Room: Math Building, 102 Minhang Campus
Speaker: 李逸 教授 （东南大学）
Title: Scalar curvature along Ricci flow
Abstract: In this talk I will discuss two interesting geometric flows, the classical Ricci flow and the coupled Kahler-Ricci flow introduced recently by Yuan Yuan, Yugang Zhang and myself. For the Ricci flow, a very recent work on the scalar curvature that is related with Hamilton's conjecture was derived, which generalizes X. D. Cao's result. For the coupled Kahler-Ricci flow, besides some basic properties, I will propose two conjectures on long time existence.
Room: Math Building, 102 Minhang Campus
Speaker: 张振雷 教授 （首都师范大学）
Title: Relatvie volume comparison of Ricci flow
Abstract: I will present a relative volume comparison of Ricci flow. It is a refinement of Perelman pseudolocality theorem. It is a joint work with Professor Tian.
Room: Math Building, 102 Minhang Campus
Speaker: 郭斌 教授 （Rutgers University）
Title: Geometric estimates for complex monge ampere equations.
Abstract: In the talk, we will prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions. This is based on a joint work with Xin Fu and Jian Song
Room: Math Building, 102 Minhang Campus
Speaker: 欧剑宇博士（上海数学中心）
Title: Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons
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.
Room: Math Building, 401, Minhang Campus
Speaker: 张永胜 （同济大学 研究员）
Title: Recent progress on the Dirichlet problem for the minimal surface system and minimal cones
Room: Math Building, 102, Minhang Campus
Speaker: 熊革 （同济大学 ）
Title: Title: New affine isoperimetric inequalities
Abstract: In this talk, we will speak on our recent work on the (dual) affine surface area. Specifically, by establishing the variational formula for the (dual) affine surface area, we define the affine (intersection or) projection measure. Taking into account the two important characters in convex geometry: the (intersection or) projection function and the ellipsoid, we establish new affine isoperimetric inequalities, which open up a new passage to attack the longstanding Lutwak conjecture on the extremum problem of affine quermassintegrals since 1980s.
Room: Math Building, 401, Minhang Campus
Speaker: 张世金 （北京航空航天大学 ）
Title: Title: Introduction to the shrinking gradient Ricci solitons
Abstract: In this talk, first I mainly introduce some important results of the shrinking gradient Ricci solitons. Secondly, I will introduce the classification of shrinking Kahler-Ricci solitons for nonnegative holomorphic bisectional (or orthogonal bisectional ) curvature. The bisectional curvature part of (3) is a joint work with Guoqiang Wu.
Room: Math Building, 401, Minhang Campus
Speaker: 李中凯 教授（上海师范大学）
Title: Boundary behaviour of generalized harmonic functions associated with the Dunkl operators
Abstract:
The Dunkl operators are substitutes of differential operators involving reflection terms, which are connected with Coxeter groups with root systems. In this talk we are concerned with the boundary behaviour of generalized harmonic functions associated with the Dunkl operators. The main results are about characterizations on local existence of non-tangential boundary values of generalized harmonic functions, which are given by non-tangential boundedness of functions and finiteness of a Lusin-type area integral in the Dunkl setting. This is a joint work with Jiaxi Jiu.
Room: Math Building, 401, Minhang Campus
Speaker: 莫小欢 （北京大学 教授）
Title: Title: Homogeneous Einstein Finsler metrics on $(4n+3)$-dimensional spheres
Abstract: In this lecture， we discuss a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ among this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.
Room: Math Building, 401, Minhang Campus
Speaker: 吴国强 （浙江理工大学）
Title: Title: Some resuls on gradient shrinking Ricci soliton
Abstract: I will start with some background on Ricci soliton, then review previous results on soliton from three aspects: the rigidigy, the local regularity and the classification results; at last I will focus on a global problem, i.e. The splitting theorem. Our main result implies that the soliton has only one end, if the Ricci curvature satisfies some pinching condition
Room: Math Building, 401, Minhang Campus
Speaker: 王建红 （ECNU）
Title: The gradient estimates for nonlinear parabolic equations and applications
Room: Math Building, 402, Minhang Campus
Speaker: Mounir Hajli （上海交通大学）
Title: Title: The holomorphic analytic torsion associated with canonical metrics on P^1
Abstract: In this talk, I shall construct singular Laplacians associated with the canonical metrics of line bundles on the complex projective line. Then, I shall show that these operators admit a non-negative, discrete and infinite spectrum. I shall explain how to compute explicitly the regularised determinant of the zeta function associated with this Spectrum.
Room: Teaching Building NO. 3, Room 418, Minhang Campus
Speaker: 吴鹏 （上海数学中心）
Title: Einstein four-manifolds of positive determinant self-dual Weyl curvature
Abstract: The question that when a four-manifold with a complex structure admits a compatible Einstein metric of positive scalar curvature has been answered by Tian, LeBrun, respectively. Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, LeBrun classified Hermitian Einstein four-manifolds with positive scalar curvature. In this talk we consider the inverse problem, that is, when a four-manifold with an Einstein metric of positive scalar curvature admits a compatible complex structure. We will show that if the determinant of the self-dual Weyl curvature is positive then the manifold admits a compatible complex structure.
Room: Math Building Room 401
Speaker: 殷浩 (USTC)
Title: Higher order neck analysis for harmonic maps and its applications
Abstract: We shall prove some refined estimate on the neck region when a sequence of harmonic maps from surfaces blow up. It generalizes the well-known energy identity and no neck theorem of harmonic maps. We then discuss applications, which include a further blow-up of the neck region and an inequality about the nullity and index of the sequence.
Room: Math Building Room 401
Speaker: 熊金刚
Title: Smoothness of weak solutions to the fast diffusion equations
Abstract: In the pioneering work of Berryman and Holland 1980, stability results for $C^{2,1}$ solutions to fast diffusion equations (FDE) in bounded domains with zero Dirichlet condition were established. However, the existence of $C^{2,1}$ solutions was not known and establishing $C^{2,1}$ a priori estimates was listed as an open problem in that paper. Global Holder regularity and interior smoothness in the subcritical range were proved later by Yazhe Chen & DiBenedetto 1988 and DiBenedetto, Kwong & Vespri 1991. I will report a recent joint work with Tianling Jin, in which we solved this regularity problem completely. We proved that weak solutions are smooth to the boundary in the subcritical and critical range. One of the novel ingredients of our proof is using an evolution equation of a quantity (generalizing the scalar curvature), by which we established $L^\infty L^q$ estimates for the time derivative.
Room: Math Building Room 402
Speaker: W. Klingenberg, Durham University
Title: On a conjecture of Toponogov on complete convex surfaces
Abstract: In 1995, Victor Andreevich Toponogov authored the following conjecture: “Every smooth strictly convex and complete classical surface of the type of a plane has an umbilic point, possibly at infinity“. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle. Namely we prove that (a) the Fredholm index of an associated Riemann Hilbert boundary problem for holomorphic discs is negative, which is an outcome of the regularity of the Cauchy-Riemann operator in presence of a symmetry group. Thereby, (b) no such solutions may exist for a generic perturbation of the boundary condition (these form a Banach manifold under the assumption that the Conjecture is incorrect). Finally, however, (c) the geometrization by a neutral metric of the associated model allows for Mean Curvature Flow with mixed Dirichlet - Neumann boundary conditions to generate a holomorphic disc from an initial spacelike disc. This completes the indirect proof of said conjecture as (b) and (c) are in contradiction.
Room: Teaching Building No. 3, Room 214
Speaker: Shi Yalong (Nanjing University)
Title: J-flow and cscK metrics on minimal models
Abstract: We use the recent theorem of Chen-Cheng to prove the existence of a family of collapsing cscK metrics on Kahler manifolds with semi-ample canonical bundles. A conjecture about the limiting behavior of these metrics will also be discussed. This is joint work with Wangjian Jian and Jian Song.
Room: Teaching Building No. 3, Room 418
Speaker: Huang Libing (Nankai University)
Title: A conclusive theorem on Finsler metrics of sectional flag curvature
Abstract: Flag curvature is the most important quantity in Finsler geometry, because it generalizes the notion of sectional curvature in Riemannian geometry. In this talk I will give a brief introduction to Finsler geometry and discuss some interesting curvature phenomena. In particular, I will prove that when flag curvature reduces to sectional curvature, then either the Finsler metric is Riemannian, or the flag curvature is isotropic (constant if the dimension is greater than two).
Room: Room 401, Math Building
Speaker: Prof. Hongwei Xu (Zhejiang University)
Title: Recent Developments in Sphere Theorems
Abstract: It plays an important role in geometry and topology of manifolds to study sphere theorems. During the past fifteen years, a number of great achievements on sphere theorems, including the Poincare conjecture, the 1/4-pinching differentiable sphere theorem, and the Willmore conjecture, have been made by several geometers. In this talk, I will discuss recent progress and new techniques developed in the study of sphere theorems. I will also propose some open problems in this area.
Room: Room 102, Math Building
Speaker: Li fengjiang (ECNU)
Title: Rigidity characterization of compact Ricci solitons
Abstract: In this talk, I will introduce the recent joint work with Jian Zhou. Firstly we define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a compact Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.
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.
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.
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.
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.
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
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.
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.
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.
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 fundament