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

Jianru DUAN 段剑儒 (Peking University)

Title: Universal L^2-torsion detects fibered 3-manifolds

Abstract: It is well-known that the Alexander polynomial of a fibered knot must be monic. But in general the converse is not true. In this talk, we introduce the universal L^2-torsion of a 3-manifold, an invariant defined in analogy with the classical torsion,  but using tools from L^2-theory. We will show that this invariant detects fibered 3-manifolds. Moreover, we extend the definition of the universal L^2-torsion to taut sutured 3-manifolds and show that it detects product sutured manifolds.

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

Place: Ningzhai (宁斋) 203

Past talks

Jiahuang Chen 陈家煌 (AMSS)

Title: The SL₂-Character Variety and Hitchin Map over Non-Archimedean Fields

Abstract:  Let Σ be a closed Riemann surface of genus at least two. In classical non-abelian Hodge theory, the SL₂(C)-character variety admits a holomorphic map to the space of quadratic differentials on Σ, known as the Hitchin map.

In this talk, I will present an analogous picture over a non-Archimedean valued field F. We define the SL₂(F)-character variety Χ_F and a non-Archimedean Hitchin map Φ:Χ_F→ H⁰(Σ,K_Σ^2). Our main results establish the continuity of Φ with respect to the natural non-Archimedean topology on Χ_F, and we prove that its image is contained in the space of Jenkins-Strebel differentials. This is joint work with Siqi He.

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

Place: Ningzhai (宁斋) 203

Hong CHANG 常洪 (PKU)

A construction of minimal coherent filling pairs

Let $S_g$ denote the genus $g$ closed orientable surface. A coherent filling pair of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its  algebraic intersection number. A minimally intersecting filling pair, $(\alpha,\beta)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this talk, we give a simple geometric procedure for constructing minimally intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus and prove that minimally intersecting coherent filling pairs exist for all $S_{g,p}, g \geq 3, p>0$. We also compare our construction with constructions from Aougab-Menasco-Neiland and Jeffreys. This is a joint work with William Menasco.

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

Place: Ningzhai (宁斋) 203

Xiayu TAN 谈夏羽 (THU)

Pseudo-isotopies in 4-mfds

In this talk I'll first go through the definition of pseudo-isotopy and construct two obstructions discovered by Hatcher and Wagoner, for two diffeomorphisms being isotopic from being just pseudo-isotopic. When dim(X)>5 these two obstructions give a complete answer, but when dim(X)=5 there are only surjections while when dim(X)=4,this is neither surjective or injective. In this paper, Olivier Singh studied the image of both obstructions and proved a “stable surjection” in dim=4 case. Using the results, we can prove that for some 4-mfds X (S^2xS^1xI or (M_1#M_2)xI when M_i=K(\pi, 1)-space), we have infinitely many connected components in Diff(X,\partial X) which are pseudo-isotopic to id but not isotopic to id.

Time: 2025-11-07, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

YuJie LIN 林毓杰 (THU)

The Boundary Dehn Twists on Punctured K3 Surfaces

In 4-manifold topology, Dehn twists along Seifert fibered 3-manifolds provide an important source fof exotic diffeomorphisms. A notable example is the boundary Dehn twist on a punctured K3 surface, which Baraglia-Konno and Kronheimer-Mrowka proved is nontrivial in the smooth mapping class group relative to boundary. I will show that despite being smoothly nontrivial, this diffeomorphism is trivial in the abelianization of the mapping class group. The proof is based on an obstruction for Spin^\mathbb{C} families due to Baraglia-Konno and the global Torelli theorem of K3 surfaces.

Time: 2025-10-31, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

Wujie SHEN 申武杰 (THU)

An Exponential Improvement for Ramsey Lower Bound

For any constant C > 1, let p_C denote the unique solution in (0, 1/2) satisfying C = log p_C /log (1 - p_C). We prove a new lower bound on the Ramsey number r(l, Cl) for sufficiently large l, showing that there exists varepsilon(C) > 0 such that r(l, Cl) ≥ (p_C^(-1/2) + varepsilon(C))^l. This provides the first exponential improvement over the classical lower bound by Erdos (1947). This is a joint work with Jie Ma and Shengjie Xie.

Time: 2025-10-24, Friday 10:00-11:30AM

Place: Ningzhai (宁斋) 203

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

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.