Fall 2022 Schedule
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Sep. 15 2022, Fangzhou Jin, Tongji University
Time: 13:30 - 15:30 Place: Room 2201, East Main Guahuang Tower Title: Milnor-Witt cycle modules and perverse homotopy heart Abstract: We define Milnor-Witt cycle modules over a base scheme and study the relations with the perverse homotopy t-structure. This is a joint work with F. Déglise and N. Feld. -
Sep. 22 2022， Andrea Bianchi, University of Copenhagen
Time: 13:30 - 15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title:Trivial and non-trivial actions of the Johnson filtration on the homology of configuration spaces Abstract:Let S=S_{g,1} be a compact, connected, orientable surface of genus g with one boundary curve, and let F_n(S) denote the space of ordered configurations of n distinct points in S. The homology groups H_*(F_n(S)) admit a natural action of the mapping class group Mod(S)=pi_0(Diff_+(S,dS)), and we are broadly interested in what kind of representations of Mod(S) arise in this way; in particular, how trivial/non-trivial the action of Mod(S) is. We consider the Johnson filtration on Mod(S) by subgroups J(0)>J(1)>...>J(i)>..., for i>=0. We will compare the following results: 1) (joint with J.Miller and J.Wilson) J(i) acts trivially on H_*(F_n(S)) for i>=n; 2) (joint with A.Stavrou) If g>=2, J(n-1) acts non-trivially on H_n(F_n(S)). 3) I will discuss the main ideas of the proofs, and I will conclude with a conjecture. -
Sep. 29 2022, Heng Xie, Sun-Yat sen University
Time: 13:30 - 15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title: The real cycle class map and its applications Abstract:What is the singular cohomology in algebraic geometry? Classically, for smooth varieties X over complex numbers, there are cycle class maps from Chow groups to singular cohomology. For smooth varieties X over real numbers, I will construct a real cycle class map from I-cohomology to singular cohomology and show that the real cycle class is compatible with pullbacks, pushforwards, and intersection products. These results are proved in a joint work with J. Hornbostel, M. Wendt, and M. Zibrowius. Together with F. Jin, we generalize the real cycle class map construction from smooth varieties to singular varieties. I will discuss some applications of the real cycle class map. -
Oct. 6 2022, NO TALK due to National holiday.
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Oct. 13 2022, Gabriel Pallier, Sorbonne Université
Time: 14:30-15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Lie groups with a small space of metric structures Abstract: In this talk we will consider a family of solvable, non-nilpotent Lie groups, including the three-dimensional group SOL. On such a group, any pair of left-invariant Riemannian metrics are found to be roughly similar: after multiplying one of them by a suitable multiplicative constant, they will differ by at most a bounded amount. This allows one to reformulate various earlier results about the quasiisometries of these groups in a common framework. I will compare this result with a recent theorem of Oregon-Reyes, giving an opposite conclusion when considering non-elementary word-hyperbolic groups: the latter are found to have large spaces of metric structures. Joint work with Enrico Le Donne and Xiangdong Xie. -
Oct. 20 2022, Indira Chatterji, Universit´e de Nice
Time: 15:00 - 17:00 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Median geometry and Hyperbolicity Abstract: I will be discussing triangles in metric spaces, and how informations on triangles can give informations on algebraic properties of a cocompact group of isometries of the metric space. This talk will be accessible to non-spacialists. -
Oct. 27 2022, Ilya Gekhtman, Technion – Israel Institute of Technology
Time: 15:00 - 17:00 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Martin, Floyd and Bowditch boundaries of relatively hyperbolic groups Abstract: Consider a transient random walk on a countable group $G$. The Green distance between two points in the group is defined to be minus the boundary of the probability that a random path starting at the first point ever reaches the second. The Martin compactification of the random walk is a topological space defined to be the horofunction boundary of the Green distance. It is a topological model for the Poisson boundary. The Martin boundary typically heavily depends on the random walk; it is thus exciting when for some large class of random walks, the Martin boundary is equivariantly homeomorphic to some well known geometric boundary of the group. Ancona showed in 1988 that this is the case for finitely supported random walks on hyperbolic groups: the Martin boundary is identified with the Gromov boundary. We generalize Ancona's results to relatively hyperbolic groups: the Martin boundary equivariantly continuously surjects onto the Gromov boundary of any hyperbolic space on which the group acts geometrically finitely (called the Bowditch boundary), and the preimage of any conical limit point is a singleton. When the parabolic subgroups are virtually abelian (e.g. for Kleinian groups) we show that the preimage of a parabolic fixed point is a sphere of appropriate dimension, so the Martin boundary can be identified with a Sierpinski carpet. A major technical tool is a generalization of a deviation inequality due to Ancona saying the Green distance is nearly additive along word geodesics, which has various other applications, including to comparing harmonic and Patterson-Sullivan measures for negatively curved manifolds and to local limit theorems for random walks. We do all this using an intermediate construction called the Floyd metric obtaining by suitably rescaling the Cayley graph and considering the associated completion called the Floyd compactification. We show that for any finitely supported random walk on a finitely generated group, the Martin boundary surjects to the Floyd boundary, which in turn by work of Gerasimov covers the Bowditch boundary of relatively hyperbolic groups. This is based on several joint works with subsets of Dussaule, Gerasimov, Potyagailo, and Yang. -
Nov. 3 2022, Marco Linton, University of Warwick
Time: 16:00 - 18:00 (Very Special Time) Zoom Id: 853 0188 1524 password: Fudan2022 Time: Hyperbolic one-relator groups Abstract: Since their introduction by Gromov in the 80s, a wealth of tools have been developed to study hyperbolic groups. Thus, when studying a class of groups, a characterisation of those that are hyperbolic can be very useful. In this talk we will turn to the class of one-relator groups. In previous work, we showed that a one-relator group not containing any Baumslag--Solitar subgroups is hyperbolic, provided it has a Magnus hierarchy in which no one-relator group with a so called `exceptional intersection' appears. I will define one-relator groups with exceptional intersection, discuss the aforementioned result and will then provide a characterisation of the hyperbolic one-relator groups with exceptional intersection. Finally, I will then discuss how this characterisation can be used to establish properties for all one-relator groups. -
Nov. 10 2022, Thomas Ng, Technion – Israel Institute of Technology
Time: 15:00 - 17:00 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Residually finite quotients via relative cubulation Abstract: Small cancellation theory is a rich source of cocompactly cubulated groups. The classical C’(1/6) condition has a natural generalization to quotients of free products. While free products exhibit much more exotic behavior than free groups, these quotients act on a Gromov hyperbolic polygonal complexes and have been used to solve embedding problems in groups. When the factor groups are assumed to act geometrically on a CAT(0) cube complex, Martin and Steenbock show that such C’(1/6) quotients are again geometrically cubulated. I will describe joint work with Eduard Einstein proving that when the free factors are residually finite every C’(1/6) quotient is again residually finite. Our proof relies on showing that the quotient groups admit relatively cubulations, a kind of improper action on cube complexes. -
Nov. 17 2022, Merlin Incerti-Medici, Karlsruhe Institute for Technology
Time: 15:00 -17:00 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Hyperbolic projections and topological invariance of Morse boundaries Abstract:When studying hyperbolic spaces, they admit a boundary at infinity that is invariant under quasi-isometries. This allows us to associate this boundary as a topological invariant to hyperbolic groups. This procedure fails quickly when moving to other spaces; already for CAT(0) spaces there are quasi-isometric spaces that have non-homeomorphic boundary at infinity. In recent years, several notions of boundaries have been introduced to remedy this. One such boundary is the Morse boundary, which, for CAT(0) spaces, can be seen as subsets of the visual boundary. However, it turns out that the topology that makes the Morse boundary quasi-isometry-invariant does not coincide with the topology of the visual boundary and the visual topology is not quasi-isometry-invariant even when restricted to Morse boundaries. In this talk, we will show that nevertheless, the visual topology on Morse boundaries has some decent invariance properties for a large class of examples (most cubulated groups). We will do so by introducing a new method to define a topology on Morse boundaries, show that this topology has good invariance properties and that it is a naturally occurring topology for many examples. -
Nov. 24 2022, Reid Monroe Harris, The Chinese University of Hong Kong,Shenzhen
Time: 13:30 - 15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title: Monodromy of Smooth Quartic Plane Curves Abstract: We consider the parameter space ${\mathcal U}_d$ of smooth plane curves of degree $d$. The universal smooth plane curve of degree $d$ is a fiber bundle $\mathcal{E}_d\to\mathcal{U}_d$ with fiber diffeomorphic to a surface $\Sigma_g$. This bundle gives rise to a monodromy homomorphism $\rho_d:\pi_1(\mathcal{U}_d)\to\mathrm{Mod}(\Sigma_g)$, where $\mathrm{Mod}(\Sigma_g):=\pi_0(\mathrm{Diff}^+(\Sigma_g))$ is the mapping class group of $\Sigma_g$. The main result of this paper is that the kernel of $\rho_4:\pi_1(\mathcal{U}_4)\to\mathrm{Mod}(\Sigma_3)$ is isomorphic to $F_\infty\times\mathbb{Z}/3\mathbb{Z}$, where $F_\infty$ is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement $\mathcal{T}_g\setminus\mathcal{H}_g$ of the hyperelliptic locus $\mathcal{H}_g$ in Teichm\"uller space $\mathcal{T}_g$ has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil-Petersson geometry of Teichm\"uller space together with results from algebraic geometry. -
Dec. 1 2022. Julien Paupert, Arizona State University
Time: 10：00-11：00 (Special Time) Zoom Id: 853 0188 1524 password: Fudan2022 Title: Presentations for cusped arithmetic hyperbolic lattices Abstract: We present a general method to compute a presentation for any cusped hyperbolic lattice $\Gamma$, applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternionic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. This is joint work with Alice Mark. -
Dec. 8 2022, Anythony genevois, University of Montpellier （CANCELLED）
Time: TBA Zoom Id: 853 0188 1524 password: Fudan2022 -
Dec. 15 2022 Giles Gardam, University of Münster
Time: 13:30 - 15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title: The Kaplansky conjectures Abstract: There is a series of four fundamental and long-standing conjectures on group rings attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group with field coefficients has no zero divisors. I will discuss these conjectures, their connections to other open questions in various areas of mathematics, and my recent disproof of the unit conjecture. -
Dec. 22 2022 Fei Han, National University of Singapore
Time: 13:30 - 15:30 Zoom Id: 853 0188 1524 password: Fudan2022 Title: title: Characteristic numbers and index theoretic invariants for 24 dimensional string manifolds Abstract: A manifolds M is called string manifold i its free loop space LM is spin. There are many studies on the string geometry. Dimension 24 is in particular interesting for string geometry. In the talk, I will report our work on the study of characteristic numbers and index theoretic invariants for 24 dimensional string manifolds and string cobordism following Mahowald-Hopkins. This represents our joint work with Ruizhi Huang
Spring 2022 Schedule
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Mar. 03 2022, Wenyuan Yang, Peking University
Time: 15:00 - 17:00 Tencent room：861-8661-7196 password: 123555 Title: Proper actions of 3-manifold groups on finite product of quasi-trees Abstract: Let M be a compact, connected, orientable 3-manifold. In this talk, I will study when the fundamental group of M acts properly on a finite product of quasi-trees. Our main result is that this is so exactly when M does not contain Sol and Nil geometries. In addition, if there is no $\widetilde{SL(2, \mathbb{R})}$ geometry either, then the orbital map is a quasi-isometric embedding of $\pi_1(M)$. This is called property (QT) by Bestvina-Bromberg-Fujiwara, who established it for residually finite hyperbolic groups and mapping class groups. The main step of our proof is to show property (QT) for the classes of Croke-Kleiner admissible groups and of relatively hyperbolic groups under natural assumptions. Accordingly, this yields that graph 3-manifold and mixed 3-manifold groups have property (QT). This represents joint work with N.T. Nguyen and S.Z. Han. -
Mar. 10 2022, Martin Palmer, Mathematical Institute of the Romanian Academy
Time: 15:00-17:00 Zoom Id: 646 617 8889 password: 123456wu Title: Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann Abstract: One of the first interesting representations of the braid groups is the Burau representation. It is the first of the family of Lawrence representations, defined topologically by viewing the braid group as the mapping class group of a punctured disc. Famously, the Burau representation is almost never faithful, but the k = 2 Lawrence representation is always faithful: this is a celebrated theorem of Bigelow and Krammer and it implies that braid groups are linear (embed into general linear groups over a field). Motivated by this, and by the open question of whether mapping class groups are linear, I will describe recent joint work with Christian Blanchet and Awais Shaukat in which we construct analogues of the Lawrence representations for mapping class groups of compact, orientable surfaces. Tools include twisted Borel-Moore homology of configuration spaces, Schrödinger representations of discrete Heisenberg groups and the Stone-von Neumann theorem. -
Mar. 17 2022, Xianchang Meng, Shandong University
Time: 15:00 - 17:00 Tencent room：861-8661-7196 password: 123555 Title: Distinct distances on hyperbolic surfaces Abstract: Erdős (1946) proposed the question of finding the minimal number of distinct distances among any N points in the plane. Guth-Katz (2015) gave almost sharp answer for this question using incidence geometry and polynomial partitioning. We consider this problem in hyperbolic surfaces associated with cofinite Fuchsian groups, i.e. the volume of the surface is finite. We prove a lower bound of the same strength as Guth-Katz. In particular, for any finite index subgroup of the modular group, we extract out the dependence of the implied constant on the index. -
Mar. 24 2022, Edgar A. Bering IV, Technion – Israel Institute of Technology
Time: 15:00 - 17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: An exhaustion of the sphere complex by finite rigid sets Abstract: A subcomplex X < C of a simplicial complex is rigid if every locally injective, simplicial map X \to C arises as the restriction of an automorphism of $C$. Curve complexes and other surface complexes have been found to exhibit remarkable rigidity properties. Aramayona and Leininger proved that the curve complex of an orientable surface can be written as an increasing union of finite rigid sets. The sphere complex of a connect sum of $n$ copies of S^1 \times S^2 is an analog the curve complex of a surface used in the study of Out(F_n). In this talk I will present joint work with C. Leininger where we prove that there is an exhaustion of the sphere complex by finite rigid sets when n >= 3. -
Mar. 31 2022，Bruno Martelli, University of Pisa
Time: 15:00 - 17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Fibrations on higher-dimensional hyperbolic manifolds Abstract: One of the most intriguing aspects in low-dimensional topology is the existence, discovered by Jorgensen in the late 70s, of hyperbolic 3-manifolds that fiber over the circle. In this talk we will briefly review some aspects of this beautiful theory, with the notable contributions of Thurston, and more recently of Agol and Wise. Then we will show that this phenomenon is not restricted to dimension 3, by exhibiting some examples in dimension 4 and 5 (in even dimension, some critical points are necessary, and we talk about perfect circle valued Morse functions instead of fibrations). As a consequence, we will deduce that a finite type subgroup of a hyperbolic group needs not to be hyperbolic, thus answering a well-known open question in geometric group theory (joint works with Battista, Italiano, and Migliorini) -
Apr. 07 2022, John R. Parker, Durham University
Time: 15:00 - 17:00
Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic
Title: Complex hyperbolic lattices
Abstract: I will give a survey on recent results about complex hyperbolic lattices. A lattice in a Lie group is a discrete subgroup whose quotient has finite Haar measure. In this talk I will concentrate on the Lie group SU(n,1), often restricting to the case where n=2. Elements of this group act as holomorphic isometries of complex hyperbolic space. Roughly speaking, there are four methods of constructing complex hyperbolic lattices. I will discuss how these are related, with particular emphasis on a family of non-arithmetic lattices.
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Apr. 14 2022, Robert Tang, Xi’an Jiaotong-Liverpool University
Time: 15:00 - 17:00
Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic
Title: Large-scale geometry of the saddle connection graph
Abstract: For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.
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Apr. 21 2022，Weiyan Chen, Tsinghua University
Time: 15:00 - 17:00
Tencent room：861-8661-7196 password: 123555
Title: Choosing points on cubic plane curves
Abstract: It is a classical topic to study structures of certain special points on smooth complex cubic plane curves, for example, the 9 flex points and the 27 sextactic points. We consider the following topological question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given integer n? This work is joint with Ishan Banerjee.
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Apr. 28 2022, Fangzhou Jin, Tongji University (Postponed)
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May 05 2022, Brita Nucinkis, Royal Holloway, University of London
Time: 15:00 - 17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Cohomological properties of Mackey functors for totally disconnected locally compact groups Abstract: Mackey functors for finite groups are well understood. In the early 2000s this was extended to infinite discrete groups, and their cohomological finiteness conditions have been expressed in terms of relative cohomology and finiteness conditions of classifying spaces for proper actions. In this talk I will indicate how one can extend the definition to totally disconnected groups, and will indicate some of the obstacles encountered here. This is ongoing work with Ilaria Castellano and Nadia Mazza. -
May 12 2022 Inheok Choi, KAIST.
Time: 15:00 - 17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Typical behavior of random mapping classes and outer automorphisms Abstract: Random walks on spaces that manifest hyperbolic properties have been studied for decades, as a means to investigate the structure of the isometry groups of such spaces. Notable results in this topic include descriptions of generic random isometries and limit laws for displacements and translation lengths. A recent technique called the pivoting technique led to further elaborations in these directions. In this talk, I will describe properties of generic random isometries of Teichmuller space or Outer space. Namely, generic mapping classes are principal pseudo-Anosovs that make displacements with many contracting subsegments. This frequently contracting property is shared with generic outer automorphisms, which are ageometric fully irreducibles and whose expansion factors differ from their inverses'. If time permits, I will explain how this description is related to (1) Qing-Rafi's sublinearly Mosre boundaries and (2) limit laws for random 3-manifolds. -
May 19 2022 Hao Liang, Sun Yat-sen University
Time: 15:00-17:00 Tencent room：861-8661-7196 password: 123555 Title: Homomorphisms to 3-manifold groups Abstract: Following Sela's theory of limit groups (of free group), we define and study limit groups of compact 3-manifold groups. We show that the family of compact 3-manifold groups are equationally noetherian, which answers a question of Agol and Liu. The main application of our result is to give a positive answer to a question of Reid, Wang and Zhou about epimorphism sequence of closed orientable aspherical 3-manifold groups. This is joint work with Daniel Groves and Michael Hull. -
May 26 2022, NO Talk due to University Anniversary
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Jun. 02 2022, Yi Jiang, Tsinghua University
Time: 15:00-17:00 Tencent room：861-8661-7196 password: 123555 Title: Involution on pseudoisotopy spaces Abstract: The involution on pseudoisotopy spaces is closely related to the homotopy type of the diffeomorphism group of a smooth compact manifold. In this talk, we will introduce some background, a result on computing the involution on pseudoisotopy spaces and its application to space of nonnegatively curved metrics on open manifolds. This is joint work with Mauricio Bustamante and Francis Thomas Farrell. -
Jun. 09 2022, Bin Sun, Oxford University
Time: 15:00 - 17:00
Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic
Title: Generalized wreath products and rigidity of their von Neumann algebras
Abstract: We construct the first positive examples to the Connes’ Rigidity Conjecture, i.e., we construct groups G with Kazhdan’s property (T) such that if H is a group with the same von Neumann algebra as G, then H is isomorphic to G. In this talk, I will focus on the group theoretic side of this result and talk about how we applied geometric group theory to solve problems from von Neumann algebra.
Fall 2021 Schedule
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Dec. 30 2021, Yang Su Chinese Academy of Sciences
Time: 15:00-17:00 Place: Tencent Meeting ID: 573-308-837 Password: 123455 Title: Geometric structure of self covering manifolds Abstract: A topological space is self-covering if the space is a non-trivial cover of itself. I will talk about a joint work with Lizhen Qin and Botong Wang on the geometric structure of self-covering manifolds. For example, we show that in dimension >5, a self-covering manifold with infinite cyclic fundamental group is a fiber bundle over the circle. We also construct examples of self-covering manifolds which are not fiber bundle over the circle, where the fundamental group contains torsion. -
Dec. 16 2021, Shengkui Ye NYU Shanghai
Time: 15:00-17:00 Place: Tencent Meeting ID: 533-917-811 Password: 123455 Title: The group of quasi-isometries of the line cannot act effectively on the line Abstract: Let G be the group of orientation-preserving quasi-isometries of the real line. We show that G is left-orderable, but not simple. Moreover, the group G cannot act effectively on the real line R.