Geometry and Complex Analysis Seminar
Student No.:20
Time:Tue, 15:20-16:55, Oct.9-Jan.15
Instructor:Tian Yin, Wu Yunhui, Xiao Jian  
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-10-9
Ending Date:2019-1-15


Speaker: Yunhui WU (Tsinghua University)

Title: Growth of the Weil-Petersson inradius of moduli space

Abstract: We study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmuller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$.


Speaker: Weiguo FU (CAS)

Title: Cartan's equivalence method and its application to CR geometry

Abstract: In this talk I will give an introduction to Cartan's equivalence method by applying it to an old problem of classifying surfaces up to orientation preserving local isometry. Then I will talk about the 1937 PhD thesis of Mohsen Hachtroudi, a student of E. Cartan, where he applied this method to a system of 2nd order PDE to obtain invariants just as we have seen in the case of classification of surfaces. These invariants later reappear as ``S-tensors" in the famous paper of Chern-Moser on the CR real hypersurfaces in complex manifolds. I will show the explicit expression of such tensors and discuss how they can be applied to CR real hypersurfaces defined by implicit functions.


Speaker: Jingcao WU (Fudan University)

Title: Symmetrization of Plurisubharmonic functions on the Fano manifolds

Abstract: Given a compact complex manifold Y with a negative line bundle L, we study the Schwarz-type symmetrization on the total space of L. We prove that this symmetrization does not increase the Monge-Amp ere energy for the fiberwise S^1-invariant plurisubharmonic functions in the "unit ball" under some assumptions. As an application we generalize the sharp Moser-Trudinger inequality on the unit ball.


Speaker: Yashan Zhang (Peking University)

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.


Speaker: Jinsong Liu (Chinese Academy of Sciences)

Title: A global estimate of discrete Riemann mappings

Abstract: In order to study low dimensional topology, W. Thurston introduced Circle Packing. And he conjectured a connection between Circle packing and the Riemann mappings. Rodin-Sullivan proved Thurston's conjecture that his scheme converges to the Riemann mapping f. If the domain $\Omega$ is a bounded quasidisk, we will give an estimate of the rate of globally uniform convergence of the approximating maps f_n to f. This is a joint work with Shiyi Lan.


Speaker: Athanase Papadopoulos (University of Strasbourg)

Title: Actions of mapping class groups on spaces of laminations and foliations

Abstract: I will explain several rigidity results of actions of mapping class groups of surfaces on spaces of foliations and laminations. The most recent result is joint work with K. Ohshika.


Speaker: Binbin Xu (University of Luxembourg)

Title: Domains of discontinuity of the outer-automorphism group action on the character variety

Abstract: The outer-automorphism group Out(F2) of a rank 2 free group F_2 acts naturelly on the PSL(2,C)-character variety of F_2. In order to study the dynamical property of this action, the Q-condition and the primitive stability were introduced by Bowditch (generalized by Tan-Wong-Zhang) and Minsky respectively. In particular, these two conditions induce two domains of discontinuity for the Out(F_2) action, which are both strictly bigger than the one induced by the convex cocompact representations. In the collaboration with Jaejeong Lee, by showing the equivalence between the two conditions, we prove that these two domains of discontinuity coincide.


Speaker: Wenyuan Yang (Peking University)

Title: Counting conjugacy classes in groups with contracting elements

Abstract: In this talk, we shall derive an asymptotic formula for the number of conjugacy classes of elements for a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(n)$ (resp. $\mathcal C'(n)$) the set of (resp. primitive) conjugacy classes of stable length at most $n$. The main result is an asymptotic formula as follows: $$\sharp \mathcal C(n) \asymp \sharp \mathcal C'(n) \asymp \frac{\exp(\omega(G))}{n}.$$ As a consequence of the formulae, the conjugacy growth series is transcendental for all non-elementary relatively hyperbolic groups, graphical small cancellation groups with finite components. As by-product of the proof, we establish several useful properties for an exponentially generic set of elements. In particular, it yields a positive answer to a question of Maher that an exponentially generic elements in mapping class groups have their Teichm\"{u}ller axis contained in the principal stratum. This is a joint work with Ilya Gekhtman (U. Toronto).


Speaker: Huabin Ge (Beijing Jiaotong University)

Title: On the realization and rigidity of hyperbolic polyhedra

Abstract: Using combinatorial Ricci flow methods, I will show a new proof of Rivin's characterizations for ideal convex hyperbolic polyhedra. I will also give some generalizationsof Rivin's results to polyhedra with infinite faces. The talks are based on some joint work with Jinsong Liu and Wenfeng Jiang, with Bobo Hua and Ze Zhou, and with Song Dai and Shiguang Ma.


Speaker: Jie LIU (Morningside Center of Mathematics, CAS)

Title: Anticanonical geometry of Fano manifolds with coindex four

Abstract: Fano manifolds are fundamental objects studied in complex geometry and algebraic geometry, and it is well-known that all Fano manifolds of given dimension form a bounded family. Thus it is natural to ask if it is possible to classify them. This was already done for Fano manifolds of coindex at most three. However, up to now the classification of Fano manifolds with coindex four is very far from known even in dimension four. In this talk, I will present our recent results on the geometry of the pluri-anticanonical systems of Fano manifolds with coindex four. In particular, I will explain its relation with polarized (singular) Calabi-Yau threefolds. If time is permitted, open questions and difficulties will also be discussed.


Speaker: Jian XIAO (Tsinghua University)

Title: Group discussions -- some problems in complex algebraic geometry


Speaker: Baohua FU (Chinese Academy of Sciences)

Title: On Fano complete intersections in rational homogeneous varieties

Abstract: Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if X=\cap_{i=1}^r D_i \subset G/P a general complete intersection of r ample divisors such that K^*_{G/P} \otimes O_{G/P}(-\sum_i D_i) is ample, then X is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.