## Seminar on Geometry

## 报告人简介

**2019-12-19**

**Title:** Lie
theory in the study of closed geodesics on Finsler spheres with K=1 and compact
homogeneous Finsler manifolds

**Speaker:** Prof. Ming Xu
(Capital Normal University)

**Abstract:** 近期R.Bryant、P.Foulon、S.Ivanov、V.Matveev、Z.Willer等5位学者在JDG上的文章按测地线特性给出常曲率芬斯勒S^2的分类。由于常曲率芬斯勒球上“对径映射”的存在，李理论在他们的研究工作中发挥了重要作用。他们的方法可以推广到高维数芬斯勒常曲率球，给出素闭测地线条数或闭素测地线轨道数下界的估计。通过齐性常曲率芬斯勒球分类的研究，我们可以证明，当常曲率芬斯勒球上只有有限个素闭测地线轨道时，它上面存在齐性全测地Randers子球。李理论的方法同样可以用于讨论其它芬斯勒流形上的闭测地线。我们猜测，一个紧致芬斯勒流形上如果只有一个素闭测地线轨道，它必然为一个赋予标准度量的秩一对称空间。利用齐性测地线的讨论和正旗曲率齐性芬斯勒空间分类的代数框架，我们证明了上述猜测对于偶数维或可反奇数维齐性芬斯勒空间是正确的。**
**

** **

**2019-10-10**

**Title:** Geometric
Application of Differential K-theory

**Speaker:** Prof. Bo Liu (East
China Normal University)

**Abstract:** In 1957, Grothendieck introduces the K-theory in algebraic geometry.
Later, its real counterpart, topological K-theory, is used to prove the famous
Atiyah-Singer index theorem. In 1990's, in Arakelov geometry and arithmetic
algebraic geometry, the K-theory is extended to the arithmetic K-theory. In
this century, motivated by the theoretical physics, people extend the
topological K-theory to the differential K-theory. In this talk, we will
compare the arithmetic K-theory and the differential K-theory. Through the
comparison, we obtain a new purely geometric result: localization formula of
eta invariants, which cannot be proved by geometrical methods until now.

** **

**2019-9-26**

**Title:** On a topology
property for the moduli space of Kapustin-Witten equations

**Speaker:** Teng Huang
(University of Science and Technology of China)

**Abstract:** In this talk, we first recall the Kapustin-Witten equations on a four-dimensional manifold which were
introduced by Kapustin and Witten. By a
compactness theorem due to Taubes, we will prove that if (A,\phi) is a smooth
solution of KW equations and the connection A is closed to a generic ASD
connection A_{\infty}, then (A,\phi) must be a trivial solution. We also prove
that the moduli space of the solutions of KW equations is non-connected if the
connections on the compactification of moduli space of ASD connections are all
generic. At last, we extend the results for the KW equations to other equations
on gauge theory such as the Hitchin-Simpson equations and Vafa-Witten on a
compact K\"{a}hler surface.

** **

**2019-9-19**

**Title:** Connected sums of
manifolds of positive scalar curvature

**Speaker:** Prof. Jozef Dodziuk
(The City University of New York, USA)

**Abstract:** I will review a refinement of the construction, due
to Gromov and Lawson, of metrics of positive scalar curvature on connected sums
of manifolds posessing such metrics. This refinement works in all dimensions
greater than or equal to three and allows constructing examples useful in
testing properties of limits of sequences of manifolds of positive scalar
curvature.

** **

**2019-9-12**

**Title:** A Miyaoka-Yau type
inequality of surfaces in characteristic p>0

**Speaker:** Xiaotao SUN (Tianjin
University)

**Abstract:** For minimal smooth projective surfaces $S$ of general type, we prove
$K^2_S\leq 32\chi(\sO_S)$ and give examples of $S$ with
$$K^2_S=32\chi(\sO_S).$$

This proves that $\chi(\sO_S)>0$ holds for all smooth projective minimal surfaces $S$ of general type, which answers completely a question of Shepherd-Barron. Our key observation is that such Miyaoka-Yau type inequality follows slope inequalities of a fiberation $f:S\to C$.

However, we will gives examples of $f:S\to C$ with non-smooth generic fibers of arithmetic genus $g\ge 2$ such that

$$K^2_{S/C}<\frac{4g-4}{g}{\rm deg}f_*\omega_{S/C},$$

which are counterexamples of Xiao's slope inequality in case of positive characteristic. This is a joint work with Gu Yi and Zhou Mingshuo.

** **

**--------------------------History--------------------------**

**2019-6-28**

**Title:** Spherical conical
metrics

**Speaker:** Xuwen Zhu (UC
Berkeley, USA)

**Abstract: **The problem of finding and classifying constant curvature metrics with
conical singularities has a long history bringing together several different
areas of mathematics. This talk will focus on the particularly difficult
spherical case where many new phenomena appear. When some of the cone angles
are bigger than $2\pi$, uniqueness fails and existence is not guaranteed;
smooth deformation is not always possible and the moduli space is expected to
have singular strata. I will give a survey of several recent results regarding
this singular uniformization problem, connecting PDE techniques with complex
analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin
Xu.

** **

**2019-6-20**

**[1]**

**Title:** Metric distortion in
the geometric Schottky problem

**Speaker:** Lizhen Ji
(University of Michigan, USA)

**Abstract: **

There are two basic moduli spaces, the moduli space M_g of compact Riemann surfaces of genus g, and the moduli space A_g of principally polarized abelian varieties of dimension g.

The period (or Jacobian) map connects naturally these two spaces \pi: M_g --> A_g. A_g is a complete noncompact locally symmetric space, and the map \pi induces a length metric on M_g. In this talk, I will discuss some results on the metric distortion of the map \pi and the large scale geometry of the induced length metric on M_g.

** **

**[2]**

**Title:** Regular domains,
constant Gauss-Kronecker curvature and Monge–Ampere equation

**Speaker:** Xin Nie (KAIST,
South Korea)

**Abstract: **We will first survey
the theory of globally hyperbolic flat spacetimes due to Geoffrey Mess, with
emphasis on the notion of regular domains in the Minkowski space and foliations
of such domains by hypersurfaces of constant Gauss-Kronecker curvature. We then
explain a joint work with Andrea Seppi which generalizes the theory to the
setting of affine differential geometry. The technical core consists of
improvements of the results of An-Min Li, Udo Simon and Bohui Chen on a real
Monge–Ampere equation.

** **

**2019-6-6**

**Title:** Blow-up formulas and
their applications

**Speaker:** Lingxu Meng (NUC)

**Abstract: **We will explicitly
write out blow-up formulas of cohomology with values in local systems and
Dolbeault cohomology with values in locally free sheaves on arbitrary complex
manifold and prove that the blow-up formula given by Rao, S., Yang, S. and
Yang, X.-D. is still an isomorphism in the noncompact case. Moreover, we give
some applications of these formulas.

** **

**2019-5-30**

**Title:** Dynamical degrees of
self-maps on abelian varieties

**Speaker:** Fei Hu (UBC)

**Abstract: **Let $X$ be a smooth
projective variety defined over an algebraically closed field of arbitrary
characteristic, and $f\colon X \to X$ a surjective morphism. The $i$-th cohomological
dynamical degree $\chi_i(f)$ of $f$ is defined as the spectral radius of the
pullback $f^*$ on the \'etale cohomology group $H^i_{et}(X, \bQ_\ell)$ and the
$k$-th numerical dynamical degree $\lambda_k(f)$ as the spectral radius of the
pullback $f^*$ on the vector space $N^k(X)_\bR$ of real algebraic cycles of
codimension $k$ modulo numerical equivalence. Truong conjectured that
$\chi_{2k}(f) = \lambda_k(f)$ for any $1 \le k \le \dim X$. When the ground
field is the complex number field, the equality follows from the positivity
property inside the de Rham cohomology of the ambient complex manifold
$X(\bC)$. We prove this conjecture in the case of abelian varieties. The proof
relies on a new result on the eigenvalues of self-maps of abelian varieties in
prime characteristic, which is of independent interest.

**Title:** Dynamical degrees of
self-maps on abelian varieties

**Speaker:** Zheng Huang (纽约城市大学CUNY)

**Abstract: **In late 1970s,
Uhlenbeck initiated a program to study minimal immersions of closed surfaces into
hyperbolic three-manifolds. Naturally bifurcation for the equations occurs and
she described the first such diagram. In this project, joint with M. Lucia
(CUNY) and G. Tarantello (Roma) we study in much depth on this bifurcation. In
particular we will determine the full range of parameter where solutions exist
and prove multiplicity results.

** **

**2019-5-23**

**Title:** The analysis and
geometry of isometric embedding

**Speaker:** Guoyi Xu (Tsinghua
University)

**Abstract: **In 1950's,
Nash-Kuiper built up the C^1 isometric embedding for any surface into
$\mathbb{R}^3$, this can be viewed as analysis side of isometric embedding. On
the other hand, there is obstruction for the existence of $C^2$ isometric
embedding of surface into $\mathbb{R}^3$ known since Hilbert, which reflects
the geometry flavor of isometric embedding. What's happening from $C^1$ to
$C^2$ (from analysis to geometry)? We will present our partial progress along
this direction. The talk will be accessible to audience with basic knowledge of
analysis and differential geometry.

**2019-5-16**

**Title:** Symplectic
Laplacians, boundary conditions and cohomology

**Speaker:** Lihan Wang
(University of Connecticut)

**Abstract: **Symplectic Laplacians
are introduced by Tseng and Yau in 2012, which are related to a system of
supersymmetric equations from physics. These Laplacians behave different from
usual ones in Rimannian case and Complex case. They contain both 2nd and 4th
order operators. In this talk, we will discuss these operators and their
relations with cohomologies on compact symplectic manifolds with boundary. For
this purpose, we will introduce new boundary conditions for differential forms
on symplectic manifolds. Their properties and importance will be discussed.

** **

**2019-5-9**

**Title:** A critical
Allard-Reifenberg type regularity theorem and the rigidity of minimal
ends

**Speaker:** 周杰(中科院数学所)

**Abstract: **The uniqueness (or
classification) of geometric objects and their tangent structures is an
important topic in geometric analysis. In this talk, we will discuss such a
result for minimal surfaces. Especially, we will show that every connected
minimal surface in 3-dimensional Euclidean space with density less than 3 but
at least two ends must be the Catenoid. An interesting key point is to find
this is related to some critical version of the well-known Allard regularity
theorem in Geometric Measure Theory.

**2019-4-25**

**Title:** Instantaneously
complete Chern-Ricci flow and K\"ahler-Einstein metrics

**Speaker:** 黄少创博士（清华大学）

**Abstract: **In this talk, we study the Chern-Ricci flow on complete
non-compact Hermitian manifold with "negative first Chern class" (We
will give the definition in the talk). Under certain general conditions (the
initial metric may be incomplete or with unbounded curvature or even only an
nonnegative Hermitian form), we prove the flow can be instantaneously complete
and has a long-time solution converging to a complete negative Kahler-Einstein
metric. In general, we can not conclude the flow tends to the initial metric
smoothly and locally, we also discuss conditions so that this is true. These
works are joint with Man-Chun Lee and Professor Luen-Fai Tam.

**2019-4-18**

**Title: **The rigidity on the
second fundamental form of projective manifolds

**Speaker:** Prof. Ping Li (Tongji University)

**Abstract: **We review some known gap phenomena related to the second
fundamental form ofthe minimal submanfolds and complex submanifolds in the unit
spheres and complex projective spaces respectively, and then present our recent
progress on them.

**2019-4-11**

**Title: **Topology of
holomorphic Lagrangian fibrations and Hodge theory of compact hyper-Kaehler
manifolds

**Speaker: **Qizheng Yin (Peking University)

**Abatract: **The P=W conjecture of de Cataldo, Hausel, and Migliorini
relates the topology of Hitchin fibrations to the Hodge theory of character
varieties via Simpson’s nonabelian Hodge theory. Here P and W stand for
Perverse and Weight. While the original conjecture remains open, the P=W
phenomenon can be observed in a much wider context. In recent joint work with
Junliang Shen, we proved a compact version of the P=W conjecture, which relates
the topology of holomorphic Lagrangian fibrations to the Hodge theory of
compact hyper-Kaehler manifolds. I will present the circle of ideas as well as
some applications of our result.

**2019-3-28**

**Title: **The remainder term in
spectral asymptotic and lattice point problem

**Speaker: **王为为博士(清华大学)

**Abstract:** For some special surface and planar domain, the spectral
counting function can be converted to the number of the lattice point within a
bounded region. Then the order of the remainder term in the spectral asymptotic
can be given by the lattice point problem. In this talk, we will discuss
this converting method and present some of our results.

**2019-3-21**

**Speaker: **邓富声教授（中国科学院大学）

**Title: **Linear invariants of complex manifolds and their variation

**Abstract: **We will present a new characterization of plurisubharmonic
functions.Combining this with a power technique, we give a new proof of the
positivity of the direct image sheafof the twisted relative canonical boundle
associated to a family of Stein manifolds of compact Kahler manifolds.We will
also show that two bounded hyperconvex domains are biholomorphic if their
associated spaces of L^p holomorphic functions are linear isometric. For a
family of bounded domains,we will show that spaces of L^p holomorphic functions
on fibers form a holomorphic vector bundle over the base space with positively
curved singular Finsler metrics.This lecture is based on joint works with
Zhiwei Wang, Liyou Zhang, and Xiangyu Zhou.

** **

**2019-3-14**

**Speaker:** Yi Liu (Peking
University)

**Title: **Virtual homological eigenvalues of surface automorphisms

**Abstract:** For any self-homeomorphism of a surface, a homological
eigenvalue refers to an eigenvalue of the induced action on homology. A virtual
homological eigenvalue refers to a homological eigenvalue for some lift of the
homeomorphism to some finite cover of the surface. How much can we infer about
the homeomorphism from the virtual homological eigenvalues? In this talk, I
will review some recent results in this direction, which are also related to
virtual properties of surface bundles over the circle.