Date:16:00,March 3
Place: Tencent Meeting:692-996-019, No password
Speakers:Martin de Borbon (King’s College London)
Title:Parabolic bundles and spherical metrics
Abstract:I will use the Kobayashi-Hitchin correspondence for parabolic bundles to derive results of Troyanov and Luo-Tian on existence and uniqueness of spherical metrics on the Riemann sphere with cone angles less than . This is joint work with Dmitri Panov.
Date:16:00pm,March 4
Place: Tencent Meeting:692-996-019, No password
Speakers:Martin de Borbon (King’s College London)
Title:Polyhedral Kahler cone metrics on Cn
Abstract:I will discuss a particular class of flat torsion free meromorphic connections on Cn with simple poles at hyperplane arrangements. The main result is that, if the holonomy is unitary, then the metric completion (of the flat Kahler metric on the arrangement complement) is polyhedral. In the case of the braid arrangement, our result extends to higher dimensions the well-known existence criterion for spherical metrics on the Riemann sphere with three (non-integer) cone points. This is joint work with Dmitri Panov.
Date:16:00pm,March 24
Place: Tencent Meeting:740-871-646 PW:0324
Speakers:Dali Shen (Tata Institute of Fundamental Research)
Title:Kaehler cone-manifolds associated with a projective arrangement
Abstract:Given a hyperplane arrangement of some type in a projective space, the Dunkl system, developed by Couwenberg, Heckman and Looijenga, is used to study the geometric structures on its complement, and as a consequence it leads to the discovery of new ball quotients when
the so-called Schwarz conditions are imposed. In this talk, I will show that the space, investigated in this system, is still of a particular type of structure, namely, the structure of a cone-manifold, when there is no Schwarz conditions imposed. I will illustrate this theory by discussing the one-dimensional example, which originates from the classical hypergeometric system.
Date:14:00-15:00pm, April 1
Place: Tencent Meeting:806-612-530, No password
Speakers:Li Sheng (Sichuan University)
Title:Extremal Metrics on Toric Manifolds and Homogeneous Toric Bundles
Abstract:An example of Apostolovet al. indicate that the condition of K-stability may not be correct one for general polarised manifolds. Szekelyhidi modified definition of K-stability by filtration and stated a variant of the Yau-Tian-Donaldson conjecture. We will talk about our proof of this variant of YTD conjecture for toric manifolds and homogeneous toric bundles. This is jointed work with Li An-Min and Lian Zhao.
Date: 15:30-16:30pm, April 7
Place: Tencent Meeting: 814-705-572, No password
Speakers: Ling Yang (Fudan University)
Title: On complete space-like stationary surfaces in 4-dimensional Minkowski space with graphical Gauss image
Abstract: Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto's theorem to complete space-like stationary surfaces in 4-dimensional Minkowski space, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in 4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish several structure theorems for complete stationary graphs in 4-dimensional Minkowski space from the viewpoint of the degeneracy of Gauss maps.
Date:14:00-15:00pm,April 14
Place: Tencent Meeting:229-114-233, No password
Speakers:Huichun Zhang (Sun Yat-sen University)
Title:A one-phase free boundary problem on non-collapsing RCD-spaces
Abstract:In this talk, we will introduce some regularity results for a one-phase free boundary problem on metric measure spaces with a generalizied lower Ricci bound, curvature-dimension condition. It contains the Lipschitz regularity of solutions and the partial regularity of the free boundary. This is based on a joint work with Chung-Kwong Chan, and Xi-Ping Zhu.
Date:9:00am,April 21
Place: Tencent Meeting:914-3694-1759 PW:202202
Speakers:Tristan Ozuch (MIT)
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
Date: 10:00-11:00am,April 28
Place: Tencent Meeting:702-074-207, No password
Speakers: Yuan Liu (State University of New York at Buffalo)
Title: Cartan's method and its applications in sheaf cohomology
Abstract: We will talk about Cartan’s original ideal in proving theorem A and B on closed cubes. Then we use this method instead of the standard Cech cohomology to prove some well-known results about sheaf cohomology on closed cubes.
Date: 14:00-15:00pm,May 6
Place: Tencent Meeting:805-270-898, No password
Speakers: Xuezhang Chen (Nanjing University)
Title: Almost sharp Sobolev trace inequalities in the unit ball under constraints
Abstract: We establish three new families of Sobolev trace inequalities in the unit ball with constraint of higher order moments of the boundary volume element. All involved Sobolev trace inequalities have been shown to be almost optimal through constructing precise test functions. Moreover, these optimal constants are closely related to cubature formulas and the Delsarte-Goethals-Seidel theory in 1977. These higher order moments Sobolev trace inequalities could be regarded as natural generalizations of Beckner and Lebedev-Milin inequalities to the Paneitz operator. This is joint work with Wei Wei and Nan Wu.
Date:9:00am,May 12
Place: Tencent Meeting:914-3694-1759 PW:202202
Speakers:Maxwell Stolarski ( Arizona State University)
Title: Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
Abstract:Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss recent work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.
Date: 9:00-10:00am, May 19
Place:Tencent Meeting:335-951-828, No password
Speakers:Alex Mramor (Johns Hopkins University)
Title:Some new applications of the mean curvature flow to the study of self shrinkers
Abstract:Self shrinkers are basic singularity models for the mean curvature flow. By perturbing the appropriately and flowing out of them, one can leverage what one knows about the behavior of the flow to say something about the original shrinker in question. In this talk I’ll discuss some recent results applying this idea to self shrinkers in R^3 and R^4
Date: 10:00-11:00am, May 26
Place:Tencent Meeting:787-047-089, No password
Speakers:Yuan Liu (State University of New York at Buffalo)
Title:On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces
Abstract:This talk focus on a generalization of the result of Katzarkov and Ramachandran from algebraic surfaces to Kähler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact Kähler surface which does not have two ends, except that we replace the p-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.
Date:14:30-15:30, June 9
Place: Tencent Meeting: 511-600-793, no password
Speakers:Wenjiao Yan (Beijing Normal University)
Title:On two questions of James
Abstract:In 1958, I. M. James defined octonionic Stiefel spaces V_k(O^n)—the space of orthonormal k-frames in O^n with natural topology and raised two fundamental questions about V_k(O^n): (1) Is the projection π:V_k(O^n) → V_q(O^n)(q < k) a fiber map ? (2) Is V_k(O^n) a manifold ? We gave partial answers to both of them. This talk is based on joint works with Professor Zizhou Tang and Professor Chao Qian.
Date:14:00-15:00, June 16
Place: Tencent Meeting:382-540-149, no password
Speakers:Chong Song (Xiamen University)
Title:Finite-time singularities of 2d V-harmonic map flow
Abstract:The V-harmonic map is a natural generalization of the harmonic map, but lack of a variational structure in general. We find that when V is conformal, the evolution and blow-up of 2d V-harmonic map flow still share same properties with the ordinary harmonic map flow. In particular, the finite-time singularities must locate at zeroes of V when the blow-up rate is sufficiently slow.
Date:9:00am, June 23
Place: Tencent Meeting:914-3694-1759 PW:202202
Speakers:Jonathan Zhu (Princeton University)
Title: Prescribed-point area estimates in space forms
Abstract: We discuss sharp area estimates for minimal submanifolds that pass through a prescribed point in a geodesic ball in a space form. The corresponding estimate in Euclidean space was first conjectured by Alexander, Hoffman and Osserman in 1974, and was previously proven in full generality by Brendle and Hung.
Date:9:00-10:30am, Jul 7,8,19,20, 2022
Place: Room 5106, the Fifth Teaching Building
Speakers:Xin Nie (Southeast University)
Title:Higgs bundles, higher Teichmüller theory and minimal surfaces (I, II, III, IV)
Abstract:1. Holomorphic vector bundles and Higgs bundles. • Symplectic reduction and its Kähler/hyperkähler versions. • Relation with G.I.T. (Kempf-Ness theorem). • Holomorphic vector bundles, Atiyah-Bott's reduction and Narasimhan-Seshadri theorem. • Higgs bundles, Hitchin's reduction and Hitchin-Kobayashi correspondence. 2. G-Higgs bundles and surface group representations. • SL(2,R)-Hitchin component and Teichmüller theory. • SL(n,R)-Hitchin component, cyclic Higgs bundles
• Theory of semsimple Lie algebras and principal 3d subalgebras. • G-Higgs bundles for a general Lie group G. • SO(p,q)-Higgs bundles. 3. Harmonic maps associated to G-Higg bundles and Labourie's conjecture
• harmonic maps, minimal surfaces and Riemannian symmetric spaces. • harmonic maps given by G-Higgs bundles. • Energy functional and Labourie's conjecture. • Strategy via Infinitesimal rigidity and the SL(3,R)-case (affine spheres). • Cyclic surfaces (Labourie Ann.Math.2017). 4. Minimal surfaces in pseudo-hyperbolic spaces. • pseudo-hyperbolic spaces. • second variation formula and maximal surfaces. • Relation with Teichmüller theory (Bonsante-Schlenker Invent.Math.2010). • Labourie's conjecture for SO(2,n) (Collier-Tholozan-Toulisse Duke.Math.J. 2019). • A-surfaces and their infinitesimal rigidity (speaker arXiv:2206.13357)