## トポロジー火曜セミナー

開催情報 火曜日　17:00～18:30　数理科学研究科棟(駒場) 056号室 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html Tea: 16:30 - 17:00 コモンルーム

### 2020年12月08日(火)

17:30-18:30   オンライン開催

The intersection polynomials of a virtual knot (JAPANESE)
[ 講演概要 ]
We define two kinds of invariants of a virtual knot called the first and second intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. We study several properties of the polynomials. By introducing invariants of long virtual knots, we give connected sum formulae of the intersection polynomials, and prove that there are infinitely many connected sums of any two virtual knots as an application. Furthermore, by studying the behavior under a crossing change, we show that the intersection polynomials are finite type invariants of order two, and find an invariant of a flat virtual knot derived from the the intersection polynomials. This is a joint work with R. Higa, T. Nakamura, and Y. Nakanishi.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年12月01日(火)

17:00-18:00   オンライン開催

Goeritz groups of bridge decompositions (JAPANESE)
[ 講演概要 ]
For a bridge decomposition of a link in the 3-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere that preserve each of the bridge sphere and link setwise. The Birman-Hilden theory tells us that this is a $\mathbb{Z} / 2 \mathbb{Z}$-quotient of a "hyperelliptic Goeritz group". In this talk, we discuss properties, mainly of dynamical nature, of this group using a measure of complexity called the distance of the decomposition. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings. This talk is based on a joint work with Susumu Hirose, Daiki Iguchi and Eiko Kin.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年11月24日(火)

17:30-18:30   オンライン開催

Intersection of Poincare holonomy varieties and Bers' simultaneous uniformization theorem (JAPANESE)
[ 講演概要 ]
Given a marked compact Riemann surface X, the vector space of holomorphic quadratic differentials on X is identified with the space of CP1-structures on X. Then, by the holonomy representations of CP1-structures, this vector space properly embeds into the PSL(2, C)-character variety, the space of representations of the fundamental group of X into PSL(2,C).

In this manner, different Riemann surfaces structures yield different half-dimensional smooth analytic subvarieties in the character variety. In this talk, we discuss some properties of their intersection. To do so, we utilize a cut-and-paste operation, called grafting, of CP1-structures.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年11月17日(火)

17:00-18:00   オンライン開催

Lefschetz fibration on the Milnor fibers of simple elliptic and cusp singularities (JAPANESE)
[ 講演概要 ]
In this talk a joint work with Naohiko Kasuya(Kyoto Sangyo U.), Hiroki Kodama(Tohoku U.), and Atsuhide Mori(Osaka Dental U.) is reported. The main result is the following.

There exist a Lefschetz fibration of the Milnor fiber of T_{pqr}-singularity (1/p + 1/q + 1/r ≦ 1) to the unit disk with regular fiber diffeomorphic to T^2.

An outline of the construction will be explained, through which, the space of 2-jets of (R^4, 0) to (R^2, 0) is analysed. This is motivated by F. Presas' suggestion that the speaker's construction of regular Poisson structures(=leafwise symplectic foliations) on S^5 might be interpreted by leafwise Lefschetz fibration''. These Lefschetz fibrations give a way to look at K3 surfaces through an extended class of Arnol'd's strange duality. These applications are introduced as well.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年10月27日(火)

17:00-18:00   オンライン開催

Vassiliev derivatives of Khovanov homology and its application (JAPANESE)
[ 講演概要 ]
Khovanov homology is a categorification of the Jones polynomial. It is known that Khovanov homology also arises from a categorical representation of braid groups, so we can regard it as a kind of quantum knot invariant. However, in contrast to the case of classical quantum invariants, its relation to Vassiliev invariants remains unclear. In this talk, aiming at the problem, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. Namely, we extend Khovanov homology to singular links so that extended ones can be seen as "derivatives" in view of Vassiliev theory. As an application, we compute first derivatives to determine Khovanov homologies of twist knots. This talk is based on papers arXiv:2005.12664 (joint work with N.Ito) and arXiv:2007.15867.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年10月20日(火)

17:00-18:00   オンライン開催

Alexandru Oancea 氏 (Sorbonne Université)
[ 講演概要 ]
A certain number of dualities between homological and cohomological invariants of free loop spaces have been observed over the years, having the flavour of Poincaré duality but nevertheless holding in an infinite dimensional setting. The goal of the talk will be to explain these through a new duality theorem, whose proof uses symplectic methods. The talk will report on joint work with Kai Cieliebak and Nancy Hingston.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年10月06日(火)

17:30-18:30   オンライン開催

[ 講演概要 ]
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index.

In a previous work, motivated by the study of lattice gauge theory, we derived a formula expressing the Atiyah-Patodi-Singer index in terms of the eta invariant of “domain-wall fermion Dirac operators” when the base manifold is a flat 4-dimensional torus. Now we generalise this formula to any even dimensional closed Riemannian manifolds, and prove it mathematically rigorously. Our proof uses a Witten localisation argument combined with a devised embedding into a cylinder of one dimension higher. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.

This talk is based on a joint paper arXiv:1910.01987, to appear in CMP, with H. Fukaya, M. Furuta, T. Onogi, S. Yamaguchi, and M. Yamashita.
[ 参考URL ]
https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

### 2020年09月29日(火)

17:00-18:00   オンライン開催

Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds (JAPANESE)
[ 講演概要 ]
In 1998, Lawrence-Zagier introduced a certain q-series and proved that its limit value at root of unity q=exp(2π i / K) coincides with the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant of the Poincare homology sphere Σ(2,3,5) at the level K. Employing the idea of Gukov-Marino-Putrov based on resurgent analysis, we generalize the result of Lawrence-Zagier for the Seifert loops (Seifert manifolds with a single loop inside). That is, for each Seifert loop, we introduce an explicit q-series (WRT function) and show that its limit value at the root of unity coincides with the WRT invariant of the Seifert loop. We will also discuss a q-difference equation satisfied by the WRT function. This talk is based on a joint work with H. Fuji, H. Murakami and Y. Terashima which is available on arXiv:2007.15872.
[ 参考URL ]
https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

### 2020年07月28日(火)

17:00-18:00   オンライン開催

Anderson Vera 氏 (京都大学数理解析研究所)
A double filtration for the mapping class group and the Goeritz group of the sphere (ENGLISH)
[ 講演概要 ]
I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere, the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of $S^3$. In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group. A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年07月21日(火)

17:00-18:00   オンライン開催

Sergei Burkin 氏 (東京大学大学院数理科学研究科)
Twisted arrow categories of operads and Segal conditions (ENGLISH)
[ 講演概要 ]
We generalize twisted arrow category construction from categories to operads, and show that several important categories, including the simplex category $\Delta$, Segal's category $\Gamma$ and Moerdijk--Weiss category $\Omega$ are twisted arrow categories of operads. Twisted arrow categories of operads are closely connected with Segal conditions, and the corresponding theory can be generalized from multi-object associative algebras (i.e. categories) to multi-object P-algebras for reasonably nice operads P.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年07月21日(火)

18:00-19:00   オンライン開催

Dexie Lin 氏 (東京大学大学院数理科学研究科)
Monopole Floer homology for codimension-3 Riemannian foliation (ENGLISH)
[ 講演概要 ]
In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation. Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年07月14日(火)

17:30-18:30   オンライン開催
Lie群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。

Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (JAPANESE)
[ 講演概要 ]
G をリー群とし,X を G-等質空間とする. X のいくつかの開集合を G 移動で貼り合わせて得られる多様体を(G,X)-多様体とよぶ. X の G 不変局所幾何構造(計量など)は(G,X)-多様体に移植可能であり, (G,X)-多様体はよい幾何構造を持った多様体の例を供給することが期待される. この意味で, (G,X)-多様体の構成は微分幾何学における重要な研究テーマの一つである.

G の離散部分群が X に固有不連続に作用するとき, その離散群を X の不連続群とよび, その作用による X の商多様体を Clifford--Klein 形と呼ぶ. Clifford--Klein 形は (G,X)-多様体である. これより G-等質空間 X 上の不連続群の構成や分類は重要な問題となる. G-等質空間 X のイソトロピーがコンパクトである場合には, Gの捻じれのない離散部分群はすべて不連続群である. しかし X のイソトロピーが非コンパクトであるような場合においては, G の捻じれのない離散群であっても, X の不連続群になるとは限らない.

[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年07月07日(火)

17:00-18:00   オンライン開催

Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor (JAPANESE)
[ 講演概要 ]
We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. This is the joint work with Masatoshi Sato and Masaaki Suzuki.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年06月30日(火)

17:00-18:00   オンライン開催

Daniel Matei 氏 (IMAR Bucharest)
Homology of right-angled Artin kernels (ENGLISH)
[ 講演概要 ]
The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E), which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E. An Artin kernel Nh(G) is defined by an epimorphism h of A(G) onto the integers. In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of Nh(G).
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年06月23日(火)

17:00-18:00   オンライン開催

Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)
[ 講演概要 ]
I will explain my recent collaboration with several groups that develops gauge theory for families
to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.
After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.
Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.
If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020年01月28日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Existence problems for fibered links (JAPANESE)
[ 講演概要 ]
It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.
There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

### 2020年01月28日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室

Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)
[ 講演概要 ]
Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

### 2020年01月14日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

SO(3)-invariant G2-geometry (JAPANESE)
[ 講演概要 ]
Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G2 and Spin(7). Many authors have studied G2- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G2-manifolds with SO(3)-symmetry. Such torsion-free G2-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold. As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold. The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.

### 2020年01月14日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室

Algebraic entropy of sign-stable mutation loops (JAPANESE)
[ 講演概要 ]
Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.
We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

### 2020年01月07日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Magnitude homology of crushable spaces (JAPANESE)
[ 講演概要 ]
The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

### 2020年01月07日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室

Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)
[ 講演概要 ]
Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

### 2019年12月17日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)
[ 講演概要 ]
シンプレクティック・ベクトル空間の（コンパクト）部分集合に対して、シンプレクティック・ホモロジー（Floer ホモロジーの一種）を用いてそのシンプレクティック容量（capacity）を定義することができる。一般に、Floerホモロジーの定義には非線形偏微分方程式（いわゆるFloer方程式）の解の数え上げが関わるため、容量を定義から直接計算したり評価したりするのは難しい。この講演では（シンプレクティック・ベクトル空間をEuclid空間の余接空間とみなしたとき）fiberwiseに凸な集合のシンプレクティック・ホモロジーおよび容量をループ空間のホモロジーから計算する公式を示し、その応用を二つ与える。

### 2019年12月10日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

q-Deformation of a continued fraction and its applications (JAPANESE)
[ 講演概要 ]
Morier-Genoud と Ovsienko によって連分数のある種の q-変形が導入された。このq-変形の最大の応用はそれを用いて向きづけられた有理絡み目の Jones 多項式がそれから直接求めることができることである。またこの連分数のq-変形は結び目理論への応用以外にも、2次無理数論、組み合わせ論への応用もあり、それについても紹介する。

### 2019年12月03日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Anton Zeitlin 氏 (Louisiana State University)
Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)
[ 講演概要 ]
I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

### 2019年11月26日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Marco De Renzi 氏 (早稲田大学)
$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)
[ 講演概要 ]
Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from finite twist non-degenerate unimodular ribbon categories. Under the additional assumption of factorizability, our renormalized invariants extend to $2+1$-TQFTs, unlike Lyubashenko’s original ones. This general framework encompasses important examples of non-semisimple modular categories which were left out of previous non-semisimple TQFT constructions.

Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.