IMS Symposium on Geometry and Topology
Oct 20, 2:30—3:30PM, Teaching Center 102, Miaomiao Zhu (Shanghai Jiao Tong University)
Title: Qualitative behavior for geometric PDEs over degenerating Riemann surfaces
Abstract: In this talk, we shall discuss the compactness of solutions to some geometric PDEs over degenerating Riemann surfaces. In particular, we investigate the asymptotic analysis and qualitative behavior for a general sequence of Sacks-Uhlenbeckα-harmonic maps from degenerating Riemann surfaces. This answers an open problem proposed by J. D. Moore, aiming at developing a partial Morse theory for closed parametrized minimal surfaces in compact Riemannian manifolds with arbitrary codimensions.
Oct 20, 4:00—5:00PM, Teaching Center 102, Xin Fu (Westlake University)
Title: Degeneration of Kahler Einstein metric with negative scalar curvature
Abstract: We survey the geometry of Kahler-Einstein with negative scalar curvature on singular variety. After reviewing some stability and asymptotics result, we discuss a recent joint work with Hein and Jiang on the degeneration of Kahler-Einstein metric on algebraic surface.
Oct 21, 9:00—10:00AM, IMS 507, Siqi He (AMSS, CAS)
Title: Rank one symmetric differentials over projective variety
Abstract: Rank one symmetric differentials, a concept introduced by Taubes, play a significant role in gauge theory and differential geometry. In this talk, we’ll dive into the world of rank one symmetric differentials over projective varieties. We’ll explore how rank one symmetric differentials are connected to Higgs bundles and a recent proposal by Chen-Ngo about the surjectivity of the Hitchin morphism. Furthermore, we will explain how rank one symmetric differentials could play a role in the Simpson integral conjecture. We will discuss a new proof using Finsler metric rigidity to prove characteristic rigidity and integrality for arithematic varieties with rank bigger than one. This talk is based on collaborated work with J.Liu and N.Mok.
Oct 21, 10:30—11:30AM, IMS 507, Jiming Ma (Fudan University)
Title: Schwartz's complex hyperbolic surface
Abstract: We show the representation $\rho_{4, 7}:G(4, 7) \rightarrow \Gamma(4, 7)$ is faithful, and determine the 4-dimensional. Let $G(4, 7)$ be an abstract group with the presentation
$$G(4,7)=\left\langle \iota_1, \iota_2, \iota_3 \Bigg| \begin{matrix} \iota_1^2= \iota_2^2= \iota_3^2=id,\\
(\iota_{1} \iota_{2})^4= ( \iota_{2} \iota_{3})^4= ( \iota_{3} \iota_{1})^4=id, \\
( \iota_{1} \iota_{3} \iota_{2}i_{3})^7= ( \iota_{2} \iota_{1} \iota_{3} \iota_{1})^7= ( \iota_{3} \iota_{2} \iota_{1} \iota_{2})^7=id\end{matrix}\right\rangle
.$$
R. Schwartz considered a representation $\rho_{4, 7}: G (4, 7) \rightarrow \mathbf{PU}(2,1)$, the image group $\Gamma(4, 7)$ is an arithmetic, geometrically finite, discrete subgroup of $\mathbf{PU}(2,1)$. R. Schwartz determined the 3-manifold at infinity of $\mathbf{H}^2_{\mathbb C}/\Gamma(4, 7)$ via a sophisticated method. More precisely, the 3-manifold at infinity is a closed hyperbolic 3-orbifold with underlying space the 3-sphere and whose singularity locus is a two-components link equipped with a $\mathbb{Z}2$-cone structure.
Oct 21, 2:30—3:30PM, IMS 507, Zhiwei Wang (Beijing Normal University)
Title: Recent progress on the positivities in several complex variables and complex geometry
Abstract: In this talk, we will introduce our recent results on the positivities in several complex variables and complex geometry. These results are based on joint works with Yinji Li and Professors Xiangyu Zhou, Fusheng Deng, Jiafu Ning, Liyou Zhang.