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: 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.