Spring 2026 Schedule

Mar. 13 Jacopo Guoyi Chen, SIMIS

Time: 16:30 - 17:30

Place: SCMS 346

Title: Cusp-transitive 4-manifolds with every cusp section

Abstract: We realize every closed flat 3-manifold as a cusp section of a complete, finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps. Moreover, for every such 3-manifold, a dense subset of its flat metrics can be realized as cusp sections of a cusp-transitive 4-manifold. Finally, we prove that there are a lot of 4-manifolds with pairwise isometric cusps, for any given cusp type.
Mar. 20 Zhongzi Wang, Peking University

Time: 16:30 - 17:30

Place: SCMS 102

Title: Finite covers of 3-manifolds which admit no orientation-reversing homeomorphism

Abstract: We proved that if a closed orientable prime 3-manifold M is neither covered by the 3-sphere nor covered by a product manifold, then M has a finite cover that admit no orientation-reversing homeomorphism. The proof uses geometric group theory after Agol-Wise’s virtual specialization. This is a joint work with Prof. Hongbin Sun
Mar. 27 Qing Liu, Nankai University

Time: 16:30 - 17:30

Place: SCMS 102

Title: Quasi-conformal Structures on Morse Boundaries

Abstract: Boundaries of hyperbolic groups have played a central role in geometric group theory. For non-hyperbolic spaces such as CAT(0) spaces, mapping class groups, and hierarchically hyperbolic groups, the Morse boundary provides a natural generalization by restricting to Morse geodesic rays. Inspired by Paulin’s classical rigidity theorem for Gromov hyperbolic groups, we investigate the conditions under which a homeomorphism between the Morse boundaries of two groups is induced by a quasi-isometry of the groups themselves. In this talk, we introduce natural quasi-conformal structures on Morse boundaries and present key rigidity results that extend Paulin’s theorem to the Morse setting.
Apr. 3 Xinlong Dong, City University of New York

Time: 16:30 - 17:30

Place: SCMS 106

Title: Quadratic differentials and random walks on pants graphs with short geodesics

Abstract: Let $X$ be an infinite Riemann surface with an upper-bounded geodesic pants decomposition and a subsequence of cuffs whose lengths converge to zero.

The vertices of the corresponding pants graph $\mathcal{G}$ are pairs of pants, and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on $X$ is ergodic if and only if the random walk on $\mathcal{G}$ is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows.

The above equivalence result uses a characterization of the measured geodesic laminations on $X$ that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on $\mathcal{G}$.

This is a joint work with Charles Bordenave and Dragomir Šarić.
Apr. 10 No Seminar due to conference in Hefei.
Apr. 17 No seminear this week, due to Mini-course by Jianru Duan, Peking university
Apr. 24 Zheng Kuang, South China University of Technology

Time: 16:30 - 17:30

Place: SCMS 102

Title: An introduction to AF-by-discrete groupoids

Abstract: AF-by-discrete groupoids form a broad class that simultaneously generalizes minimal Cantor systems and several important families of self-similar groups, including groups generated by bounded automata and spinal groups. A particularly interesting subclass, the groupoids of bounded type, gives rise to groups with remarkable properties such as amenability, torsion, and intermediate growth. In this talk, I will give a gentle introduction to AF-by-discrete groupoids, explain the notion of bounded type, and discuss some results on their asymptotic behavior, focusing on amenability and intermediate growth.
Apr. 28, Wenyuan Yang, Peking University (joint with SCMS colloquium, Special Time)

Place: Gu Lecture Hall

Time: 16:00 - 17:00

Title: Dimensions of escaping and recurrent geodesics

Abstract: In this talk, we investigate the asymptotic behavior of geodesics—such as escaping and recurrent trajectories—on Riemannian manifolds. Recurrent geodesics are characterized by their endpoints in the visual boundary, which correspond to conical limit points for the fundamental group. The Hausdorff dimension of (uniformly) conical points has been extensively studied, beginning with Patterson’s work (1975) on Fuchsian groups, followed by Sullivan (1979) for geometrically finite Kleinian groups, and later Bishop-Jones (1996) for general Kleinian groups. In this talk, we extend these classical results by computing the Hausdorff dimensions of two other key subsets of the limit set: The Myrberg limit set (a distinguished subclass of non-uniformly conical points) and the non-conical limit set. This is based on joint work with Mahan Mj (TIFR).
May 8 Foling Zou, Chinese Academy of Sciences

Place: SCMS 102

Time: 16:00 - 17:00

Title: Equivariant Steenrod Operations

Abstract: Representation graded Bredon G-equivariant cohomology theories are represented by genuine G-spectra. We introduce the concept of REulerian sequences for an equivariant commutative ring spectra R. Each R-Eulerian sequence corresponds to a stable R-cohomology operation. We apply our theory to equivariant ordinary cohomology to produce genuine equivariant lifts of the classical Steenrod operations for all finite groups. This is joint work with Prasit Bhattacharya, Alex Waugh and Mingcong Zeng.
May 15
May 22 Xin Gu, Westlake University

Place: SCMS 346

Time: 16:00 - 17:00

Title:Topological complexity of enumerative problems in algebraic geometry

Abstract: Typical enumerative problems in algebraic geometry include finding the d roots of a generic polynomial in one variable of degree d, finding the 27 lines on a smooth cubic surface, and their higher dimensional analogs. We introduce the concept of topological complexity of enumerative problems, which is a positive integer that measures the least possible number of “branches” in the algorithms that solves an enumerative problem up to an ϵ error. We are interested in the lower bounds of the topological complexity of enumerative problems. We introduce finite covering spaces associated to the enumerative problems and the concept of Schwarz genus of a covering space, which produces lower bounds of the topological complexity, and can be detected by cohomology. Finally, we present lower bounds of three enumerative problems. This is a joint work with Weiyan Chen.

May 29 Carl-Fredrik Nyberg Brodda， KIAS

Place: SCMS 102

Time: 16:30 - 17:30

Title: Profinite rigidity and the second homology group

Abstract: A finitely generated residually finite group is said to be profinitely rigid if it is determined in the class of finitely generated residually finite groups by its set of finite quotients. Much remains unknown: for example, it is an open problem whether free groups are profinitely rigid. In the past few years, much progress has been made on constructing examples of profinitely rigid groups, with a landmark paper by Bridson, McReynolds, Reid, and Spitler producing examples of profinitely rigid hyperbolic 3-manifold groups. In recent work, Bridson & Reid have used a technique via fiber products due to Bass & Lubotzky for constructing failures of profinite rigidity (in the form of “Grothendieck pairs”) which relies on being able to guarantee the vanishing of the second homology group of some hyperbolic 3-orbifold groups. In my talk, I will give an introduction to this fascinating blend of areas. I will also give a sketch of an algorithmic procedure for computing the second homology group of any hyperbolic 3-orbifold covered by a rational homology 3-sphere, and a useful criterion, both of which can be used to answer two questions of Bridson & Reid.

and a mini-course by Yongquan Zhang, College of William and Mary

Mon, May 25 | 4:00–6:00 PM | SIMIS 1510 Tue, May 26 | 4:00–6:00 PM | SIMIS 1610 Wed, May 27 | 9:30–11:30 AM | SIMIS 1010 Fri, May 29 | 9:30–11:30 AM | SIMIS 1010 Abstract:

An isometrically immersed totally geodesic hyperbolic plane in a hyperbolic 3-manifold is simply called a geodesic plane. The topological, geometric and dynamical behaviors of geodesic planes have been extensively studied. In this series of talks, I will explain some recent progress on these problems.

Independently due to Ratner and Shah, any geodesic plane in a hyperbolic 3-manifold of finite volume is either closed or dense. Recently, McMullen-Mohammadi-Oh generalized these results to convex-compact, acylindrical hyperbolic 3-manifolds of infinite volume if we restrict to the interior of the convex core. Similar generalizations have also been obtained for some geometrically finite acylindrical 3-manifolds by Benoist-Oh. This leads to three natural questions.

Does the “closed or dense” dichotomy still hold if we look at the convex core itself?
Can we classify geodesic planes outside convex cores?
What about other geometrically finite acylindrical 3-manifolds not covered by Benoist-Oh?

For the first two talks, I will give a counterexample to Question 1, describe some partial progress for Question 3 for a very specific example, and focus on an essentially complete classification for Question 2, in joint work with Tina Torkaman.

Closed geodesic planes themselves exhibit interesting collective behavior. Notably, any sequence of distinct closed geodesic planes in a finite-volume hyperbolic 3-manifold equidistributes in the sense of measure theory with respect to the hyperbolic volume measure. At the same time, it is well-known that the collection of all closed geodesics also equidistributes. Motivated by recent progress on equidistribution of intersection points of geodesics on hyperbolic surfaces, it is natural to ask the distribution of intersection points between closed geodesic planes and closed geodesics in a hyperbolic 3-manifold, and more generally, intersection points between closed totally geodesic submanifolds of complementary dimensions in higher dimensions.

For the next two talks, I will briefly discuss several different approaches for this problem in the case of surfaces, and then focus on recent joint work with Tina Torkaman on equidistribution of intersection points in hyperbolic 3-manifolds of finite volume.

Jun. 5， Matteo Migliorini, Karlsruher Institut für Technologie

Time: 16:30 17:30

Zoom id: 646 617 8889 Passcode: 123456wu

Title: The Dehn function of Thompson’s group T

Abstract: Thompson’s groups, introduced by Thompson in 1965, have had a lot of attention in the recent years. Being finitely presented, a natural question is to compute their Dehn function. All three groups are conjectured to have quadratic Dehn function; this conjecture was confirmed for Thompson’s group F by Guba in 2006. During this talk, we show how to deduce from Guba’s result that Thompson’s group T has a quadratic Dehn function as well.
Jun. 12
Jun. 19 Wenzhao Chen， ShanghaiTech
Jun. 22, Shulin Chen, University of Chicago

Fall 2025 Schedule

Sep. 12 There will be no seminar this week, instead we will have a mini-course by Hiroki Matui, Chiba University

Time:

Sep. 9th. Tuesday 15:00 - 17:00, SCMS 102 Sep. 10. Wednesday 15:00 - 17:00 SIMIS 710 Sep. 12th. Friday 15:00 - 17:00 SCMS 102

Title: Introduction to topological full groups

Abstract: In this mini-course, we will introduce various properties and examples of topological full groups, which are discrete groups naturally associated with etale groupoids whose unit spaces are Cantor sets. Over the past two decades, these groups have been extensively studied and are now known to exhibit a rich array of features, such as simplicity, amenability, and finite generation, making them central objects at the intersection of group theory, topology, and operator algebras.

Starting from the definition of topological full groups, we will discuss basic examples arising from minimal Z-actions and AF groupoids. We will also explain the connections to groupoid homology, and the reconstruction theorem which claims the group structure of the topological full group remembers the groupoid itself.

A significant part of the theory revolves around the dichotomy of ample groupoids into two classes: almost finite and purely infinite groupoids. With this distinction in mind, we will discuss structural results such as simplicity of the commutator subgroup, amenability, and finite generation of topological full groups. Depending on time constraints, we may select certain topics to discuss in more detail.
Sep. 19 No talk this week
Sep. 26 Jiawen Zhang, Fudan Univeristy

Time: 16:00 - 17:00

Place: SCMS 102

Title: A weight-free characterisation for yu’s property A

Abstract: The notion of Property A was introduced by G. Yu as a coarseanalogue of amenability, which plays a key role to attack the coarse Baum-Connes conjecture. In his original definition, the family of Property A sets of a metric space X are allowed to taken in X\times\mathbb{N}. We show that the Property A sets can indeed be chosenin X itself for any discrete metric space of bounded geometry. Thekey idea is to use a smearing map, inspired fromuniformlyfinitehomology introduced by Block and Weinberger. This is a joint workwith G. Niblo, N. Wright and J. Zhu.
Oct. 8 No talk this week.
Oct. 17 Robert Tang, Xi’an Jiaotong-Liverpool University

Time: 16:00 - 17:00

Place: SCMS 106

Title: Straightening structures on surfaces

Abstract: Triangulations are ubiquitous throughout low-dimensional topology and geometry. It is a well-known result that any two triangulations of a topological surface are related by a finite sequence of flips. The analogous result is also known to be true for geometric triangulations for surfaces equipped with certain geometric structures, for example, Euclidean cone metrics or hyperbolic surfaces with a fixed vertex set. However, the prevailing methods for studying triangulations and flip sequences, in the geometric setting, tend to be very specific to the type of geometry involved.

In this talk, I will introduce straightening structures on surfaces. This provides an axiomatic framework which models the behaviour of geodesic paths on non-positively curved surfaces. Moreover, it permits a notion of ‘straight triangulation’, generalising the geometric triangulations from the Euclidean or hyperbolic settings. Our main result is that the ‘straight flip graph’ associated to a straightening structure is non-empty, connected, and quasi-isometrically embedded in the flip graph of the underlying topological surface. I will also describe other classes of geometric structures that give rise to straightening structures.

This is joint work with Valentina Disarlo.
Oct. 24 Daxun Wang, Tsinghua University

Time: 16:00 -17:00

Place: SMCS 102

Title: Boundary actions of graph of groups and application to C*-simplicity

Abstract: In this talk, we will discuss the actions of the fundamental group of graph of groups on trees and on its associated boundaries. By studying such group actions, we provide a family of examples of fundamental groups of graph of groups that are C*-simple. This is joint work with Xin Ma and Wenyuan Yang.
Oct. 31 Lizhen Qin, Nanjing University

Time: 16:00 - 17:00

Place: SCMS 102

Title: Fiber self-covering manifolds over tori

Abstract: A manifold is called self-covering if it can be identified with a nontrivial cover of itself, where the identification may be a homeomorphism, a homotopy equivalence, or another type of map. Manifolds of this class include the circle as the simplest case and, more generally, the n-dimensional torus and its products with any manifold. We consider the question: Is every closed self-covering manifold with an abelian fundamental group necessarily a fiber bundle over a torus? Moreover, does the self-covering map of such a manifold arise directly from a self-covering map of the torus? This talk will present our current findings on this question, which is a joint work with Yang Su (苏阳) and Botong Wang (王博潼).
Nov. 3 (special time) Richard Webb, University of Manchester

Time: 16:40 - 17:40

Place: SCMS 102

Title: Equators of the 2-sphere and area-preserving homeomorphisms

Abstract: The curve complex is an important tool in the study of mapping class groups of surfaces, and other areas of low-dimensional topology such as Teichmueller space and hyperbolic 3-manifolds. In this talk, we introduce analogues of the curve complex in order to study the group of area-preserving homeomorphisms (or Hamiltonian diffeomorphisms) of the 2-sphere. How this relates to the dynamics of area-preserving homeomorphisms, and symplectic geometry, is particularly curious, especially because the original curve complex tells us lots of information about the dynamics of mapping classes of surfaces. Using our tools, we are able to construct new quasimorphisms on the group(s) above. I will discuss an application of this regarding the geometry of simple closed curves that separate the sphere into two components of equal area (i.e. equators of the sphere), and how this relates to the Equator Conjecture. Joint work with Yongsheng Jia.
Nov. 14 Jiaochao Wu, Fudan University

Time: 16:00 - 17:00

Place: SCMS 102

Title: Almost elementary groupoids and classifiable C*-algebras

Abstract: This talk is based on collaborations with Xin Ma, where we introduce and study a notion of almost elementary étale groupoids (possibly non-ample), which unifies the notions of almost finiteness and pure infiniteness, and implies strict comparison of the groupoid. Our main motivations lie in the connections to the theory of C-algebras. Most importantly, we show that an almost elementary minimal étale groupoid gives rise to a tracially Z-stable C-algebra. On the other hand, we verify that many classifiable C*-algebras (including all of the strongly self-absorbing ones) have almost elementary étale groupoid models. We also make progress toward the conjecture that the Jiang-Su algebra Z does not arise from a transformation groupoid.
Nov. 21 Guchuan Li, Peking University

Time: 16:00 - 17:00

Place: SCMS 106

Title: Transfer maps of profinite groups and their application in chromatic homotopy theory

Abstract: For a finite cover of spaces from E to B, there is a “wrong way map”, the transfer map, from the cohomology of E to the cohomology of B. This agrees with the transfer map in group cohomology. We will define a similar transfer map for a profinite group with finite cohomology dimension. As an application, after K(1)-localization, the classical J-homomorphism can be interpreted as a profinite transfer map. In joint work in progress with Ningchuan Zhang, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava E-theory for closed subgroups of the Morava stabilizer group. This defines analogs of the J-homomorphism at higher chromatic heights.
Nov. 28 Mark Pengitore， IMPAN

Time: 16:00 - 17:00

Place: SCMS 102

Title: Linearity criteria for automorphism groups of hyperbolic groups

Abstract: In this talk, we can give a general criterion for the automorphism group of nonelementary hyperbolic groups to be linear in terms of certain epimorphisms from the hyperbolic group to finite products of finite simple groups of Lie type. Potential applications include the question of linearity of mapping class groups.
Dec. 5 No talk this week due to workshop on Geometric group theory at East China Normal University.
Dec. 12 No talk this week due to speical week on hyperbolic geometry at SIMIS.
Dec. 19 Weiyan Chen, Tsinghua University

Time: 16:00 - 17:00

Place: SCMS 102

Title: The normal closure of a bounding pair map in its mapping class group

Abstract: The mapping class group of a surface consists of all self-homeomorphisms up to isotopy. It contains a mysterious subgroup, the Torelli group, which consists of mapping classes that act trivially on the homology of the surface. In the 1980s, Dennis Johnson proved a series of groundbreaking theorems about the Torelli group, one of which stated that it is normally generated by a single genus 1 bounding pair map. Extending Johnson’s result, we give two descriptions of the normal subgroup generated by a genus n bounding pair map. We characterize this geometrically defined subgroup using the algebraic invariants of Chillingworth and Casson-Morita. This work is joint with Lei Chen and Justin Lanier.
Dec. 26 Hana Jia Kong, Zhejiang University

Place: SCMS 102

Time: 16:30 - 17:30

Title: Some computations in genuine $C_3$ -equivariant homotopy theory

Abstract: I will report on some computations in genuine $C_3$-equivariant stable homotopy theory, motivated by attempts to understand $C_3$- equivariant analogues of Brown–Peterson theory. I will talk about computational input coming from the $C_3$-equivariant dual Steenrod algebra and related lifting/extension phenomena in $k$- invariant–style constructions. The talk is intended as a progress report on joint work with Angelini-Knoll, Behrens and Johnson, highlighting what is currently understood and what remains unclear.

Spring 2025 Schedule

The regular meeting time for our seminar this semester will be Friday 4:30pm to 5:30pm. The seminar is joint with SIMIS.

Feb. 21, Yue Gao, Anhui Normal University

Time: 16:30 - 17:30

Place: SIMIS 1510

Title: Shape of Thurston’s filling systole subset in surface moduli space

Abstract: In this talk, I am going to talk about the sparseness of Thurston’s subset. Sparseness is a geometric concept on Thurston’s subset first raised by Anderson-Parlier-Pettet in 2016. We have proved the sparseness of Thurston’s subset in the sense of Teichmüller distance and Weil-Petersson distance. More precisely, most surfaces in genus g surface moduli space have Teichmüller distance $\frac{1}{5}\log\log g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}−\sqrt{7\log\log g})$ to the Thurston’s subset.
Feb. 28 Kejia Zhu, UC riverside

Time: 16:30 - 17:30

Place: SCMS 102

Title: CAT(0) Geometry of Complex Curve Complements and Families

Abstract: Motivated by the question of whether braid groups are CAT(0), we investigate the CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If C is the branch locus of a generic projection of a smooth, complete intersection surface to P^2 , we show that the fundamental group of P^2 ∖ C is CAT(0). In the other direction, we prove that the fundamental group of the universal family associated with the singularities of type E6, E7, and E8 is not CAT(0). This is joint work with C. Bregman and A. Libgober.
Mar. 7 Ben Lowe, Unversity of Chicago

Time: 8:30 - 9:30 (Friday Morning unusual time)

Meeting ID: 924 3954 7729 Passcode: 710422

Zoom Link: https://zoom.us/j/92439547729?pwd=Wr62uDCIukxKy5erL9bs3ylvIqmaUT.1

Title: Minimal Surfaces in Negative Curvature

Abstract: It follows from work by Kahn-Markovic that every closed negatively curved 3-manifold contains essential minimal surfaces in great abundance. Since then the goal of better understanding these minimal surfaces has been a focus of activity, both in analogy to the geodesic flow one dimension lower and the more positive-curvature-centric min-max theory of minimal surfaces. This talk will survey recent developments in this area, which brings together techniques from dynamical systems, geometric analysis, and hyperbolic geometry.
Mar. 11 Fernando AI Assal, Wisconsin Madison

Time: 20:00 - 21:00

Zoom Meeting ID: 944 6083 3287 Passcode: 199950

Title: Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold

Abstract: Let M be a closed hyperbolic 3-manifold and let Gr(M) be its 2-plane Grassmann bundle. We will discuss the following result: the weak-* limits of the probability area measures on Gr(M) of pleated or minimal closed connected essential K-quasifuchsian surfaces as K goes to 1 are all convex combinations of the probability area measures of the immersed closed totally geodesic surfaces of M and the probability volume (Haar) measure of Gr(M).
Mar. 21 Shi Wang, Shanghaitech University

Time: 16:30 - 17:30

Place: Simis 1710

Title: Nonpositively curved 4-manifolds with zero Euler characteristic

Abstract: The well-known Chern-Gauss-Bonnet theorem shows that the Euler characteristic of a closed nonpositively curved 4-manifolds is always nonnegative. In this talk, I will present a geometric characterization of the equality case–at each point, it is either Ricci degenerate or locally foliated by totally geodesic flats. I’ll also mention the connection to Gromov’s two conjectures regarding on simplicial volume. This is joint work with Chris Connell and Yuping Ruan.
Mar. 28 Xiaolei Wu

Time: 16:30 - 17:30

Place: Simis 1510

Title: Embedding groups into simple groups

Abstract: Simple groups has played an important role in the studying of groups. The Boone-Higman Conjecture says a finitely generated group can be embedded into a finitely presented simple group if and only if it has solvable word problem. The conjecture has now been proved for many classes of groups, including hyperbolic groups, RAAGs and linear groups over the rationals. In this talk, we will first give an overview of the conjecture. Then we discuss a strategy that can be used to embed groups acting faithfully on locally finite trees to finitely presented simple groups. In particular we verify the conjecture for the Baumslag-Solitar groups and free-by-cyclic groups. The talk is based on a joint work with Kai-Uwe Bux and Claudio Llosa Isenrich.
Apr. 1, Jiawei Zhou, Bimsa

Time: 16:00 - 17:00

Place: Simis 1510

Title: Some results in non-simply connected rational homotopy theory

Abstract: Certain statements that are straightforward in simply connected rational homotopy theory become surprisingly subtle for the non-simply connected case and remain open problems. In this talk I will address two such questions: first, proving that minimal Sullivan model realizations still preserve rational cohomology in the non-simply connected setting; second, establishing an upper bound for the Lusternik-Schnirelmann category of a relative Sullivan algebra in terms of the LS categories of its base algebra and fiber algebra.
Apr. 11, Wei Wang, Shanghai Ocean University

Time: 16:30 - 17:30

Place: SCMS 102

Title: Some Gromoll filtration groups and homotopy groups of diffeomorphism groups

Abstract: In 1964, Gromoll gave a filtration of the group of homotopy n+1-spheres for n>5 and connected this filtration to the pinching of homotopy spheres. On the other hand, this filtration is closely related to homotopy groups of diffeomorphism groups of smooth manifolds.

In this talk, we will first give some background on Gromoll filtration groups and diffeomorphism groups from a topological point of view. Then we will introduce some methods of calculation and give our recent results. We will also discuss some further applications to certain smooth fibre bundles.
Apr. 18 José Pedro Quintanilha, University of Heidelberg

Time: 16:30 - 17:30

Zoom id: 646 617 8889 Passcode: 123456wu

Title: Geometric invariants for locally compact groups

Abstract: The features of a group being finitely generated or finitely presented are, respectively, the n=1 and n=2 cases of the finiteness property F_n. Towards the end of the last century, the question of when these finiteness conditions descend to subgroups of G led to the discovery of the sets Sigma^n(G) (and their homological counterparts Sigma^n(G;A), for A a Z[G]-module). Each set Sigma^n(G) is a collection of homomorphisms G –> R, refining property F_n in the sense that G has type F_n precisely if Sigma^n(G) contains the zero map. In the literature, Sigma-sets are also often called BNSR-invariants, due to Bieri, Neumann, Strebel and Renz, who pioneered the theory.

Another direction in which to generalize finiteness properties is to consider groups G with a locally compact Hausdorff topology. In that setting, Abels and Tiemeyer introduced the compactness properties C_n, which specialize to F_n for G discrete (though this fact is far from obvious at a first glance). In joint work with Kai-Uwe Bux and Elisa Hartmann, we have refined these properties C_n to sets Sigma^n(G), with our definition recovering the classical Sigma sets in the discrete case. We have also generalized various results of classical Sigma-theory to the setting of locally compact groups. In my talk, I will give an introduction to the theory of Sigma sets and explain some of these results.
Apr. 25 No talk this week due to Mini-course by Xin Li from Glasgow.
May 9, Yu Zhang, Tianjin University

Time: 16:00 - 17:00

Place: SCMS 102

Title: Extensive Computations Of the Adams Spectral Sequence E_2-Pages at Odd Primes

Abstract: In this talk, I will present joint work with Weinan Lin that overcomes fundamental computational limitations in computing the E 2-page ofthe Adams spectral sequence at odd primes Leveraging a novel generalization of Grobner basis theory, we develop an eflicient framework that allows us to determine complete additive and multiplicative structures in significantly larger degree ranges than previously possible. Notably, our method achieves these results in practical time frames. circumventing the rapid growth inherent in carlier techniques. These advances not only advance thealgorithmic toolkit for spectral sequences but also lay the groundwork for targeted investigations into stable homotopy groups of spheres and related spectra.
May 16, Ke Feng， University of Electronic Science and Technology of China

Time: 16:30 - 17:30

Place: Simis 1510

Title: Some progress on geometric ideal triangulation

Abstract: Gluing ideal tetrahedra plays a crucial role in the construction of hyperbolic 3-manifolds. It is still not known that whether a hyperbolic 3-manifold admits a geometric ideal triangulation. In this talk, I will show some progress to hyperbolize and further obtain geometric triangulations of 3-manifolds. I will show the rigidity of hyperbolic polyhedral metrics on 3-manifolds, and the connections between 3D-combinatorial Ricci flows and Thurston’s geometric ideal triangulations, which is a joint work with Huabin Ge. Further, I will show a new existence result of the geometric ideal triangulation which is a joint work with Huabin Ge and Yunpeng Meng.
May 23, Jianru Duan, Peking Universtiy

Time: 16:00 - 18:00

Place: SCMS 102

Title: The L^2 Alexander torsion for 3-manifolds and sutured manifold theory

Abstract: The L^2-Alexander torsion is an invariant associated to a 3-manifold and an 1-cohomology class. This invariant is a real function with many properties similar to the classical Alexander polynomial. In this talk, I will first review the basics of L^2-theory of 3-manifolds (e.g. L^2-betti numbers, L^2-torsions), then discuss the “leading coefficient” of the L^2-Alexander torsion and show its connection with Gabai’s sutured manifold theory and the guts theory recently developed by Agol-Zhang.
May 30, Thorben Kastenholz, Karlsruhe Institut of Technology

Time: 16:30 - 17:30

Zoom id: 646 617 8889 Passcode: 123456wu

Title: Non Vanishing of the Fourth Bounded Cohomology of Free Groups and Codimension 2 Subspaces

Abstract: Bounded cohomology is a powerful albeit very hard to compute invariant. Nothing encapsulates that more than the as of yet mysterious bounded cohomology of free groups. During this talk I will give a very brief introduction to bounded cohomology, further motivate why one should care about the bounded cohomology of free groups and then explain how to show that it is non-zero in degrees two, three and four.
Jun. 6, Not talk this week due to mini-course by Bin Sun from Michgan State University

Fall 2024 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5pm. Starting from this semester, the seminar is joint with SIMIS.

Sep. 13 Yang Shen, Fudan University

Time: 16:00 - 17:00

Place: SIMIS 1610

Title: Eigenvalues of random hyperbolic surfaces

Abstract: The theory of random hyperbolic surfaces of the Weil-Petersson model was first introduced by Mirzakhani. It is well-developed in recent years. In this talk, we mainly talk about the behaviors of closed geodesics and eigenvalues of random hyperbolic surfaces. This talk is based on some recent works of the speaker, Yunhui Wu, Yuxin He and Yuhao Xue.
Sep. 19, Zhenghao Rao,Rutgers University.

Time: 21:00 - 22:00 (Special Time)

Zoom meeting id: 283 044 9723 Password: tUd1sA

Title: Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups

Abstract: Kahn and Markovic proved the Surface Subgroup Conjecture for compact hyperbolic 3-manifolds over a decade ago, constructing surface subgroups that can be nearly Fuchsian. However, a compact hyperbolic 3-manifold may also contain surface subgroups that are significantly not totally geodesic. We extend this by showing that given any genus-2 quasi-Fuchsian group Γ and any cocompact Kleinian group G, for any K>1, there exists a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F<Γ. As an application, we demonstrate that the Hausdorff dimensions of the limit sets of surface subgroups of G are dense in the interval [1,2].
Sep. 27. Andrea Seppi, the Institut Fourier of Grenoble.

Time: 16:00 -17:00

Zoom Meeting ID: 944 6083 3287 Passcode: 199950

Title: Uniqueness and non-uniqueness for the Asymptotic Plateau Problem in hyperbolic three-space. Abstract: The Asymptotic Plateau Problem in the hyperbolic space is the problem of existence of minimal surfaces with a prescribed Jordan curve as a boundary “at infinity”. Since the work of Anderson in the 1980s, it is known to have a solution, which is however in general not unique. In this talk, I will give an overview of the subject, present examples of “pathological” non-uniqueness, and describe some criteria for uniqueness.
Oct. 11 Xiaolei Wu, Fudan University.

Time: 16:00 - 17:00

Place: SIMIS 1610

Title: Compactly-supported classes in the homology of big mapping class groups

Abstract:I will start the talk with an overview about the theory of big mapping class groups. Then I will discuss some recent advances on the homology of big mapping class groups. In particular, I want to talk about some of my recent work with Martin Palmer on when the homology of the big mapping class groups could have elements with compact support.
Oct. 18 Jiming Ma, Fudan University.

Time: 16:00 - 17:00

Place: SIMIS 1610

Title: Figure-eight knot is always over there

Abstract:It is well known that the complex hyperbolic triangle group $\Delta(3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mathbf{PU}(2,1)$ has a 1-dimensional moduli space. Deforming the representations from the classical real Fuchsian one to $\Delta(3,3,4; \infty)$, that is, when $I_3I_2I_1I_2$ is accidental parabolic, the 3-manifolds at infinity change, from a Seifert 3-manifold to the figure-eight knot complement.

When $I_3I_2I_1I_2$ is loxodromic, there is an open set $\Omega \subset \partial \mathbf{H}^2_{\mathbb C}= \mathbb S^3$ associated to $I_3I_2I_1I_2$, which is a subset of the discontinuous region. We show the quotient space $\Omega/ \Delta(3,3,4)$ is always the figure-eight knot complement in the deformation process. This gives the topological/geometrical explanation that the 3-manifold at infinity of $\Delta(3,3,4; \infty)$ is the figure-eight knot complement. In particular, this confirms the conjecture of Falbel-Guilloux-Will. This is joint work with Baohua Xie.
Oct. 25 Yanqing Zou, East China Normal University.

Time: 16:00 - 17:00

Place: SIMIS 1610

Title: Some results on distance two Heegaard splittings

Abstract:In 2000, Hempel introduced the Heegaard distance for studying a Heegaard splitting. Since then, many classical results have been obtained. Among them, one result is that distance at least three Heegaard splitting is of a hyperbolic 3-manifold. In this talk, we will share some recent results on distance two Heegaard splittings. This is a joint work with Wenjie Diao, Ruifeng Qiu.
Nov. 6 Zihao Liu, Rice University.

Time: 10:00 - 11:00

Zoom Meeting No.: 910 3139 9342 Passcode: 957825

Title: Scaled homology and topological entropy

Abstract:In this talk, I will introduce a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology as well as its connection to classic singular homology theory. In addition, after briefly introducing topological entropy, I will discuss how to generalize one of the existing results of entropy conjecture, relaxing the smooth manifold restrictions on the compact metric spaces, by using lc-homology groups. This is joint work with Bingzhe Hou and Kiyoshi Igusa.

Nov. 8 Shi Wang, Shanghai Tech University.

Time: 16:00 - 17:00

Place: SIMIS 1610

Title: The natural flow and the critical exponent

Abstract:For a complete Riemannian manifold of nonpositive curvature, we introduce a flow. We give an upper bound on the k-Jacobian of the flow in terms of the critical exponent of the fundamental group. We also give several applications connecting the geometry and topology of the manifold, which includes the linear isoperimetric inequality, the homological vanishing theorem and the non-existence of compact complex subvarieties in certain complex hyperbolic manifolds. This is joint work with Chris Connell and Ben McReynolds.
Nov. 15, Kailin Pan, Peking University.

Time: 13:30 - 15:00

Place: SCMS 102

Title: Ring operads and symmetric bimonoidal categories

Abstract: We generalize the classical operad pair theory to a newmodel for E_\infty ring spaces, which we call ring operad theory, and stablishaconnection with the classical operad pair theory, allowingtheclassical multiplicative infinite loop machine to be appliedtoalgebras over any E_\infty ring operad. As an application, we showthat classifying spaces of symmetric bimonoidal categories are directlyhomeomorphic to certain E_\infty ring spaces in the ring operadsense. Consequently, this provides a simpler construction of the algebraicK-theory as an E_\infty ring spectra.

Nov. 22, Zhenguo Huangfu, ShanghaiTech University.

Time: 16:00 - 17:00

Place: SCMS 102

Title: Relative Bounded Cohomology on Groups with Contracting Elements

Abstract: Let $G$ be a countable group acting properly on a metric spacewith contracting elements and ${H_i:1\le i\le n}$ be a finite collection of Morse subgroups in $G$. We prove that each $H_i$ has infinite index in $G$ if and only if the relative second boundedcohomology $H^{2}b(G, {H_i}{i=1}^n; \mathbb{R})$is infinite-dimensional. In addition, we also prove that for any contracting element $g$, there exists $k>0$ such that $H^{2}_b(G, \llangle g^k\rrangle; \mathbb{R})$ is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups and produce some new results on the (relative) second bounded cohomology of groups. Under the same conditions, we also prove a Gap Theorem stating that any $C$-contracting element $g$ in $G$ either has a power which is conjugate to its inverse, or else the stable commutator length of $g$ is at least equal to some constant $\tau=\tau(C)>0$. This generalizes the Gap Theorem obtained by Calegari-Fujiwara for hyperbolic groups and mapping class groups. Joint work with Renxing Wan.

Nov. 29, Tengren Zhang, Tengren Zhang

Time: 16: - 17:00

Zoom Meeting id: 942 4677 6586 Passcode 451690

Title: Patterson-Sullivan measures for relative Anosov groups

Abstract: Relative Anosov groups are a class of subgroups of a semisimple Lie group G that include all Anosov subgroups, and all geometrically finite subgroups when G has rank one. We prove that under some mild conditions on G, the Poincare series associated to a relative Anosov subgroup (and a linear function on the Cartan subspace of G) diverges at its critical exponent if its critical exponent is finite. As a consequence of this, we deduce uniqueness and ergodicity results for the associated Patterson-Sullivan measure whose dimension is the critical exponent. This is joint work with Richard Canary and Andrew Zimmer.
Dec. 6, Renxing Wan, East China Normal University

Time: 16:00 - 17:00

Place: SCMS 102

Title: This talk presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting.

For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This is joint work with Xiaoyu Xu and Wenyuan Yang.

Dec. 13, Gabriele Viaggi, University of Rome

Time: 16:00 - 17:00

Zoom Meeting id: 283 044 9723 Passcode: tUd1sA

Title: Length bounds for hyperbolic Heegaard splittings

Abstract: In theory, by Mostow rigidity, the geometry of a hyperbolic 3-manifold is a function of its topology, but how to predict explicitly geometric features from a combinatorial presentation of the 3-manifold? By groundbreaking work of Minsky and Brock, Canary, and Minsky such a formula exists for the class of hyperbolic 3-manifolds fibering over the circle. Their model allows to understand explicitly quantities such as volume, length spectrum, and Laplace spectrum directly from topological data. It is desirable to develop a similar picture for Heegaard splittings as it would apply to all 3-manifolds. In this talk, I will present some joint work with Alessandro SIsto and Peter Feller where we develop an explicit purely topological formula for the length of a (short) geodesic in a hyperbolic Heegaard splitting.

Dec. 20, Zheng Kuang, Texas A&M

Time: 10:10 - 11:10

Zoom Meeting ID: 646 617 8889 Passcode: 123456wu

Title: Growth of groups with finitely many incompressible elements

Abstract: Groups of intermediate growth has been an active researchtopic in geometric and asymptotic group theory since 1980 whenR. Grigorchuk constructed the first example of such a group, namely, the Grigorchuk group. In this talk, I will describe a new class of groups of intermediate growth. I will define the class of groups of bounded type from tile inflations, and show that if the set ofincompressible elements of a group in this class is finite, then this group has intermediate growth.
Dec. 20, Ruojing Jiang, MIT

Time: 16:00 - 17:00

Place: SIMIS 1610

Tilte: Minimal Surface Entropy of Hyperbolic Manifolds

Abstract: I will discuss the definition of minimal surface entropy and review the results for both closed hyperbolic 3-manifolds and the finite volume case. On one hand, for metrics with sectional curvature no greater than -1, the entropy achieves its minimum value if and only if it is hyperbolic. On the other hand, among all metrics with scalar curvature bounded from below by -6, the entropy reaches its maximum at the hyperbolic metric.

Dec. 27, Abdul Zalloum, Harbin Institute of Technology

Time: 16:00 - 17:00

Place: SCMS 102

Title: Globally stable cylinders for residually finite hyperbolic groups

Abstract: In 1995, Rips and Sela asked if torsion-free hyperbolic groups admit globally stable cylindres. I will discuss recent work with Petyt-Spriano were we show that all residually finite hyperbolic groups admit such.

Spring 2024 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5pm.

Mar. 8 2024 Claudio Llosa Isenrich, Karlsruhe Institute of Technology

Time: 16:00 - 17:00

Place: SCMS 102

Title: Dehn functions of central products of nilpotent groups

Abstract: The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class c is bounded above by n^{c+1}. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every k>2, we obtain many pairs of finitely presented k-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups. This talk is based on joint works with García-Mejía, Pallier and Tessera.
Mar. 14 2024 (Special Date, joint with SCMS Colloquium) Kai-Uwe Bux, Bielefeld University

Time: 17:00 - 18:00

Place: Gu Lecture Hall SCMS

Title: Telling groups apart by measures of complexity

Abstract: It is hopeless to classify infinite groups up to isomorphism. There are several invariants one can use to chart the vast area inhabited by such groups. I shall discuss several numerical group invariants coming from topology, homology, and geometry:
```
   * finiteness properties
   * (co)homologicial and geometric dimensions
   * isoperimetric inequalities
```
I shall illustrate these concepts by means of examples; and the main source of examples for groups in this talk will be arithmetic groups, e.g., the group of invertible integer n-by-n matrices.
Mar. 22 2024,
Mar. 29 2024
Apr. 5 2024
Apr. 12 2024 Biao Ma, Tongji University

Time: 16:00 - 17:00

Place: SCMS 102

Title: Profinite rigidity of the multivariable Alexander polynomials of links

Abstract: The profinite completion of a group encodes the set of all finite quotients of the group. An interesting question in low dimensional topology is what kind of properties of a 3-manifold can be determined by the profinite completion of its fundamental group. In 2018, Ueki showed that for a knot in $S^3$, the Alexander polynomial of the knot is determined by the profinite completion of its knot group. For a link in $S^3$, one would like to know if the multivariable Alexander polynomial is also determined by its link group. In this talk, I will explain why this question is complicated and report some results on what we know. This talk is based on an ongoing joint work with Jun Ueki.
Apr. 19 2024
Apr. 26 2024
May 10 2024,
May 17 2024 Youlin Li, Shanghai Jiao Tong University

Time: 16:00 - 17:00

Place: SCMS 102

Title: Algebraically overtwisted tight contact 3-manifolds from contact +1 surgeries

Abstract: In this talk, we find many algebraically overtwisted and tight 3-manifolds by contact +1 surgeries. In particular, we show that a contact 1/k surgery on the standard contact 3-sphere along any Legendrian positive torus knot with the maximal Thurston–Bennequin invariant yields an algebraically overtwisted and tight 3-manifold, where k is a positive integer. This is joint work with Zhengyi Zhou.
May 24 2024
May 31 2024 Mohamed-Lamine Messaci, Université Côte d’Azur

Time: 16:00 - 17:00

Zoom id: 646 617 8889 Password:123456wu

Title: Median spaces

Abstract: Median spaces have garnered attention in the field of geometric group theory due to the characterization that they give to the Kazhdan property (T) and to the Haagerup property. They also give a common framework for studying isometric actions on R-trees and CAT(0) cube complexes. In the first part of the talk, we introduce these spaces, give examples and recall some results. In the second part, we will consider the case of locally compact median spaces of finite rank.
Jun. 7 2024

Fall 2023 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5:30pm.

Sep. 15 2023, Jingbang Guo, Fudan University

Time: 16:00 - 17:30

Place: SCMS 106

Title: Topological Hochschild homology and Prismatic Cohomology

Abstract: The theory of prismatic cohomology is a universal p-adic cohomology theory, in the sense that it specializes to the classical p-adic cohomology theories such as the de Rham cohomology and the p-adic etale cohomology. Originally, this universal p-adic cohomology theory was expected by the calculation of topological Hochschild homology for (integral) perfectoid rings. In this seminar, the basic ideas of prismatic cohomology and topological Hochschild homology will be introduced, and between which the delicate interconnection will be emphasized. Especially the general theory of relative topological Hochschild homology and the phenomena of Bokstedt periodicity, which might benefit the calculation of (absolute) topological cyclic homology through the descent technique, are to be discussed.
Sep. 22 2023

Hongyu Wang,Yangzhou University

Time: 15:00 - 16:00

Place: East Main Guahuang Tower 1801

Title: Donaldson question for tamed closed almost complex four-manifolds

Abstract: TBA

Soumen Sarkar, Indian Institute of Technology Madras

Time: 16:00 - 17:00

Place: East Main Guahuang Tower 1801

Title: Cohomology of quasitoric manifolds over a vertex cut of a finite product of simplices

Abstract: In this talk, I’ll classify the characteristic matrices associated to quasitoric manifolds over a vertex cut of a finite product of simplices satisfying a `sign condition’. I’ll discuss the integral cohomology rings of these quasitoric manifolds with possibly minimal generators and show several relations among the products of these generators. Then, I’ll classify integral cohomology rings (up to isomorphism as graded rings) of the quasitoric manifolds over the vertex cut of a finite product of simplices. This is a joint work with Subhankar Sau.
Sep. 29 2023 No talk due to Mid-Autumn Festeival.
Oct. 6 2023, No talk due to National holiday.
Oct. 13 2023 Bingbing Liang, Soochow University

Time: 16:00 - 17:30

Place: SCMS 102

Title: A general version of Kaplansky‘s direct-finiteness conjecture

Abstract: The classical Kaplansky’s direct-finiteness conjecture says that if a, b are two elements of a group ring KG for a filed K and a group G satisfying that ab=1, then ba=1. Kaplansky proves the case that K is a field of characteristic zero and G is any group. Joint with Hanfeng Li, we prove a general case that K is replaced with a left Noetherian unital ring R and G is a sofic group. The tools of the this proof are refined mean length functions defined on the RG-modules in connection with some category language
Oct. 19 2023 Li Cai, Xi’an Jiaotong-Liverpool University

Time: Oct. 19， 15:00 - 16:00

Place: East Main Guahuang Tower 2001

Title: On graph products of spaces, groups and Hopf algebras

Abstract: In this talk we show that, the loop space X of a graph product of spaces is given by (up to weak homotopy equivalence of topological monoids) a graph product of simplicial groups, using Kan’s construction of loop spaces on simplicial sets. As a consequence, we obtain a theorem of Dobrinskaya: when X is connected, its homology (with coefficients from a field) is a graph product H of connected Hopf algebras. Some properties of the commutator of H will also be discussed.

Fangting Zheng, Xi’an Jiaotong-Liverpool University

Time: 16:00 -17:00

Place: East Main Guahuang Tower 2001

Title: Hyperbolic orbifolds of small volume

Abstract: Volume is a natural measure of complexity of a hyperbolic manifold, especially for the one of odd dimension where the Euler characteristic vanishes. In this talk, I will give a sketchy panorama about simple, i.e., small volume, hyperbolic manifolds and orbifolds.
Oct. 27 2023
Nov. 3 2023
Nov. 10 2023 Paul-Henry Leemann, Xi’an Jiaotong-Liverpool University

Time: 16:00 - 17:30

Place: SCMS 102

Title: Cayley graphs with few automorphisms

Abstract: Let G be a group and S a generating set. Then the group G naturally acts on the Cayley graph Cay(G,S) by left multiplications. The group G is said to be rigid if there exists an S such that the only automorphisms of Cay(G,S) are the ones coming from the action of G. Equivalently, a group G is rigid if there exists a graph X with G=Aut(X) acting simply transitively on the vertices of X. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.