduce an optimal Noether inequality for almost all complex irregular threefolds of general type. This is a joint work with Y. Hu.
Past Talks:
Title: Index and total curvature of minimal surfaces in noncompact symmetric spaces and wild harmonic bundles
Speaker: 李琼玲，南开大学陈省身数学研究所
Time: 16:00-17:30 on Friday, May 31th, 2024
Venue: B627, Shuangqing Complex Building
Abstract: We prove two main theorems about equivariant minimal surfaces in an arbitrary nonpositively curved symmetric spaces extending classical results on minimal surfaces in Euclidean space. First, we show that a complete equivariant branched immersed minimal surface in a nonpositively curved symmetric space of finite total curvature must be of finite Morse index. It is a generalization of the theorem by Fischer-Colbrie, Gulliver-Lawson, and Nayatani for complete minimal surfaces in Euclidean space. Secondly, we show that a complete equivariant minimal surface in a nonpositively curved symmetric space is of finite total curvature if and only if it arises from a wild harmonic bundle over a compact Riemann surface with finite punctures. Moreover, we deduce the Jorge-Meeks type formula of the total curvature and show it is an integer multiple of $2\pi/N$ for $N$ only depending on the symmetric space. It is a generalization of the theorem by Chern-Osserman for complete minimal surfaces in Euclidean n-space. This is joint work with Takuro Mochizuki (RIMS).
Title: Twisted de Rham-Higgs comparison in positive characteristic
Speaker: 张泽宝，重庆理工大学
Time: 10:00-11:30 am on Friday, May 24th, 2024
Venue: C654, Shuangqing Complex Building
Abstract: Using their nonabelian Hodge theory in positive characteristic, Ogus-Vologodsky and Schepler established a de Rham-Higgs comparison between the de Rham complex of a suitable flat bundle and the Higgs complex of its Cartier transform. Inspired by a result of Sabbah on the twisted de Rham complexes of mixed Hodge modules, we generalize the aforementioned de Rham-Higgs comparison to Landau-Ginzburg models in positive characteristic.
Title: A survey of positivity results in characteristic p and applications
Speaker: 张磊，中国科学技术大学
Time: 10:00-11:30 am on Friday, May 17th, 2024
Venue: C654, Shuangqing Complex Building
Abstract: We give a survey of recent progresses of positivity results on (rel-)pluricanonical sheaves in characteristic p. And we introduce some applications in some problems: subadditivity of Kodaira dimension, canonical bundle formula and the classification of certain varieties with nef anti-canonical divisors.
Title: Slope inequalities and a Miyaoka-Yau type inequality
Speaker: 周明铄，天津大学数学学院/应用数学中心
Time: 16:00-17:30 on Wednesday, April 24th, 2024
Venue: C654, Shuangqing Complex Building
Zoom Meeting ID: 271 534 5558
Passcode: YMSC
Abstract: After a review about Xiao’s approach of Slope inequalities and its generalization to positive characteristic, a Miyaoka-Yau type inequality for a minimal smooth surface of general type is given as an application. This is based on some work with Professor Yi Gu, Hao Sun and Xiaotao Sun.
Title: Counting l-adic local systems on a curve over a finite field
Speaker: 余红杰，中国科学院晨兴数学中心
Time: 16:00-17:30 on Wednesday, April 17th, 2024
Venue: C654, Shuangqing Complex Building；Zoom Meeting ID: 271 534 5558 Passcode: YMSC
Abstract: In 1981, Drinfeld enumerated the number of irreducible l-adic local systems of rank two on a projective smooth curve fixed by the Frobenius endomorphism. Interestingly, this number looks like the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld's result. By the Langlands correspondence, it is equivalent to count certain cuspidal automorphic representations over a function field. In this talk,I will present some counting results where we connect counting to the number of stable Higgs bundles using Arthur's non-invariant trace formula.
Title: Existence of Stable Higgs-de Rham Flow for Principal Bundles
Speaker: 孙浩，华南理工大学
Time: 16:00-17:30 on Wednesday, April 10th, 2024
Venue: C654, Shuangqing Building
Abstract: Lan-Sheng-Zuo established the Hitchin-Simpson correspondence in characteristic p. In this correspondence, semistable graded Higgs bundles with vanishing chern classes are related to crystalline representations with the help of periodic Higgs-de Rham flow. In this talk, we will give the construction of Higgs-de Rham flows for principal bundles and discuss the existence of such flows under stable conditions.
Title: Local equations defining stable map moduli, arbitrary singularities, and resolution
Speaker: 胡毅，亚利桑那大学
Time: 16:00-17:30 on Wednesday, March 20th, 2024
Venue: C654, Shuangqing Building; Zoom Meeting ID: 4552601552 Passcode: YMSC
Abstract:
I will explain the matrix local equations defining the moduli spaces of stable maps of arbitrary genus, found jointly by Jun Li and the speaker. These equations already guided us to find explicit global resolutions for these moduli spaces in the cases when the genera are one and two. By Murphy’s law, stable map moduli possess arbitrary singularities. Turning to this, I will explain Lafforgue’s version of Mnev’s universality, how it leads to standard local equations for arbitrary singularity types, and how it should guide to resolve arbitrary singularities.
Bio：
胡毅，美国亚利桑那大学教授，大湾区大学（筹）访问教授，博士毕业于麻省理工学院，主要研究方为代数几何，在几何不变量、曲线与稳定映射的模空间、奇点解消等问题上做了一系列重要的工作，获得1996年美国数学会百年纪念奖。
Title: Specialization of Linear Differential Equations
Speaker: 冯如勇，中国科学院数学与系统科学研究院系统科学研究所
Time: 16:00-17:30 on Wednesday, March 13th, 2024
Venue: C654, Shuangqing Building； Zoom Meeting ID: 4552601552 Passcode: YMSC
Abstract:
In this talk, we will discuss recent advancements in the specialization of linear differential equations. Given a linear differential equation with parameters, our focus is on how the algebraic properties of its solutions change as these parameters vary over an affine variety. For example, we inquire about the values of parameters for which linear differential equations can have a basis of algebraic solutions, assuming the original equation lacks such a basis. By extending a result of Hrushovski, we demonstrate that the set of such values is indeed ``small" in a meaningful sense. As an application, we establish Matzat’s conjecture in its entirety: The absolute differential Galois group of a one-variable function field, equipped with a non-trivial derivation, is the free proalgebraic group. The talk is based on joint work with Michael Wibmer from the University of Leeds, UK.
Bio：
冯如勇, 中国科学院数学与系统科学研究院研究员、博士生导师，主要研究方向为：符号计算与微分差分伽罗华理论，特别是微分差分方程的符号求解、构造性线性微分（差分）方程伽罗华理论及其在特殊函数理论与组合数学等领域的应用。目前已在符号计算领域国际会议ISSAC,以及期刊Journal of Symbolic Computation, Advances in Applied Mathematics, Mathematics of Computation, Transactions of the AMS等发表论文20多篇。
