Seminar on Geometry

Time: Thu 15:30-17:00,2019-9-12 ~ 12-26
Venue:Conference Room 3, Jin Chun Yuan West Building

Speaker Introduction


Title: Determination of isometric real-analytic metric and spectral invariants for elastic Dirichlet-to-Neumann map on Riemannian manifolds

Speaker: 刘跟前 Liu Genqian (Beijing Institute of Technology)

Abstract: For a compact Riemannian manifold $(\Omega, g)$ with smooth boundary $\partial \Omega$, we explicitly give local representation and full symbol expression for the elastic Dirichlet-to-Neumann map $\Xi_g$ by factorizing an equivalent elastic equation. We prove that for a strong convex or extendable real-analytic manifold $\Omega$ with boundary, the elastic Dirichlet-to-Neumann map $\Xi_g$ uniquely determines metric $g$ of $\Omega$ in the sense of isometry. We also give a procedure by which we can explicitly calculate all coefficients $a_0, a_1 \cdots, a_{n-1}$ of the asymptotic expansion $\sum_{k=1}^\infty e^{-t \tau_k}\sim \sum\limits_{m=0}^{n-1} a_m t^{m+1-n} +o(1)$ as $t\to 0^+$, where $\tau_k$ is the $k$-th eigenvalue of the elastic Dirichlet-to-Neumann map (i.e., $k$-th elastic Steklov eigenvalue). The coefficients $a_m$ are spectral invariants and provide precise information for the boundary volume ${\mbox{vol}} (\partial \Omega)$ of the elastic medium $\Omega$ and the total mean curvature as well as other total curvatures of the boundary $\partial \Omega$. These conclusions give answer to two open problems.



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.BryantP.FoulonS.IvanovV.MatveevZ.Willer5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。


Title: Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons

Speaker: Jianyu Ou (Fudan University)

Abstract: In this talk, I will show a rigidity theorem for gradient shrinking Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in R^n. I will give a brief introduction to Ricci soliton, and then talk about the obtained result.


Title: Codimension one foliations with trivial first Chern class

Speaker: Wenhao Ou (CAS)

Abstract: We describe sturctures of codimension one foliations with trivial first Chern classes on projective varieties.


Title: M-regular decompositions in Generic Vanishing

Speaker: Zhi Jiang (Fudan University)

Abstract: We will discuss a M-regular decomposition formula (also called Chen-Jiang decomposition in some references) for pushforward of pluricanonical bundles of complex varieties to abelian varieties. If time permits, we will also show a few geometric applications of this formula.


Title: Vanishing homology and the topological Abel–Jacobi map

Speaker: 傅二娟 Fu Erjuan (YMSC) 



Title: Mapping class group of manifolds which look like 3d complete intersections

Speaker: 苏阳 Su Yang(中科院,CAS

Abstract: In this talk I will introduce a class of 6-dimensional manifolds which share topological properties of complex 3-dimensional complete intersections. Then I will presesent a compuatation of the mapping class group of these manifolds. Finally I will discuss group theoretical properties of these mapping class groups, and compare them with the classical mapping class group. This is a joint work with M.Kreck.



Title: Character varieties in low-dimensional topology

Speaker: 陈海苗 Chen Haimiao

Abstract: Character varieties in low-dimensional topology.pdf 



Title: On systoles and ortho spectrum rigidity

Speaker: Hidetoshi Masai (Tokyo Institute of Technology)

Abstract: We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. In fact we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.

This is joint work with Greg McShane.




Title: Parabolic Hitchin maps and their generic fibers

Speaker: 苏晓羽Su Xiaoyu (YMSC)

Abstract: Hitchin in 1987 introduced the map now named after him, and showed that it defines a completely integrable system in the complex-algebraic sense. Subsequently Beauville, Narasimhan and Ramanan constructed a correspondence -- indeed nowadays refered to as the BNR correspondence -- which among other things characterizes the generic fiber of a Hitchin map as a compactified Jacobian. In this talk, we will review the geometric properties of Hitchin maps and introduce their generalizations to parabolic (Higgs) bundles over an algebraically closed field.


Title: 4d Yang-Mills gauge theory, 2d CP^{N-1} sigma model, anomalies and cobordisms

Speaker: 万喆彦 Wan Zheyan (YMSC)

Abstract: We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic verification. First, we propose a set of ’t Hooft higher anomalies of 4d time-reversal symmetric pure SU(N)-Yang-Mills (YM) gauge theory at \theta = \pi, via 5d cobordism invariants. Second, we propose a set of ’t Hooft anomalies of 2d CP^{N-1}-sigma models at \theta = \pi, by enlisting all possible 3d cobordism invariants and selecting the matched terms. We derive a correspondence between 5d and 3d new invariants. Thus we broadly prove a potentially complete anomaly-matching between 4d SU(N) YM and 2d CP^{N-1} models at N = 2, and suggest new (but maybe incomplete) anomalies at N = 4. This is my joint work with Juven Wang and Yunqin Zheng (arXiv: 1812.11968, 1904.00994, ...).




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.



Title: On impedance of finite and infinite networks

Speaker: Alexander Grigoryan(University of Bielefeld, Geometry)

Abstract: We discuss mathematical aspects of the notion of effective impedance of AC networks consisting of resistances, coils and capacitors. Mathematically such a network is a locally finite graph whose edges are endowed with complex valued weights depending on a complex parameter $\lambda$ (by the physical meaning, $\lambda = i\omega $, where $omega$ is the frequency of the AC). For finite networks,  the effective impedance is obtained by solving Kirchhoff's equations and is a rational function of $\lambda$. For infinite networks, the sequence of effective impedances of finite graph approximations converges in certain domains in $\mathbb{C}$ to a holomorphic function of $\lambda$, which allows to define the effective impedance of the infinite network."



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.



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.



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.



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.




Title: Metric distortion in the geometric Schottky problem

Speaker: Lizhen Ji (University of Michigan, USA)


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.



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.



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.



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.



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.


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.



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.


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.


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.


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.


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.


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.



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.

  • Contact
  • Yau Mathematical Sciences Center, Jing Zhai,
    Tsinghua University, Hai Dian District, Beijing China
  • +86-10-62773561
  • +86-10-6277356
©2018 YMSC, Tsinghua University. All Rights Reserved