The G2T2 (Geometry, Group Theory, Topology) seminar is a seminar run (predominantly) by postdocs and students, with (predominantly) postdoc and student speakers. The organisers (in alphabetical order) consist of: Yifei CAI 蔡逸飞, Xiao CHEN 陈啸, Diptaishik CHOUDHURY, Qiliang LUO 罗琪亮, Tuo SUN 孙拓, Ivan TELPUKHOVSKIY, and Daxun WANG 王大洵 (*). 

Upcoming talks

Ling ZHOU 周玲 (Duke)

Cohomological Invariants in TDA: Persistent Cup-Length and Cup Modules

In topological data analysis, cohomological ideas have recently been incorporated to enhance computations of persistence diagrams in software like Ripser. The cup product endows cohomology with a graded ring structure that captures richer topological information than the vector space structure alone. The cup-length, defined as the maximum number of cocycles with nontrivial cup product, serves as a useful invariant of the cohomology ring.

In this talk, we lift cup-length to a persistent invariant that tracks the evolution of the cohomology ring structure across a filtration. We provide a polynomial-time algorithm for its computation from representative cocycles and prove its stability under interleaving distances. To further organize the information encoded by cup products, we introduce the persistent cup module, a two-parameter persistence module indexed by filtration and cup-length, and show its stability. In addition, we compare the distinguishing power of these invariants with persistent homology and with each other, highlighting cases where they better distinguish between spaces.

Time: 2025-05-23, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Past talks

Kairui LIU (PKU)

Martin boundary and geometric boundaries of groups

In this talk, we will introduce the Martin boundary of groups - a compactification arising from random walks on groups that completely characterizes all non-negative harmonic functions on groups. We focus on the connections between this probability-related boundary and various geometric boundaries of groups.

We begin by reviewing some results in hyperbolic and relatively hyperbolic groups. Then we introduce Ancona inequalities, which describe the multiplicative behavior of Green functions along geodesics. We establish Ancona inequalities for Morse sets under specific geometric conditions. A key result we present is the construction of an injective mapping from a full-measure subset of the Roller boundary to the Martin boundary for some right-angled Coxeter groups. For general groups with contracting elements, most aspects of this construction remain valid. This is based on a joint work with Wenyuan Yang.

Time: 2025-05-16, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Josiah OH (SCMS, Fudan)

The number of ends of big mapping class groups

Mapping class groups of finite-type surfaces are well-studied objects in geometric group theory. More recently, there is growing interest in mapping class groups of infinite-type surfaces, the so-called “big”mapping class groups. A common theme in geometric group theory is to study the large-scale geometry of finitely generated groups. Big mapping class, though, are far from being finitely generated, and it is not clear a priori how one might try to view them as geometric objects. However, thanks to work of Rosendal and Mann-Rafi, there is now a classification of (many) of the big mapping class groups that have a well-defined quasi-isometry type, allowing the study of their large-scale geometry. In this talk we give a light survey of all of the above concepts, as well as discuss recent work with Yulan Qing and Xiaolei Wu on determining the number of ends of some of these big mapping class groups.

Time: 2025-04-25, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Kanako OIE (Nara Women's University)

On finiteness of the geodesics joining a pair of points in curve complexes

The curve complex C(S) of an orientable surface S is the simplicial complex whose vertices correspond to isotopy classes of simple closed curves on S. The structure of C(S) is difficult to analyze due to its local infiniteness, and many of its properties remain unknown. Examples where the number of geodesics connecting two points in C(S) is infinite have been previously found, and uniqueness results are known from the work of Ido-Jang-Kobayashi. Regarding finiteness, the speaker has shown that for geodesics of length 2, if the number of geodesics is finite, then it is bounded above.

Time: 2025-04-18, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Qing LAN 蓝青 (PKU)

A Generalization of Seifert Geometry Based on the Siegel Upper Half-Space

The \widetilde{SL(2,R)}-geometry is one of Thurston's eight geometries, which fibers over the hyperbolic plane. Generalizing this geometry, we construct a geometry fibering over the Siegel upper half-space, and provide a volume formula for some manifolds with this geometry.

For n=2, a prototype is first constructed via the normal bundle of an equivariant embedding into a Grassmannian manifold. It turns out that this geometry is the homogeneous space given by a central extension of \widetilde{Sp(2n,R)}, modulo its maximal compact subgroup. After fixing a convention for the invariant measure, the volume of a "Seifert-like" closed manifold of this geometry is given by the length of the fiber circle times the Euler characteristic of the base manifold, up to a sign.

Time: 2025-04-11, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Abdul ZALLOUM (Harbin Institute of Technology)

Hierarchical hyperbolicity from a cubical perspective

The theory of hierarchically hyperbolic spaces (HHSs) emerged from the observation that many cocompact CAT(0) cube complexes (CCCs) exhibit a structure analogous to that of mapping class groups and Teichmüller spaces. In particular, the powerful machinery of subsurface projections, which plays a fundamental role in the study of mapping class groups and Teichmüller spaces, extends to a broad class of CCCs. This insight not only led to the definition of HHSs but also raised the natural question of whether the reverse perspective holds: Can techniques from CCCs be leveraged to study mapping class groups, Teichmüller spaces, and HHSs more generally? In this talk, I will define these key objects and discuss recent developments that shed light on this question.

Time: 2025-03-28, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Yu HUANG 黄煜 (PKU)

Profinite rigidity and geometric topology of hyperbolic 3-manifolds

Profinite completion of fundamental groups of 3-manifolds encodes a lot of topological and geometrical information. Further more, it is interesting to ask whether the profinite completion can identify uniquely the homeomorphism type of a 3-manifold. Particularly, the hyperbolic manifold case is a long-term open question.

In fact, the profinitely rigid finite-volumn hyperbolic manifolds form a closed set under the geometric topology. From this observation, we produce plenty of examples of profinitely rigid cusped hyperbolic manifolds using bubble-drilled construction. The big rigid family includes Whitehead link complement and Borromean ring complement. If time permits, we will introduce how to handle the bubble-drilled construction manually.

Time: 2025-03-21, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Jumpei YASUDA (Osaka University / Osaka Metropolitan University)

2-bridge knots and 2-plat 2-knots

A 2-bridge knot is an embedded loop in the 3-sphere S^3 with at most two maximal points. For a rational number a/p, a 2-bridge knot K(a/p) is constructed based on this rational number. The double branched cover of S^3 along K(a/p) is diffeomorphic to the lens space L(a/p). Schubert classified 2-bridge knots by applying the classification of lens spaces.

A 2-knot is a (smoothly) embedded 2-sphere in the 4-sphere S^4. In this talk, we introduce a new class of 2-knots, called 2-plat 2-knots. We provide a normal form F(a/p) for 2-plat 2-knots and compute several invariants. Additionally, we propose questions concerning the classification of 2-plat 2-knots.

Time: 2025-03-14, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

(*): Yi Huang also occasionally helps out a tiny bit.