Spectral Geometry Seminar @ USTC

Speaker: 龚禹霖 （清华大学）

Time：

June 21, 16:00-17:00

Place：

5205

Title：

16:00-17:00

Abstract：

We investigate the spectral distribution of the twisted Laplacian associated with uniform square-integrable bounded harmonic 1-form on typical hyperbolic surfaces of high genus. First, we estimate the spectral distribution by the supremum norm of the corresponding harmonic form. Subsequently, we show that the square-integrable bounded harmonic form exhibits a small supremum norm for typical hyperbolic surfaces of high genus. Based on these findings, we prove a uniform Weyl law for the distribution of real parts of the spectrum on typical hyperbolic surfaces.

Speaker: 来米加（上海交通大学）

Time：

May 31, 16:00-17:00

Place：

2204

Title：

Hamilton's pinching condition under conformal deformation

Abstract：

Hamilton's pinching conjecture asserts that if a three dimensional manifold satisfies a Ricci pinching condition (Ric-\epsilon Rg\geq 0, for some small \espsilon>0), then M must be compact unless it is flat. This conjecture was recently proved by Lee and Topping. In this talk, I will first talk about the origin of this conjecture, which is a result of Hamilton on hypersurfaces in Euclidean space with pinched second fundamental form. Then I shall present a result joint with Guoqiang Wu, which investigates the pinching condition in higher dimensional locally conformally flat manifolds.

Speaker: 葛建全（北京师范大学）

Time：

May 17, 10:00-11:00

Place：

2507

Title：

Integral-Einstein hypersurfaces and Simons-type inequalities in spheres

Abstract：

We introduce a generalization, the so-called Integral-Einstein (IE) submanifolds, of Einstein manifolds by combining intrinsic and extrinsic invariants of submanifolds in Euclidean spaces, in particular, IE hypersurfaces in unit spheres. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres, which is the main object of the Chern conjecture: such hypersurfaces are isoparametric. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with $g\leq2$ principal curvatures. In particular, all the minimal isoparametric hypersurfaces with $g\geq3$ principal curvatures are IE hypersurfaces. As applications, we also obtain some spherical Bernstein theorems. A universal lower bound for the average of squared lenth of second fundamental form of non-totally geodesic minimal hypersurface in unit spheres is established, partially proving the Perdomo Conjecture.

Speaker: 孙林林 （广西师范大学）

Time：

May 15, 14:30-15:30

Place：

5107

Title：

Some results related to the Kazdan-Warner equations

Abstract：

The Kazdan-Warner equation on surface comes from the prescribed Gaussian curvature problem, and also appears in various contexts such as the abelian Chern-Simons-Higgs models. I shall talk about some results related to the Kazdan-Warner equations on surfaces or finite graphs, including the elliptic method and parabolic approach to the Kazdan-Warner equations on surfaces with sign-changed prescribing function, as well as the topological method to the existence of Kazdan-Warner equations on surfaces or finite graphs.

Speaker: Stéphane NONNENMACHER (Université Paris-Saclay)

Time：

April 17, 10:00-11:00

Place：

2205

Title：

Delocalization of the Laplace eigenmodes on Anosov surfaces

Abstract：

The eigenmodes of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$ can exhibit various localization properties in the high frequency regime, which strongly depend on the properties of the geodesic flow. We will focus on situations where this flow is strongly chaotic (Anosov), e.g. if the sectional curvature of $(M,g)$ is negative.The Quantum Ergodicity theorem then states that almost all the eigenmodes become equidistributed on $M$, in the the high frequency limit. The Quantum Unique Ergodicity conjecture claims that this behaviour admits no exception. What can be said about possible exceptional eigenstates? In the case of compact surfaces with Anosov geodesic flow, we prove that all eigenmodes fully delocalize across $M$: for any open set $\Omega$ on $M$, the $L^2$ mass on $\Omega$ of any eigenstate is uniformly bounded from below. This is in contrast with, e.g., the case of eigenstates on the round sphere, which may be strongly concentrated along a closed geodesic. The proof uses various methods of semiclassical analysis, the structure of stable and unstable manifolds of the Anosov flow, and a Fractal Uncertainty Principle due to Bourgain-Dyatlov. Joint work with S.Dyatlov and L.Jin.

Speaker: 金龙（清华大学）

Time：

April 10, 17:00-18:00

Place：

5207

Title：

Counting Pollicott-Ruelle resonances for Axiom A flows

Abstract：

In 1980's, Pollicott and Ruelle independently introduced the concept of resonances for hyperbolic dynamical systems, for example, Smale's Axiom A flows. They are the poles of the meromorphic continuation of the Laplace transform of the correlation function and thus connected to the mixing property of the system. They are also closely related to the zeros and poles of the dynamical zeta function which is connected to the distribution of periods for closed orbits in the system. In the special cases of Anosov flows, their distributions have been well studied since the work of Faure-Sjostrand in 2010. In this talk, we present the first counting result on Pollicott-Ruelle resonances for general Axiom A flows satisfying strong transversal condition. In particular, we give a polynomial upper bound and a sublinear lower bound on the number of resonances in strips. This is based on joint work with Tao Zhongkai.

Schedule of 2023：Past Talks

Speaker: 王芳（上海交通大学）

Time：

December 29, 10:00-11:00

Place：

2408

Title：

A positive mass theorem for fractional GJMS operators

Abstract：

In this talk, I will first give some pointwise comparison formulae for fractional Q-curvatures of different orders, which implies the comparison theorem for Green functions. Based on this, a positive mass theorem for GJMS operators of order between 2 and 4 is given. This is joint work with Huihuang Zhou.

Speaker: Jian WANG （The University of North Carolina at Chapel Hill）

Time：

July 5, 16:00-17:00

Place：

2302

Title：

Damping for Fractional Wave Equations

Abstract：

Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models. Joint work with Thomas Alazard and Jeremy Marzuola.

Speaker: Hao XU （Zhejiang University）

Time：

June 15, 16:00-17:00

Place：

5406

Title：

Spectral geometry of Kahler manifolds

Abstract：

First we survey known results on spectral geometry of Kahler manifolds. Then we study the question of the spectral characterization of CP^n. Namely for each fixed nonnegative integers p, if a compact Kahler manifold M of complex dimension n has the same p-spectra as CP^n equipped with the Fubini-Study metric, we give explicit range of n such that this Kahler manifold is holomorphically isometric to CP^n . This extends previous works of Tanno, Chen-Vanhecke, Goldberg for p<=2 and Ping Li for even p. This is joint work with K. Liu, X. Huang and Y. Zhi.

Speaker: Xianchao WU (Wuhan University of Technology)

Time：

June 1, 16:00-17:00

Place：

5406

Title：

Distribution of Schr\"{o}dinger eigenfunctions

Abstract：

In this talk, we consider the distribution of eigenfunctions of the semiclassical Schr\"{o}dinger operator on a compact manifold. Their behaviors in forbidden regions and classically allowed regions are dramatically different.
In forbidden regions, we will introduce a partial converse to the Agmon estimates (ie. exponential lower bounds for the eigenfunctions) in terms of Agmon distance under a control assumption. Then by considering a Dirichlet problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we will show a sharp reverse Agmon estimate on a hypersurface in the analytic setting.
However, in classically allowed regions, the behavior of Schr\"{o}dinger eigenfunctions is much more complicated. Intuitively, the more time a packet spends near a hypersurface the more concentration we would expect to see there. How to describe it quantitively? We show that if the defect measure $\mu$ associated to a sequence of Schr\"{o}dinger eigenfunctions is $\epsilon_0$- tangentially diffuse with respect to the hypersurface, then one can get $o(1)$ improvement of the well known $O(h^{-1/4})$ restriction bounds.

Speaker: Yongqiang ZHAO (West Lake University)

Time：

May 25, 16:00-17:00

Place：

5406

Title：

Eigenvalue mutiplicities and vanishing sums of roots of unity

Abstract：

It is well known that the standard flat torus T^2=R^2/Z^2 has arbitrary large Laplacian-eigenvalue multiplicies. Consider the discrete torus C_N * C_N with the discrete Laplacian operator; we prove, however, its eigenvalue multiplicities are uniformly bounded for any N, except for the eigenvalue one when N is even. Our main tool to prove this result is the beautiful theory of vanishing sums of roots of unity. In this talk, we will give a brief introduction to this theory and outline a proof of the uniformly boundedness multiplicity result.

Speaker: Cheng ZHANG (Tsinghua University)

Time：

April 28, 16:00-17:00

Place：

5505

Title：

Sharp Lp estimates and size of nodal sets of generalized Steklov eigenfunctions

Abstract：

We prove sharp Lp estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their L2 norms on the boundary. We prove it by establishing Lp bounds for the harmonic extension operators as well as the spectral projection operators on the boundary. Moreover, we derive lower bounds on the size of nodal sets for a variation of the Steklov spectral problem. We consider a generalized version of the Steklov problem by adding a non-smooth potential on the boundary but some of our results are new even without potential.

Speaker: Yakun XI (Zhejiang University)

Time：

April 27, 16:00-17:00

Place：

5205

Title：

Can you hear your location on a manifold?

Abstract：

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point x on the manifold, up to symmetry, from its pointwise Weyl counting function. This problem has a physical interpretation. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echos and reverberations? Our main result provides an affirmative answer to this question for a generic class of metrics.

Speaker: Dong ZHANG (Peking University)

Time：

April 26, 16:30-17:30

Place：

5307

Title：

Spectral duality as a tool for studying the nonlinear graph eigenvalue problems

Abstract：

Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining and network science. In this talk, I will show that one can move from the primal to the dual nonlinear eigenvalue formulation maintaining the spectrum, the variational spectrum as well as the corresponding multiplicities unchanged. Applications to the spectral theory of graph p-Laplacians and Cheeger inequalities on simplicial complexes, will be discussed.

Speaker: Hongyi CAO (Peking University)

Time：

April 06, 14:30-15:30

Place：

5407

Title：

Localization for quasi-periodic Schr\"odinger operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Potentials

Abstract：

Anderson Localization (pure point spectrum with exponentially decaying eigenfunctions) is an important phenomenon in spectrum theory for quasi-periodic (QP) operators. In this talk, we will discuss lattice QP Schr\"odinger operators on $\mathbb{Z}^d (d\geq1)$ with $C^2$-cosine like potentials. We will show quantitative Green's function estimates and the arithmetic version of Anderson localization for such QP Schr\"odinger operators. This talk is based on a joint work with my advisors Zhifei Zhang (Peking University) and Yunfeng Shi (Sichuan University).

Speaker: Zhenhao LI (University bielefeld )

Time：

March 31, 16:00-17:00

Place：

5106

Title：

On synthetic Ricci curvature bounds on metric-measure spaces

Abstract：

Abstract: The setting of metric-measure spaces with synthetic Ricci bounds (e.g. CD/RCD spaces) has attracted not only analysts but also quite many geometers. In this talk, I will first present the classical Lott-Villani-Sturm theory on the synthetic lower Ricci bound relying on the convexity of entropy functionals. Also I will mention another description by the contraction of Wasserstein distances between heat flows, in the spirit of Bakry-Emery. If time permits, I will talk about recent topics on generalising above two viewpoints to the theory of Ricci flows.

Speaker: Guoyi XU (Tsinghua University)

Time：

March 20, 10:00-11:00

Place：

5307

Title：

The first Dirichlet eigenvalue and the width

Abstract：

There are a lot of result about the sharp lower bound of the first Dirichlet eigenvalue or Neumann eigenvalue under different restriction (back to Faber-Krahn and Payne-Weinberger for Euclidean domain, Li-Yau and Zhong-Yang for manifolds case). Generally, the sharp lower bounds of those eigenvalues are achieved on disk (sphere, for the case that boundary is empty, corresponding non-collapsed case) or line segment (circle, corresponding collapsed case). Recent years, there are research to characterize the difference between the domain and the model space (mentioned above), by the gap between the eigenvalue and its sharp bound (quantitative Faber-Krahn inequality etc). And the results along this direction obtained so far, in the spirit, are close to the quantitative isoperimetric inequality established during the last decade (Fusco-Maggi-Pratelli etc). The common point is that the model space is “homogenous in any direction (disk or sphere etc) and is non-collapsed. In this talk, we present our recent result, which gives the explicitly quantitative inequality, linking the width of the domain with the gap between the first Dirichlet eigenvalue and its sharp lower bound. One novel thing is that our model space is collapsed line segment. Most part of this talk only requires the basic PDE and Riemannian geometry knowledge.

Schedule of 2022：Past Talks

Speaker: Olaf Post (University of Trier)

Time：

Dec 21, 16:00-17:00

Place：

ZOOM：976 2601 7096 会议密码：137769

Title：

Some recent results on graph perturbations and the effect on their spectrum

Abstract：

In this talk I will talk about joint works with Fernandó Lledo (Madrid) and John Fabila (Edinburgh) on the spectrum of discrete Laplacians on weighted magnetic graphs and its behaviour under perturbations such as removing edges, contracting vertices etc. One application is an easy spectral criterion whether a graph is Hamiltonian or not. Another application is a construction of families of isospectral graphs.

Speaker: Xiaolong Hans HAN (Yau Mathematics Science Center, Tsinghua University)

Time：

Dec 14, 10:00-11:00

Place：

腾讯会议：588-250-072 会议密码：202223

Title：

Large Steklov eigenvalue on random hyperbolic surfaces

Abstract：

We construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue tends to infinity, using the connection between eigenvalues and Cheeger's / Jammes's constants, and the recent work of Xin Nie, Yunhui Wu, and Yuhao Xue. Using the Weil-Petersson metric, we also show that the probability that a random Riemann surface has a large 1st normalized Steklov eigenvalue is asymptotically one.

Speaker: Jing MAO （Hubei University）

Time：

Dec 07, 16:00-17:00

Place：

腾讯会议：249-108-389 会议密码：202222

Title：

Polya-type inequalities on spheres and hemispheres

Abstract：

Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on n-spheres and -hemispheres, we characterise those with the lowest and highest orders which equal $\lambda$ and for which Polya's conjecture holds and fails. We further derive Polya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. This talk is based on a joint-work with Prof. Pedro Freitas AND Prof. Isabel Salavessa.

Speaker: Changwei XIONG (Sichuan University)

Time：

October 26, 10:00-11:00

Place：

腾讯会议：693-367-593 会议密码：202221

Title：

Some estimates on an exterior Steklov eigenvalue problem

Abstract：

In this talk we will discuss a Steklov eigenvalue problem on an exterior Euclidean domain. We will present sharp lower and upper bounds for its first eigenvalue under various conditions on the domain. Time permitting, we shall discuss an upper bound for its second eigenvalue.

Speaker: Yong LIN（Tsinghua University）

Time：

October 12, 10:00-11:00

Place：

腾讯会议：449-488-884 会议密码：202220

Title：

Normalized discrete Ricci flow and community detection

Abstract：

我们证明了由图上离散的Ricci曲率定义的曲率流方程解的存在唯一性。同时我们利用这种曲率流方程研究图分割问题，并且比较了我们的图分割方法和其他经典的图分割方法。

Speaker: Longzhi LIN （UC Santa Cruz）

Time：

September 28, 10:30-11:30

Place：

腾讯会议：419-604-062 会议密码：202219

Title：

Energy convexity and uniqueness of conformal-harmonic maps

Abstract：

In this talk we will survey some recent results on the energy convexity for weakly harmonic and biharmonic maps and the applications. We will then introduce a conformally invariant analogue of the intrinsic biharmonic map that we call conformal-harmonic map, which is a critical point of a conformally invariant energy functional in four dimension and satisfies a conformally invariant fourth order Paneitz-type PDE. A version of energy convexity and uniqueness of conformal-harmonic maps that we showed in a most recent joint work with J. Zhu will be discussed.

Speaker: Bobo HUA（Fudan University）

Time：

September 14, 16:00-17:00

Place：

1418

Title：

Some results of semilinear PDEs on lattice graphs

Abstract：

Yamabe type semilinear PDEs have been well studied on R^n. In this talk, we discuss some recent results on the lattice graph Z^n. Open questions are more than what we know. This is based on joint work with Ruowei Li and Florentin Muench.

Speaker: Asma HASSANNEZHAD（University of Bristol）

Time：

June 8, 16:00-17:00

Place：

ZOOM ID: 945 8387 4680 Passcode: 695038

Title：

Nodal counts for the Dirichlet-to-Neumann operators with potential

Abstract：

The zero set of an eigenfunction is called the nodal set and the connected components of its complement are called the Nodal domains. The well-known Courant nodal domain theorem gives an upper bound for the nodal count of Laplace eigenfunctions on a compact manifold. We consider the harmonic extension of eigenfunctions of the Dirichlet-to-Neumann operators with potential. When the potential is zero, these harmonic extensions are called the Steklov eigenfunctions. It has been known that the Courant nodal domain theorem holds for Steklov eigenfunctions. We discuss how we can get a Courant-type bound for the nodal count of the Dirichlet-to-Neumann operator in the presence of a potential. This is joint work with David Sher.

Speaker: Lingzhong ZENG （Jiangxi Normal University）

Time：

June 1, 16:00-17:00

Place：

腾讯会议：184-674-728 会议密码：202216

Title：

Universal Inequalities of Laplacian on Riemannian Manifolds and Extensions

Abstract：

In this talk, we would like to review some universal inequalities of Laplacian on Riemannian Manifolds. Furthermore, we consider the eigenvalues of Laplacian on the closed Riemannian manifolds. As an application, we give a very sharp upper bound for the second eigenvalue of Laplacian on the isoparametric hypersurface embeded in the unit sphere. Finally, we also give some universal bounds of Xin Laplacian on the translators in the sense of MCF. This is a work joint with Zhouyuan Zeng.

Speaker: Francesco TUDISCO (Gran Sasso Science Institute, L'Aquilla, Italy)

Time：

May 25, 16:00-17:00

Place：

ZOOM ID: 942 6323 4612 Passcode: 713982

Title：

Nodal domain count of the generalized p-Laplacian on graphs

Abstract：

We consider a generalized p-Laplacian operator on discrete graphs which generalizes the linear Schrödinger operator (obtained for p=2). We consider a set of variational eigenvalues of this operator and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous p-Laplacian, we prove that the variational spectrum of the discrete generalized p-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer the Weyl’s inequalities for the Laplacian operator to the nonlinear case and thus we prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-Laplacian on graphs, including those corresponding to variational eigenvalues. When applied to the linear case p=2, the new results imply well-known properties of the linear Schrödinger operator as well as novel ones.

Speaker: Yuhua SUN (Nankai University)

Time：

May 18, 10:00-11:00

Place：

腾讯会议：229-902-342 会议密码：202214

Title：

Some progress on semilinear elliptic inequalities involving gradient terms on weighted graphs

Abstract：

We study existence and nonexistence of nontrivial positive solutions to the following semilinear elliptic inequalities involving gradient terms on weighted graphs $$\Delta u+u^p|\nabla u|^q\leq 0,$$ Where $(p, q)\in \mathbb{R}^2$. This talk is based on joint works with Qingsong Gu, Lu Hao, Xueping Huang.

Speaker: Xi CHEN （Shanghai Center for Mathematical Sciences）

Time：

May 11, 10:00-11:00

Place：

腾讯会议：120-414-316 会议密码：202213

Title：

锥流形上波的衍射现象

Abstract：

在具有锥奇性的黎曼流形上，波传播到锥点时会发生衍射现象，即波的传播分离成几何波和衍射波。我们用拟微分算子和傅里叶积分算子完整地刻画波的衍射现象。

Speaker: Genqian LIU (Beijing Institute of Technology)

Time：

April 27, 16:00-17:00

Place：

腾讯会议：757-475-957 会议密码：202212

Title：

Heat trace asymptotic expansions for Lame elastic operator and the Stokes flow operator

Abstract：

Spectral asymptotics for partial differential operators have been the subject of extensive research for over a century. It has attracted the attention of many mathematicians and physicists. In this talk, we will give a survey about the spectral geometric problem for some famous differential operators. We then will discuss the heat trace asymptotic expansions for the Lame elastic operator and Stokes flow operator.

Speaker: Bobo HUA (Fudan University)

Time：

April 20, 10:00-11:00

Place：

腾讯会议：182-624-802 会议密码：202211

Title：

Steklov eigenvalues on graphs

Abstract：

In this talk, we introduce the Steklov eigenvalue problems on graphs, and estimate the eigenvalues using geometric quantities.

Speaker: Yaoping HOU (Hunan Normal University)

Time：

April 13, 10:00-11:00

Place：

腾讯会议：807-608-186 会议密码：202210

Title：

On spectra of signed graphs

Abstract：

In this talk, we I will introduce some recent results on eigenvalues of signed graphs, such as signed graphs with few disitinct eigenvalues, integral subcubic signed graphs, the muliplicity of eigenvalues, and the signed graphs with spectral radius does not exceed $\sqrt{2+\sqrt{5}}.$

Speaker: Guoyi XU（Tsinghua University）

Time：

April 6, 10:00-11:00

Place：

腾讯会议：981-472-543 密码：202204

Title：

The sharp estimates of functions and the related rigidity, stability

Abstract：

Since Cheng-Yau proved the gradient estimate of harmonic functions in 1975, their method played important role in geometric analysis. Its philosophy was generalized to prove the lower bound of eigenvalues and parabolic Harnack estimate by Li-Yau. In this talk, we will discuss the sharp gradient estimate for harmonic functions, sharp Dirichlet eigenvalues in geodesic ball. Furthermore, we present the corresponding sharp estimate for Green's function and heat kernel, and the rigidity and stability of those estimates will also be discussed. This is a survey report based on my former work and the joint work with Haibin Wang---Jie Zhou, and Qixuan Hu----Chengjie Yu. Only basic Riemannian geometry and PDE knowledge is enough to understand most part of the talk.

Speaker: Xiaodong ZHANG (Shanghai Jiaotong University)

Time：

March 30, 10:00-11:00

Place：

腾讯会议：910-896-042 密码：202203

Title：

The Discrete Faber-Krahh Inequality of Graphs

Abstract：

The Faber-Krahn inequality states in spectral geometric theory that the ball has smallest first Dirichlet eigenvalue among all bounded domains with the fixed volume in $\mathbb{R}^n$. In this talk, the Dirichlet eigenvalues of graphs with boundary condition were introduced. Then some similar Faber-Krahn inequalities were proved to be held for some classes of graphs, for example the set of all trees and connected unicyclic (bicyclic) graphs with a given graphic unicyclic (bicyclic) degree sequence $\pi$ under minor conditions. Moreover, the extremal unicyclic (bicyclic) graph is unique and possess spiral like ordering and can be regarded as ball approximations. This talk is joined with Guang-Jun Zhang (张光军, 青岛科技大学), Jie Zhang (张杰， 上海立信会计金融学院)

Speaker: Chengjie YU (Shantou University)

Time：

March 23, 16:00-17:00

Place：

腾讯会议：530-921-280 密码 202203

Title：

Minimal Steklov Eigenvalues on Combinatorial Graphs

Abstract：

In this talk, we will present some details of our recent work on extending Jeol Friedman's theory on nodal domains for Laplacian eigenfunctions on combinatorial graphs to Steklov eigenfunctions and applying it to solve an minimum problem on Steklov eigenvalues for combinatorial graphs which is also an extension of Friedman's theory on solving a similar minimum problem on Laplacian eigenvalues. This talk is based on a joint work with Yingtao Yu.

Speaker: Linlin SUN (Wuhan University)

Time：

March 16, 16:00-17:00

Place：

腾讯会议 101-578-059 Passcode 202203

Title：

Rigidity results of CSL submanifolds in the unit sphere

Abstract：

I will talk about the rigidity of contact stationary Legendrian (CSL) submanifolds in the unit sphere based on the joint works with Prof. Luo Yong and Dr. Yin Jiabin. We prove some optimal rigidity results of closed CSL submanifolds and obtain a new characterization of the minimal Calabi torus in the unit sphere.

Speaker: Shicheng XU (Capital Normal University)

Time：

March 09, 10:00-11:00

Place：

腾讯会议 126-609-398 密码 202203

Title：

Total squared mean curvature of immersed submanifolds in a negatively curved space

Abstract：

Let n≥2 and k≥1 be two integers. Let M be an isometrically immersed closed submanifold of dimension n and co-dimension k, which is homotopic to a point, in a complete manifold N, where the sectional curvature of N is no more than δ<0. We prove that the total squared mean curvature of M in N and the first non-zero eigenvalue λ_1(M) satisfies λ_1(M)≤ n(δ +Vol^(-1)(M) ∫ |H|^2 dvol. The equality implies that M is minimally immersed in a geodesic sphere after lifted to the universal cover of N. This completely settles an open problem raised by E. Heintze in 1988.

Speaker: Dong ZHANG （Max-Planck Institute for Mathematics in the Sciences, Leipzig）

Time：

March 02, 16:00-17:00

Place：

腾讯会议：385544473 Passcode: 202202

Title：

Some progresses on spectra of discrete structures

Abstract：

Simplicial complexes, graphs and hypergraphs are typical discrete structures that can be studied by employing methods in spectral theory. In this talk, I will present some recent developments on spectra of discrete structures. This includes new results on maximal gap intervals for the graph Laplacian; a refined analysis of the variational and non-variational eigenvalues of the graph p-Laplacian; the nonlinear spectral duality for convex and homogeneous function pairs, and its applications to hypergraph p-Laplacians, and Cheeger inequalities on simplicial complexes.

Speaker: Norbert PEYERIMHOFF（Durham University）

Time：

February 25, 16:00-17:00 (Beijing Time)

Place：

（UPDATED） Zoom: 98661278861 Passcode: 980641

Title：

Some applications of an improved Cheeger inequality by Kwok et al

Abstract：

In 2013, Kwok, Lau, Lee, Oveis Gharan and Trevisan gave an improved Cheeger inequality in the graph theoretical setting. This inequality involves, besides the first positive eigenvalue and the Cheeger constant, also higher Laplace eigenvalues. In this talk, I will discuss this improved Cheeger inequality and present some applications of its counterpart in the smooth setting of Riemannian manifolds. The talk will cover results by Shiping Liu and also results derived in collaboration with him and Matthias Keller.

Speaker: Zhiqin LU（UC Irvine）

Time：

January 21, 10:00-11:00 (Beijing Time)

Place：

腾讯会议：772909850 Passcode: 202201

Title：

On manifold with positive spectrum

Abstract：

In this talk, I will first present my joint work with Bobo Hua on manifold with positive spectrum with respect to a Schrodinger operator. On a relevant topic, I will then present the discontinuity phenomena of eigenfunctions on singular space and its applications in constructing manifold with essential spectrum gaps.

Speaker: Juergen JOST（Max-Planck Institute for Mathematics in the Sciences, Leipzig）

Time：

January 14, 16:00-17:00 (Beijing Time)

Place：

Zoom：91088937796 Passcode: 545110

Title：

Spectra of graphs and hypergraphs

Abstract：

In this talk, after recalling results about the spectrum of the normalized Laplacian of a graph, I shall introduce a Laplace operator for a class of hypergraphs that model chemical reaction systems, and I shall present the corresponding spectral theory. This will include Cheeger type inequalities. The talk will represent joint work with Raffaella Mulas and Zhang Dong.

• ORGANIZERS

  • Prof. Shiping LIU (USTC)
  • Prof. Zuoqin WANG (USTC)