Tuesday Seminar on Topology

[Japanese]   [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo (with live streaming on Zoom)
or
17:00 -- 18:00 Online seminar on Zoom.

Last updated June 25, 2025
Information :
Kazuo Habiro
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

Pre-registration is required.
Once you register, you can attend all our seminars until the end of March, 2024.
The zoom meeting will be opened 15 minutes before start.

Making audio and video recordings is prohibited.

April 8, 17:00-18:30 -- Room 056 with live streaming on Zoom

Asuka Takatsu (The University of Tokyo)

Concavity and Dirichlet heat flow

Abstract: In a convex domain of Euclidean space, the Dirichlet heat flow transmits log-concavity from the initial time to any time. I first introduce a notion of generalized concavity and specify a concavity preserved by the Dirichlet heat flow. Then I show that in a totally convex domain of a Riemannian manifold, if some concavity is preserved by the Dirichlet heat flow, then the sectional curvature must vanish on the domain. The first part is based on joint work with Kazuhiro Ishige and Paolo Salani, and the second part is based on joint work with Kazuhiro Ishige and Haruto Tokunaga.

April 15, 17:00-18:30 -- Room 056 with live streaming on Zoom

Kento Sakai (The University of Tokyo)

Harmonic maps and uniform degeneration of hyperbolic surfaces with boundary

Abstract: If holomorphic quadratic differentials on a punctured Riemann surface have poles of order >1 at the punctures, they correspond to hyperbolic surfaces with geodesic boundary via harmonic maps. This correspondence is known as the harmonic map parametrization of hyperbolic surfaces. In this talk, we use this parametrization to describe the degeneration of hyperbolic surfaces via Gromov-Hausdorff convergence. As an application, we study the limit of a one-parameter family of hyperbolic surfaces in the Thurston boundary of Teichmüller space.

April 22, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar] -- Room 056 with live streaming on Zoom

Takayuki Okuda (Hiroshima University)

Coarse coding theory and discontinuous groups on homogeneous spaces

Abstract: Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map \[ R : M \times M \to \mathcal{I}. \] For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying \[ R(C \times C) \cap \mathcal{A} = \emptyset. \] This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.

May 13, 17:00-18:30 -- Room 056 with live streaming on Zoom

Yuichi Ike (The University of Tokyo)

Interleaving distance for sheaves and its application to symplectic geometry

Abstract: The Interleaving distance was first introduced in the context of the stability of persistent homology and is now used in various fields. It was adapted to sheaves by the pioneering work of Curry, and later in the derived setting by Kashiwara and Schapira. In this talk, I will explain that the interleaving distance for sheaves is related to the energy of Hamiltonian actions on cotangent bundles. Moreover, I will show that the derived interleaving distance is complete, which enables us to treat non-smooth objects in symplectic geometry using sheaf-theoretic methods. This is based on joint work with Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, and Claude Viterbo.

June 3, 17:00-18:30 -- Room 056 with live streaming on Zoom

Tatsuo Suwa (Hokkaido University)

Localized intersection product for maps and applications

Abstract: We define localized intersection product in manifolds using combinatorial topology, which corresponds to the cup product in relative cohomology via the Alexander duality. It is extended to localized intersection product for maps. Combined with the relative Cech-de Rham cohomology, it is effectively used in the residue theory of vector bundles and coherent sheaves. As an application, we have the functoriality of Baum-Bott residues of singular holomorphic foliations under certain conditions, which yields answers to problems and conjectures posed by various authors concerning singular holomorphic foliations and complex Poisson structures. This includes a joint work with M. Correa.

References
[1] M. Correa and T. Suwa, On functoriality of Baum-Bott residues, arXiv:2501.15133.
[2] T. Suwa, Complex Analytic Geometry - From the Localization Viewpoint, World Scientific, 2024.

June 10, 17:00-18:30 -- Room 056 with live streaming on Zoom

Takayuki Morifuji (Keio University)

Bell polynomials and hyperbolic volume of knots

Abstract: In this talk, we introduce two volume formulas for hyperbolic knot complements using Bell polynomials. The first applies to hyperbolic fibered knots and expresses the volume of the complement in terms of the trace of the monodromy matrix. The second provides a formula for the volume of general hyperbolic knot complements based on a weighted adjacency matrix. This talk is based on joint work with Hiroshi Goda.

June 17, 17:00-18:30 -- Room 056 with live streaming on Zoom

Taketo Sano (RIKEN iTHEMS)

A diagrammatic approach to the Rasmussen invariant via tangles and cobordisms

Abstract: Rasmussen's s-invariant is an integer-valued knot invariant derived from Khovanov homology, and it has remarkable applications in topology, such as providing a combinatorial reproof of the Milnor conjecture. Although the s-invariant is defined using the quantum filtration of the homology group, it is difficult to interpret it geometrically. In this talk, we give a cobordism-based interpretation of the s-invariant based on Bar-Natan's reformulation of Khovanov homology via tangles and cobordisms. This interpretation allows for the computation of the s-invariant from a tangle decomposition of the knot. As an application, we demonstrate that the s-invariants of a certain infinite family of pretzel knots can be determined by hand.

Preprint: https://arxiv.org/abs/2503.05414

June 24, 17:00-18:30 -- Room 056 with live streaming on Zoom
June 26, 15:30-17:00 -- Room 122 with live streaming on Zoom

Danny Calegari (The University of Chicago)

Universal circles and Zippers

Abstract: If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle-the circle at infinity of the universal cover of the fiber-preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles-a circle with a faithful action of the fundamental group preserving a pair of invariant laminations-and those universal circles play a key role in relating the dynamical structure to the geometry of M. In these two talks, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers-and their associated universal circles-may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures. This is joint work with Ino Loukidou.

July 1, 17:00-18:00 -- Room 056 with live streaming on Zoom

Genki Sato (Fcuro, Inc.)

Presentation of finite Reedy categories as localizations of finite direct categories

Abstract: In this talk, we present a novel construction that, for a given Reedy category $C$, produces a direct category $\operatorname{Down}(C)$ and a functor $\operatorname{Down}(C) \to C$, exhibiting $C$ as an $(\infty,1)$-categorical localization of $\operatorname{Down}(C)$. This result refines previous constructions in the literature by ensuring that $\operatorname{Down}(C)$ is finite whenever $C$ is finite-a property not guaranteed by existing approaches, such as those by Lurie or by Barwick and Kan. As an intended future application, this finiteness property is expected to be useful for embedding the construction into the syntax of a (non-infinitary) logic. In particular, I expect that the construction may be used to develop a meta-theory of finitely truncated simplicial types and other finite Reedy presheaves for homotopy type theory, thereby extending Kraus and Sattler's unfinished approach. This talk is based on arXiv:2502.05096.

July 8, 17:00-18:30 -- Room 056 with live streaming on Zoom

Hiroki Ishikura (The University of Tokyo)

Stallings-Swan's Theorem for Borel graphs

Abstract: A Borel graph is a simplicial graph on a standard Borel space X such that the edge set is a Borel subset of X^2. Such objects have been studied in the context of countable Borel equivalence relations, and recently there are many attempts to apply the ideas of geometric group theory to them. Stallings-Swan's theorem states that groups of cohomological dimension 1 are free groups. We will talk about an analog of this theorem for Borel graphs: A Borel graph on X with uniformly bounded degrees of cohomological dimension 1 is Lipschitz equivalent to a Borel acyclic graph on X. This is proved by establishing a criterion for certain decomposition of Borel graphs, which is inspired by Dunwoody's work on accessibility of groups.

June 15, 17:00-18:30 -- Room 056 with live streaming on Zoom

Anastasiia Tsvietkova (Rutgers University)

Polynomially many genus g surfaces in a hyperbolic 3-manifold

Abstract: For a low-dimensional manifold, one often tries to understand its intrinsic topology through its submanifolds, in particular of co-dimension 1. For example, it was noticed before that presence of embedded essential surfaces in a 3-manifold can give information about that manifold. However to construct, classify or count such surfaces is a non-trivial task. We will discuss a universal upper bound for the number of non-isotopic genus g surfaces embedded in a hyperbolic 3-manifold, polynomial in hyperbolic volume. The surfaces are all closed essential surfaces, oriented and connected. This is joint work with Marc Lackenby.

June 22, 17:00-18:30 -- Room 056 with live streaming on Zoom

Alexis Marchand (Kyoto University)

Sharp spectral gaps for scl from negative curvature

Abstract: Stable commutator length is a measure of homological complexity of group elements, with connections to many topics in geometric topology, including quasimorphisms, bounded cohomology, and simplicial volume. The goal of this talk is to shed light on some of its relations with negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.