Geometry and Analysis Seminar, Fall 2024

Organizers: Siqi He, Guohuan Qiu, Jian Wang, Jinmin Wang

We welcome everyone who is interested in speaking and attending our seminar. If you are interested in giving a seminar talk, please tell us by sending an email to jinmin@amss.ac.cn

Our seminar holds weekly on Tuesdays. The location is No.55 Zhongguancun East Road, Beijing, China, and the precise seminar room will be updated below.

Schedule

The schedule is shown here and you can find the details of the planned talks below this table.

Abstract

  1. Speaker: Peter Smillie (Max Planck Institute)

    Title: The mapping class group action on SL(2,R) representations.

    Time & Venue: 2024.9.10（周二）16:00-17:00 晨兴 410

    Abstract: Let X(G) be the character variety of representations of a surface group into a Lie group G, so that the mapping class group acts on X(G). When G is compact, this action is ergodic by work of Goldman and Pickrell-Xia, and a well-known conjecture of Goldman is that for G=PSL(2,R) (and genus at least 3), the action is ergodic on each non-Fuchsian topological component of X(G). This turns out to be essentially equivalent to a conjecture of Bowditch that every non-elementary non-Fuchsian representation in X(PSL(2,R)) sends some simple closed curve loop to a non-hyperbolic element. By studying the action of the mapping class group on the tangent cone to the subvariety of nontrivial diagonal representations, we prove that Bowditch's condition holds in a neighborhood of the nontrivial diagonal representations in the Euler number zero component. This is joint work with James Farre and Martin Bobb.

  2. Speaker：Jintian Zhu (Westlake University)

    Title: Towards a further comprehension for mass inequalities

    Time & Venue: 2024.9.24（周二）10:30-11:30 晨兴 110

    Abstract: In this talk, I will start with an introduction on asymptotically flat manifolds and classical mass inequalities including Riemannian positive mass theorem and Riemannian Penrose inequality. Then we make a review on recent developments on mass inequalities for asymptotically flat manifolds with arbitrary ends and mention the most general mass-systole conjecture raised by myself. Finally, I present my recent works on the mass-systole conjecture as evidence for its validity.

  3. Speaker: Haizhong Li (Tsinghua University)

    Title: Summation for two sets in hyperbolic space

    Time & Venue: 2024.10.15（周二）​16:00-17:00 晨兴410

    Abstract: The classical Brunn-Minkowski theory studies the geometry of convex bodies in Euclidean space by use of the Minkowski summation. In this talk, we introduce a summation for two sets in hyperbolic space, and we call it the hyperbolic summation. Then we develop a Brunn-Minkowski theory in hyperbolic space by use of our hyperbolic summation. This is joint work with Botong Xu.

  4. Speaker: ​Guo Chuan Thiang (Peking University)

    Title: Large-scale geometry obstructs localization

    Time & Venue: 2024.10.22（周二）​​10:30-11:30 晨兴410

    Abstract: Given a Riemannian manifold, we may choose a discretization and find a localized basis for the Hilbert space of square-integrable functions. However, there exist natural Hilbert subspaces which do not admit such a discretely-localized description. This is based on the concept of topological insulators in physics, as quantum systems with no atomic limit. I will explain that the obstruction to localizing a Hilbert subspace is a coarse-geometric invariant, which can be computed by an index formula with concrete physical meaning. The lesson is that discretization of space has a very different meaning in quantum mechanics, with invariants going beyond the classical topological ones associated to triangulations of the manifold.

  5. Speaker: ​Zhenxiao Xie (Beihang University)

    Title: Willmore surfaces in 4-dimensional conformal manifolds

    Time & Venue: 2024.11.5（周二）​​10:30-11:30 晨兴110

    Abstract: In this talk, we show the first and second variational formulas of the Willmore functional for closed surfaces in 4-dimensional conformal manifolds. As an application, the Clifford torus in CP^2 is proved to be strictly Willmore-stable. This provides a strong support to the conjecture of Montiel and Urbano, which states that the Clifford torus in CP^2 minimizes the Willmore functional among all tori or all Lagrangian tori. In 4-dimensional locally symmetric spaces, by constructing some holomorphic differentials, we prove that among all minimal 2-spheres only those super-minimal ones can be Willmore. This is a joint work with Prof. Changping Wang.

  6. Speaker: ​Shijie Gu (Northeastern University)

    Title: BNPC manifolds of dimension at most four are Euclidean

    Time & Venue: 2024.11.12（周二）​​10:30-11:30 晨兴110

    Abstract: In 1981, Gromov asked whether there exist simply connected topological manifolds, other than Euclidean space, that admit a metric of non-positive curvature in a synthetic sense. Since CAT(0) spaces are contractible, it follows from the classification of surfaces that any CAT(0) 2-manifold is Euclidean. In dimension 3, by combining results of Brown and Rolfsen, CAT(0) manifolds are homeomorphic to R^3. Recently, Lytchak, Nagano, and Stadler proved that CAT(0) 4-manifolds are Euclidean. In this talk, I will discuss Gromov's question and introduce spaces of (global) non-positive curvature in the sense of Busemann, abbreviated as BNPC spaces. I will show that the results above can be extended to BNPC manifolds. This is joint work with Tadashi Fujioka.

  7. Speaker: ​Dashan Yan (Stony Brook University)

    Title: A Gluing Theorem For Collapsing Warped-QAC Calabi-Yau Manifolds

    Time & Venue: 2024.12.17（周二）​​10:30-11:30 晨兴110

    Abstract: We carry out a gluing construction for collapsing warped-QAC Calabi-Yau manifolds in \(C^{n+2} n>1\). This gluing theorem verifies a conjecture by Yang Li on the behavior of the warped QAC Calabi-Yau metrics on affine quadrics when two singular fibers of a holomorphic fibration go apart. We will also discuss a bubble tree structure for those metrics.

  8. Speaker: ​Zhihan Wang (Cornell University)

    Title: Generic Regularity of Minimal Submanifolds with Isolated Singularities

    Time & Venue: 2024.12.24（周二）​​10:30-11:30 晨兴110

    Abstract: Singularities are commonly found in geometric variational objects, such as minimal submanifolds, where they are locally modeled on minimal cones. Despite the abundance of singularity models constructed in the literature, it is conjectured that in generic settings, they are significantly simpler. In this talk, we present a characterization of minimal cones that can serve as singularity models for minimal submanifolds with isolated singularities in a generic Riemannian manifold, without imposing additional constraint on dimension or codimension. As an application, we shall discuss a generic finiteness result of low area minimal hypersurfaces in nearly round 4-spheres. This is based on the joint work with Alessandro Carlotto and Yangyang Li.