The ECNU Differential Geometry Seminar takes place usually on Thursdays at 1:00 pm in Teaching Building NO. 4, Room 414 on Minhang Campus.

- Date: Thursday, Jan. 3, 2019, 1:00-2:00pm

Room: Room 102, Math Building

Speaker: Dr. Casey Blacker (ECNU)

Title: Polysymplectic Reduction and the Moduli Space of Flat Connections

Abstract: In a landmark paper, Atiyah and Bott showed that the moduli space of flat connections on a principal bundle over an orientable closed surface is the symplectic reduction of the space of all connections by the action of the gauge group. By appealing to polysymplectic geometry, a generalization of symplectic geometry in which the symplectic form takes values in a given vector space, we may extend this result to the case of higher-dimensional base manifolds. In this setting, the space of connections possesses a natural polysymplectic structure, and the polysymplectic reduction of this space by the action of the gauge group yields the moduli space of flat connections equipped with a 2-form taking values in the cohomology of the base manifold. In this talk, based on the recent preprint arXiv:1810.04924, I will first review the polysymplectic formalism and then outline its role in obtaining the moduli space of flat connections.

- Date: Thusday, Dec. 27, 2018, 1:00-2:00pm
- Date: Thursday, Dec. 20, 2018, 1:00-2:00pm
- Date: Thursday, Dec. 6, 2018, 1:00-2:00pm

Room: Room 401, Math Building

Speaker: Dr. Bo Liu (ECNU)

Title: Family version of the Kirillov formula

Abstract: The Kirillov formula is an important formula in representation theory which represents the character of a representation as an integral over a certain orbit of the group in its coadjoint represention. In 80's, this formula was extended as an index formula with infinitesimal group action by Berline-Vergne and Bismut. In this talk, we will extend this formula to the family version. This is a joint work with Xiaonan Ma recently.

Room: Room 102, Math Building

Speaker: Dr. Lina Chen (ECNU)

Title: A Geometric Approach to the Modified Milnor Problem

Abstract: The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency Conjecture in Riemannian geometry: given $n, d>0$, there exists a constant $\epsilon(n,d)>0$ such that if a compact Riemannian $n$-manifold $M$ satisfies that Ricci curvature $\op{Ric}_M\ge -(n-1)$, diameter $d\ge \op{diam}(M)$ and volume entropy $h(M)<\epsilon(n,d)$, then the fundamental group $\pi_1(M)$ is virtually nilpotent. We will verify the Nilpotency Conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e., if $h(M)<\epsilon(n,d)$, then $h(M)=0$. This is a joint work with Professor Xiaochun Rong and Shicheng Xu.

Room: Teaching Building No. 4, Room 414

Speaker: Dr. Huan Wang (ECNU)

Title: The growth of dimension of cohomology of semipositive line bundles on complex manifolds

Abstract: We study the dimension of cohomology of semipositive line bundles on complex manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. We will introduce the background and recent progresses based on arXiv: 1810.09881v1.

- Date: Monday, Nov. 19, 2018, 3:00-4:00pm
- Date: Nov. 22, 2018, 1:00-2:00pm
- Date: Nov. 26, 2018 （Monday), 1:00-2:00pm

Room: Math Building, Room 102

Speaker: Prof. Pak-Tung HO (Sogang University, Seoul, South Korea)

Title: Q-curvature in conformal geometry

Abstract: Q-curvature is a generalization of the Gaussian curvature. In this talk, I will explain about the definition of Q-curvature and some of its properties. Then I will talk about some problems related to Q-curvature, including the problem of prescribing Q-curvature.

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Haozhao Li (University of Science and Technology of China)

Title: On the multiplicity-one conjecture for mean curvature flow

Abstract: In this talk, I will explain recent progress on Ilmanen's multiplicity-one conjecture for closed smooth embedded mean curvature flow with type I mean curvature. This is joint work with Bing Wang.

Room: Math Building, Room 401 (Minhang Campus)

Speaker: Prof. Qing Ding (Fudan University)

Title: Almost Complex Structures on S^6 and Related Schrodinger Flows

Abstract: In this talk, we report our recent result on the G_2 binormal motion of curves in R^7

associated to the almost complex structure on S^6 by using the $G_2$-structure from the octonions O.

Some related problems are also discussed.

- Date: Nov. 29, 2018, 1:00-2:00pm

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Yuxing Deng (Beijing Institute of Technology)

Title: Gradient steady Ricci solitons with linear curvature decay

Abstract: In this talk, we will talk about steady Ricci solitons with nonnegative sectional curvature and linear scalar curvature decay. In particular, we will give a classification of 3-dimensional steady Ricci solitons with linear decay and positively curved 4-dimensional noncollapsed steady Ricci solitons with linear decay.

- Date: Wednesday, Nov. 14, 2018, 1:00-2:00pm
- Date: Thursday, Nov. 15, 2018, 3:30-4:30pm

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Shiping Liu (University of Science and Technology of China)

Title: What are discrete spheres?

Abstract: The Bonnet-Myers theorem states that an n-dimensional complete Riemannian manifold M with Ricci curvature lower bounded by a positive number (n-1)K is compact, and its diameter is no greater than $\pi/\sqrt{K}$. Moreover, Cheng’s rigidity theorem tells that the diameter estimate is sharp if and only if M is the n-dimensional round sphere. In this talk, I will discuss discrete analogues of round spheres in graph theory via exploring discrete Bonnet-Myers-Cheng type results. This talk is based on joint works with Cushing, Kamtue, Koolen, Muench, and Peyerimhoff.

Room: Math Building, Room 401

Speaker: Prof. Huai-Dong Cao (Lehigh University)

Title: Singularities of the Ricci flow and Ricci solitons

Abstract: The Ricci flow, introduced by R. Hamilton in 1982, evolves the initial geometry of a given space by the parabolic Einstein equation. One of the most important issues in the study of the Ricci flow is to understand the formation of singularities. It turns out generic singularities of the Ricci flow are essentially modeled by Ricci solitons. In this talk, I will discuss the phenomena of singularity formation in the Ricci flow and present some recent progress on Ricci solitons

- Date: Nov. 8, 2018, 1:00-2:00pm

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Chengjie Yu (Shantou University)

Title: Li-Yau Type Gradient Estimates on Hyperbolic Spaces

Abstract: In this talk, I will present my recent joint works with Ms. Feifei Zhao on finding sharp Li-Yau type gradient estimates on manifolds with a nonzero Ricci curvature lower bound. Indeed, we only obtained some partial results on hyperbolic spaces.

- Date: Nov. 1, 2018, 1:00-2：00pm

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Bin Shen (Southeast University)

Title: On variation of action integral in Finsler gravity

Abstract: In this talk, we will introduce a generalized action integral of both gravity and matter defined on the sphere bundle over Finsler space-time manifold $M$ with a Lorentz-Finsler metric. The Euler-Lagrange equation of this functional, a generalization of the Riemann-Einstein gravity equation with a defined cosmological constant, is obtained by using some divergence theorems. Fibers of the sphere bundle are unbounded according to the pseudo-Finsler metric. Moreover, solutions of vacuum Finsler gravity equation under the weakly Landsberg condition are discussed and some concrete examples are provided. At last, we raise some questions for further study.

- Date: Thur. Oct. 25, 2018, 1:00-2:00pm

Room: Teaching Building No. 4, Room 414

Speaker: Dr. Yashan Zhang (Peking University, BICMR)

Title: Generalized Kaehler-Einstein metrics on Riemann surfaces and applications

Abstract: In this talk, we plan to discuss Song-Tian's (possibly singular) generalized Keahler-Einstein metric on the canonical models of projective manifolds with semi-ample canonical line bundle. When the canonical model is one dimensional (i.e. a Riemann surface), we give the metric asymptotics of the generalized Kaehler-Einstein metric near its singular points, implying a special case of a conjecture of Song and Tian. Then we present some applications of this result in studying infinite-time singularities of the Kaehler-Ricci flow.

- Date: Oct. 18, 2018, 1:00-2:00pm
- Date: September 11, 2018 10:00-11:00
- Place: Math Building 401

Room: Teaching Building No. 4, Room 414

Speaker: Prof. Shicheng Xu (Capital Normal University)

Title: Quantitative rigidity for domains and immersed hypersurfaces in a Riemannian manifold

Abstract: A classical isoperimetric inequality by A. D. Alexandrov says that for any simply-connected domain Ω on a surface, L^2>=4π*A-K*A^2, where L is the length of boundary, A the area of Ω, and K the upper bound of Ω's Gaussian curvature. Moreover, "=" holds if and only if Ω is a geodesic ball of constant curvature K. For domains in higher dimensional Riemannian manifolds, however, such isoperimetric-typed rigidity with respect to the upper sectional curvature bound is rarely known.

In this talk, we consider a similar rigidity via Heintze-Reilly's inequality for immersed hypersurface M^n in a convex ball B(p,R) of a (n+1)-manifold N: λ_1(M)<= n(K+max H), where λ_1 is 1st eigenvalue of Laplacian on M, H the mean curvature of immersion, and K=max K_N the upper sectional curvature bound of N.

We prove its quantitative rigidity: under some natural restrictions on R, vol(M), mean curvature H and L^q norm (q>n) of 2nd fundamental form of M, if λ_1(M)>= n(K+max H)(1-ε), then not only M is embedded, diffeomorphic and C^α-close to a round sphere, but also the whole enclosed domain Ω is C^{1,α}-close to a geodesic ball of constant curvature K.

Such quantitative rigidity is known before only in simply-connected space forms or the infinitesimal case that diam M goes to 0. We construct counterexamples to show that both the bound of 2nd fundamental form's L^q-norm (q>n) and the convexity of B(p,R) are necessary. Our proof is based on tools from comparison Riemannian geometric, geometric analysis and metric geometry, such as, Moser iteration, Cheeger-Gromov's convergence theorem, and C^α convergence of pointwise non-collapsing manifolds with a L^p integral Ricci curvature bound in Cheeger-Colding's theory. This is a joint work with Yingxiang Hu.

Speakers: Shi Yuguang (Beijing University)

Title: Quasi-local mass and uniqueness of isoperimetric surfaces in asymptotically hyperbolic manifolds

Abstract: Quasi-local mass is a basic notion in General Relativity. Geometrically, it can be regarded as a geometric quantity of a boundary of a 3-dimensional compact Riemannian manifold. Usually, it is in terms of area and mean curvature of the boundary. It is interesting to see that some of quasi-local masses, like Brown-York mass, Hawking mass and isoperimetric mass have deep relation with classical isoperimetric inequality in asymptotically flat (hyperbolic) manifolds. In this talk, I will discuss these relations and finally give an application in the uniqueness of isoperimetric surfaces in asymptotically Ads-Schwarzschilds manifold with scalar curvature. This talk is based on my recent joint works with M.Echmair, O.Chodosh and my Ph.D student J. Zhu .

## Short course:

- Title: Introduction to Ricci flow
Speaker: Prof. Bennett Chow （UCSD）

Classroom: Mathematics Department, Minhang

Schedule:

Lecture 1: September 14, 10:00-11:40 Room 126

Lecture 2: September 17, 10:00-11:40 Room 102

Lecture 3: September 19, 10:00-11:40 Room 102

- Date: July 12, 2018 10:00-11:00
Place: Math Building 402

Speakers: Pan jiayin (Rutgers University)

Title: On the Milnor conjecture

Abstract: In 1968, Milnor conjectured that any open n-manifold of non-negative Ricci curvature has a finitely generated fundamental group. This conjecture remains open today. We will talk about the history and recent progress on this conjecture.

## Summer school:

- Title: On complex Monge-Ampere equation and Calabi Conjecture
Speaker: Professor Zhiqin Lu, University of California, Irvine

Abstract: We will talk about Yau’s solution of Calabi Conjecture, with an introduction of working knowledge of PDE in Kahler geometry along the way. The techniques will be using are: Moser iteration, Alexsandroff maximum principle, Krylov-Safarov weak Harnarck inequality, Krylov-Evans estimate in the context of complex Monge-Ampere equations.

Classroom: Mathematics Department, Room 402, Minhang

Schedule:

Lecture 1: July 2, 13:30-15:30

Lecture 2: July 3, 13:30-15:30

Lecture 3: July 4, 13:30-15:30

Lecture 4: July 5, 13:30-15:30

Lecture 5: July 9, 13:30-15:30

Lecture 6: July 10, 13:30-15:30

- Date: July 10, 2018 10:00-11:00
Place: Math Building 402

Speakers: Li Nan (CUNY—New York City college of technology)

Title: Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Abstract: We study the quantitative singular sets $\mathcal S^k_\epsilon$ for collapsed Alexandrov spaces. We prove a new covering theorem and the packing estimates for $\mathcal S^k_\epsilon$. We also show that $\mathcal S^k_\epsilon$ are $k$-rectifiable, and for every $1\le k\le n-2$, we construct examples for which $\mathcal S^k_\epsilon$ is a Cantor set with positive $\mathcal H^k$-measure. This is a joint work with Aaron Naber.

- Date: June 15, 2018 9:30-10:30
Place: Math Building 126

Speakers: Dr. Zhang yongjia (UCSD)

Title: On Perelman's no shrinking breather theorem

Abstract: We prove Perelman's no shrinking breather theorem in the complete and noncompact case. Out proof uses the idea of Lu and Zheng of constructing an ancient solution, and removes a technical assumption made by them.

- Date: June 15, 2018 10:30-11:30
Place: Math Building 126

Speakers: Prof. Bennett Chow （UCSD）

Title: Introduction to problems and conjectures on 4-dimensional Ricci flow

Abstract: A basic question is whether Ricci flow is related to 4-dimensional topology. The answer to this question is unknown. However, one may begin to speculate on what 4-dimensional Ricci flow might look like geometrically during finite time. We discuss the potential impact on singularity theory of the work of Munteanu and Wang and others on gradient Ricci solitons.

- Date: June 7, 2018
Title: Fredholm Conditions on Singular Manifolds

Speaker: Qiao Yu (Shaanxi Normal University)

Abstract: A classical theorem states that if M is a compact manifold and P is a pseudodifferential operator on M, then P is Fredholm if and only if P is elliptic. This theorem is no longer true for singular or non-compact manifolds. In this talk, we would like to extend this theorem to singular setting via Lie groupoid techniques. First of all, we recall the notion of manifolds with corners (following the work of Melrose). Then we present the concept of a Fredholm Lie groupoid, which is a class of Lie groupoids for which certain characterization of Fredholm operators is valid, and then adopt b- calculus, scattering calculus, and edge calculus in the frame work of Fredholm Lie groupoids. Finally, we discuss briefly the relation between Fredholm Lie groupoids and index theory. This is joint work with Catarina Carvalho and Victor Nistor.

- Date: May 17, 2018
Title: Nonparametric Mean Curvature Type Flows of graphs and its applications

Speaker: Zhou Hengyu (Chongqing University)

Abstract: In this talk we discuss nonparametric mean curvature type flows of graphs in product manifolds. The speed of such flow is the mean curvature minus a smooth function. In the case of the capillary terms, we show that such flow exists for all times and converges uniformly to a smooth solution to the Capillary problem. Some applications to the translating mean curvature flow will be discussed.

- Date: May 10, 2018
Title: The growth of dimension of cohomology of semipositive line bundles on Hermitian manifolds

Speaker: Wang Huan (ECNU)

Abstract: We study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we generalize Berndtsson's estimate on compact manifolds to some non-compact cases, including covering manifolds, 1-convex manifolds, pseudo-convex domains, weakly 1-complete manifolds and complete manifolds.

- Date: April 26, 2018 13：00-14：00
Title: Contructing entropy formulas in curvature flows via the Boltzmann entropy

Speaker: Guo Hongxin (Wenzhou University)

Abstract: In this talk, we will present a method to construct entropy formulas in curvature flows starting from the classical Boltzmann entropy for positive solutions of the heat equation.

- Date: April 26, 2018 14：00-15：00
Title: 几何图像

Speaker: 彭亚新 (上海大学)

Abstract: 报告主要介绍数据分析和处理中的几何理论、方法和应用。首先，介绍基于李群线性化和统计学习方法的数据集匹配算法及其应用；然后，介绍微分同胚群上的内蕴优化算法在医学影像标准化中的使用；最后，针对图像检索，识别和分类中所关心的相似度量问题，介绍线性度量学习方法及如何通过度量矩阵群流形上的内蕴优化算法求解最优的度量。最后，推广到非线性度量和局部度量等情形。

- Date: April 19, 2018
Title: Geometric and topological properties of gradient Einstein manifolds

Speaker: Wang linfeng (Nantong University)

Abstract: Gradient Einstein manifolds are complete manifolds with special metric structure. In this report we plan to consider four questions on gradient Einstein manifolds. 1) Estimates for various geometric quantities, in particular the estimates of the scalar curvature and the growth of the potential function play important roles in the study of gradient Einstein manifolds. 2) Eigenvalue estimates of the weighted Laplace operator and the lower bound estimate for the weighted fundamental gap on gradient -Einstein solitons, the spectral gap and compact resolvent for the weighted Hodge Laplace operator on gradient Ricci solitons and quasi Einstein manifolds with lower dimension. 3) The geometric or topological properties at infinity for gradient Einstein manifolds. 4) Classification questions for gradient Einstein manifolds under suitable conditions.

- Date: April 12, 2018 13:00-14:00
Title: Rigidity of Einstein four-manifolds with positive sectional curvature

Speaker: Wu Peng (Fudan University)

Abstract: Einstein metrics are most natural Riemannian metrics on differentiable manifolds. In dimensions 2 and 3, they must have constant sectional curvature, while in dimension 4, they are much more complicated. For the complex setting, in 1990 Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, and in 2012 LeBrun classified Hermitian, Einstein four-manifolds with positive scalar curvature. For the real setting, however less is known, even assuming a (strong) condition of positive sectional curvature. In this talk I will first talk about some background on Einstein manifolds, then I will focus on Einstein four-manifolds with positive curvature. If time permitted, I will also talk about my recent attempts of attacking this problem via k-positive curvature operator.

- Date: April 12, 2018 14:00-15:00
Title: Existence of solutions of a boundary value problem for Dirac-harmonic maps

Speaker: Zhu Miaomiao (Shanghai Jiao Tong University)

Abstract: In this talk, we shall present some recent progresses on a heat flow approach to the existence of solutions of a boundary value problem for Dirac-harmonic maps.

- Date: March 29, 2018
Title: Manifold with positive curvature

Speaker: Wu Guoqiang (Zhejiang Sci-Tech University)

Abstract: In this talk,I will talk about the geometry and topology of manifold with positive curvature. I will start with quite basic fact in Riemannian geometry, then focus on the Sphere theorem proved by other People in the past thirty years. At last, my work on manifold with positive isotropic curvature will be mentioned.

- Date: March 22, 2018
Title: Holomorphic Morse Inequalities revisited

Speaker: Wang Huan (ECNU)

Abstract: In this talk we recall the characterization of projective manifolds and Moishezon manifolds through the positivity of holomorphic line bundles. We will revisit the Siu-Demailly’s solution of Grauert-Riemenschneider conjecture and introduce Demailly’s Holomorphic Morse Inequalities on compact complex manifolds and its generalizations by Bonavero and others.

- Date: March 8, 2018 (Math Building 401)
Title: Finsler warped product metrics of Douglas type

Speaker: Mo Xiaohuan (Beijing University)

Abstract: In this lecture we discuss the warped structures of Finsler metrics. We obtain the differential equation that characterizes the Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics.

- Date: December 14, 2017
Title: Diameter rigidity for Kähler manifolds with positive bisectional curvature

Speaker: Yuan Yuan (Syracuse University)

Abstract: I will discuss the recent work with Gang Liu on the diameter rigidity for Kähler manifolds with positive bisectional curvature.

- Date: December 7, 2017
Title: The existence of Kahler-Einstein metrics on K-polystable Q-Fano varieties with non-positive discrepancies

Speaker: Feng Wang (Zhejiang University)

Abstract: We will prove the YTD's conjecture for $Q-$Fano varieties X which has a log smooth resolution $M$ with non-positive discrepancies. At first, we extend Tian's work to the log smooth case. After proving the log K-stability, we get the existence of conic KE metrics on $M$. Then we show that these metrics converges to the singular KE metric on X. This is a joint work with Professsor Tian and Chi Li.

- Date: November 30, 2017
Title: Harmonic maps into CAT(1) spaces

Speaker: Yingying Zhang (Yau Mathematical Science Center, Tsinghua University)

Abstract: In this talk, we will discuss existence and regularity results of harmonic maps into CAT(1) space. In their famous work, Sacks-Uhlenbeck discovered a "bubbling phenomena" for harmonic maps from a Riemann surface to a compact Riemannian manifold. We generalized this result when the target space is a compact locally CAT(1) space. Our proof adapted a local harmonic replacement technique.

We also discuss regularity results of harmonic maps from a Riemannian polyhedra into a CAT(1) space. (The talk is based on the joint work with C. Breiner, A. Fraser, L. Huang, C. Mese, and P. Sargent.)

- Date: November 23, 2017
Title: Introduction to Elliptic genus

Speaker: Bo Liu (East China Normal University)

Abstract: In this talk, I'll introduce the elliptic genus from the point of view of cobordism, elliptic function and the quantum field theory and explain the relations of it with modular forms and the Monster moonshine. At last, we prove the rigidity of elliptic genus for Z/k manifolds, which is a conjecture in 1996. This is a joint work with Yu Jianqing.

- Date: November 16, 2017
Title: Some universal inequalities for Dirichlet eigenvalues on subgraphs of Lattices

Speaker: Bobo Hua (Fudan University)

Abstract: In this talk, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice Zn. This is a joint work with Yong Lin and Yanhui Su.

- Date: November 9, 2017
Title: Li-Yau gradient estimates without Ricci curvature lower bound

Speaker: Meng Zhu (East China Normal University)

Abstract: In their celebrated work, P. Li and S.-T. Yau proved the famous Li-Yau gradient estimate for positive solutions of the heat equation on manifolds with Ricci curvature bounded from below. Since then, Li-Yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of manifolds.

In this talk, we will present our recent works on Li-Yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely, |Ric_| in L^p for some p>n/2 or certain Kato type of norm of |Ric_| being bounded together with a Gaussian upper bound of the heat kernel. We also study the Li-Yau estimate for the heat equation under the Ricci flow with bounded scalar curvature. These assumptions allow the lower bound of the Ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results on Li-Yau bounds. These are joint works with Qi S. Zhang.

- Date: November 2, 2017
Title: Existence of Ricci flow on noncompact manifolds

Speaker: Fei He (Xiamen University)

Abstract: Though the Ricci flow has been extensively studied, its short-time existence on noncompact manifolds with potentially unbounded curvature is still not fully understood. There has been some important progress on this problem by many authors. We will discuss recent results and their applications.

- Date: October 26, 2017
Title: The Determinant of Laplace Operators and the Analytic Torsion

Speaker: Bo Liu (East China Normal University)

Abstract: In this talk, we introduce the Ray-Singer analytic torsion as the determinant of Laplace operators and the extended Cheeger-Mueller Theorem by Bismut-Zhang which gives the explicit relation between the Ray-Singer analytic torsion and the Reidemeister torsion. Note that the Reidemeister torsion is the first topological invariant in the history distinguishing the homotopy equivalent but not homeomorphic manifolds. At last, we explain the complex valued torsion, Burghelea-Haller torsion and Cappell-Miller torsion, and the resent results by Liu-Yu and Su-Zhang.

- Date: September 27, 2017 (A Special Time)
Title: Quantitative maximal local rewinding volume rigidity with Ricci curvature bounded below II

Speaker: Lina Chen (East China Normal University)

Abstract: For a metric ball $B_r(x)$ in a Riemannian manifold, its local rewinding volume is the volume of $B_r(x^*)$, where $B_r(x^*)\subset (U^*, x^*)$, the (uncomplete) Riemannian universal cover of $(B_r(x), x)$. A compact manifold with Ricci curvature bounded below by $(n-1)H$ is isometric to a space form with constant curvature $H$ if and only if every $\rho$-ball ($\rho$ fixed) achieves the maximal local rewinding volume. In this talk, we will prove that if a compact manifold $M$ with Ricci curvature lower bound $(n-1)H$ satisfies that the universal cover space $\tilde M$ is non-collapsing (there exist a positive lower bound of the volume of a unit ball in $\tilde M$) and each $\rho$-ball almost achieves the maximal local rewinding volume, then this manifold is diffeomorphic and close to a space form with $H$-constant curvature. This is joint work with Xiaochun Rong and Shicheng Xu.

- Date: September 20, 2017 (A Special Time)
Title: Quantitative maximal local rewinding volume rigidity with Ricci curvature bounded below

Speaker: Lina Chen (East China Normal University)

Abstract: For a metric ball $B_r(x)$ in a Riemannian manifold, its local rewinding volume is the volume of $B_r(x^*)$, where $B_r(x^*)\subset (U^*, x^*)$, the (uncomplete) Riemannian universal cover of $(B_r(x), x)$. A compact manifold with Ricci curvature bounded below by $(n-1)H$ is isometric to a space form with constant curvature $H$ if and only if every $\rho$-ball ($\rho$ fixed) achieves the maximal local rewinding volume. In this talk, we will prove that if a compact manifold $M$ with Ricci curvature lower bound $(n-1)H$ satisfies that the universal cover space $\tilde M$ is non-collapsing (there exist a positive lower bound of the volume of a unit ball in $\tilde M$) and each $\rho$-ball almost achieves the maximal local rewinding volume, then this manifold is diffeomorphic and close to a space form with $H$-constant curvature. This is joint work with Xiaochun Rong and Shicheng Xu.