The ECNU Differential Geometry Seminar takes place usually on Thursdays at 1:00 pm in Room 318 Building No. 3 on the Minhang Campus.

- Date: September 11, 2018 10:00-11:00
Place: Math Building 401

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.