## 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

### 2021年06月29日(火)

17:00-18:00   オンライン開催

Stability of non-proper functions (JAPANESE)
[ 講演概要 ]
In this talk, we will give a sufficient condition for (strong) stability of non-proper functions (with respect to the Whitney topology). As an application, we will give a strongly stable but not infinitesimally stable function. We will further show that any Nash function on the Euclidean space becomes stable after a generic linear perturbation.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年06月22日(火)

17:00-18:30   オンライン開催

On infinite presentations for the mapping class group of a compact non orientable surface and its twist subgroup (JAPANESE)
[ 講演概要 ]
An infinite presentation for the mapping class group of any compact orientable surface was given by Gervais, and then a simpler one by Luo. Using these results, an infinite presentation for the mapping class group of any compact non orientable surfaces with boundary less than or equal to one was given by Omori (Tokyo University of Science), and then one with boundary more than or equal to two by Omori and the speaker. In this talk, we first introduce an infinite presentation for the twisted subgroup of the mapping class group of any compact non orientable surface. I will also present four simple infinite presentations for the mapping group of any compact non orientable surface, which are an improvement of the one given by Omori and the speaker. This work includes a joint work with Omori.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年06月15日(火)

17:00-18:00   オンライン開催

Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval (JAPANESE)
[ 講演概要 ]
In this talk, we consider subgroups of the group PLo(I) of piecewise linear orientation-preserving homeomorphisms of the unit interval I = [0, 1] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of PLo(I) which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application we give a necessary and sufficient condition for any two subgroups of the R. Thompson group F that are stabilizers of finite sets of numbers in the interval (0, 1) to be isomorphic.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年06月08日(火)

17:00-18:00   オンライン開催

Graphs whose Kronecker coverings are bipartite Kneser graphs (JAPANESE)
[ 講演概要 ]
Kronecker coverings are bipartite double coverings of graphs which are canonically determined. If a graph G is non-bipartite and connected, then there is a unique bipartite double covering of G, and the Kronecker covering of G coincides with it.

In general, there are non-isomorphic graphs although they have the same Kronecker coverings. Therefore, for a given bipartite graph X, it is a natural problem to classify the graphs whose Kronecker coverings are isomorphic to X. Such a classification problem was actually suggested by Imrich and Pisanski, and has been settled in some cases.

In this lecture, we classify the graphs whose Kronecker coverings are bipartite Kneser graphs H(n, k). The Kneser graph K(n, k) is the graph whose vertex set is the family of k-subsets of the n-point set {1, …, n}, and two vertices are adjacent if and only if they are disjoint. The bipartite Kneser graph H(n, k) is the Kronecker covering of K(n, k). We show that there are exactly k graphs whose Kronecker coverings are H(n, k) when n is greater than 2k. Moreover, we determine their automorphism groups and chromatic numbers.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年06月01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。

On the discrete decomposability and invariants of representations of real reductive Lie groups (JAPANESE)
[ 講演概要 ]

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

### 2021年05月25日(火)

17:00-18:00   オンライン開催

On a characteristic class associated with deformations of foliations (JAPANESE)
[ 講演概要 ]
A characteristic class for deformations of foliations called the Fuks-Lodder-Kotschick class (FLK class for short) is discussed. It seems unknown if there is a real foliation with non-trivial FLK class. In this talk, we show some conditions to assure the triviality of the FLK class. On the other hand, we show that the FLK class is easily to be non-trivial for transversely holomorphic foliations. We present an example and give a construction which generalizes it.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年05月18日(火)

17:00-18:00   オンライン開催

Geoffrey Powell 氏 (CNRS and University of Angers)
On derivations of free algebras over an operad and the generalized divergence (ENGLISH)
[ 講演概要 ]
This talk will first introduce the generalized divergence map from the Lie algebra of derivations of a free algebra over an operad to the trace space of the appropriate associative algebra. This encompasses the Satoh trace (for Lie algebras) and the double divergence of Alekseev, Kawazumi, Kuno and Naef (for associative algebras). The generalized divergence is a Lie 1-cocyle.

One restricts to considering the positive degree subalgebra with respect to the natural grading on the Lie algebra of derivations. The relationship of the positive subalgebra with its subalgebra generated in degree one is of particular interest. For example, this question arises in considering the Johnson morphism in the Lie case.

The talk will outline the structural results obtained by using the generalized divergence. These were inspired by Satoh's study of the kernel of the trace map in the Lie case. A new ingredient is the usage of naturality with respect to the category of free, finite-rank abelian groups and split monomorphisms. This allows global results to be formulated using 'torsion' for functors on this category and extends the usage of naturality with respect to the general linear groups.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年05月11日(火)

17:00-18:00   オンライン開催

トポロジカルとは限らない invertible QFT の分類問題と, Anderson dual の differential なモデル (JAPANESE)
[ 講演概要 ]
Freed and Hopkins conjectured that the deformation classes of non-topological invertible quantum field theories are classified by a generalized cohomology theory called the Anderson dual of bordism theories. Two of the main difficulty of this problem are the following. First, we do not have the axioms for QFT's. Second, The Anderson dual is defined in an abstract way. In this talk, I will explain the ongoing work to give a new approach to this conjecture, in particular to overcome the second difficulty above. We construct a new, physically motivated model for the Anderson duals. This model is constructed so that it abstracts a certain property of invertible QFT's which physicists believe to hold in general. Actually this construction turns out to be mathematically interesting because of its relation with differential cohomology theories. I will start from basic motivations for the classification problem, reportthe progress of our work and explain future directions. This is the joint work with Yosuke Morita (Kyoto, math) and Kazuya Yonekura (Tohokku, physics).
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021年04月27日(火)

17:00-18:00   オンライン開催

On a singular de Rham complex in diffeology (JAPANESE)
[ 講演概要 ]
Diffeology gives a complete, co-complete, cartesian closed category into which the category of manifolds embeds. In the framework of diffeology, the de Rham complex in the sense of Souriau enables us to develop de Rham calculus. Moreover,Iglesias-Zemmour has been introduced homotopical concepts such as homotopy groups, cubic homology groups and fibrations in diffeology. Thus one might expect differential homotopy theory'. However, the de Rham theorem does not hold for Souriau's cochain
complex in general. In this talk, I will introduce a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJUpcOCppzwpGd3r_XqdszQ1XN6FvXpNURbj

### 2021年04月20日(火)

17:00-18:00   オンライン開催

Realisation of measured laminations on boundaries of convex cores (JAPANESE)
[ 講演概要 ]
I shall present a generalisation of the theorem by Bonahon-Otal concerning realisation of measured laminations as bending laminations of geometrically finite groups, to general Kleinian surface groups which might be geometrically infinite. Our proof is based on analysis of geometric limits, and is independent of the technique of hyperbolic cone-manifolds employed by Bonahon-Otal. This is joint work with Shinpei Baba (Osaka Univ.).
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJUpcOCppzwpGd3r_XqdszQ1XN6FvXpNURbj

### 2021年04月13日(火)

17:00-18:00   オンライン開催

Quantitative Birman-Menasco theorem and applications to crossing number (JAPANESE)
[ 講演概要 ]
Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of Birman-Menasco finiteness theorem; an estimate of the crossing number of knots in terms of genus and braid index. As applications, we give various supporting evidences of various conjectural properties of the crossing number of knots.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJUpcOCppzwpGd3r_XqdszQ1XN6FvXpNURbj

### 2021年01月12日(火)

17:00-18:00   オンライン開催

Bounded cohomology of volume-preserving diffeomorphism groups (JAPANESE)
[ 講演概要 ]
Let M be a complete Riemannian manifold of finite volume. Brandenbursky and Marcinkowski proved that the third bounded cohomology of the volume-preserving diffeomorphism group of M is infinite dimensional when the fundamental group of M is "complicated enough". For example, if M is two-dimensional, the above condition is satisfied if the Euler characteristic is negative. Recently, we have extended this result in the following two directions.

(1) When M is two-dimensional and the Euler characteristic is greater than or equal to zero.
(2) When the volume of M is infinite.

In this talk, we will mainly discuss (1). The key idea is to use the fundamental group of the configuration space of M (i.e., the braid group), rather than the fundamental group of M. If time permits, we will also explain (2). For this extension, we introduce the notion of "norm controlled cohomology".
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020年12月15日(火)

17:00-18:00   オンライン開催

Braids, triangles and Lissajous curve (JAPANESE)
[ 講演概要 ]
The purpose of this talk is to introduce Lissajous 3-braids. Suppose we have a closed curve on the plane, and we consider the periodic motion of n points along the closed curve. If the motion is collision-free, then we get a braid obtained from the trajectory of the set of n points in question. In this talk, we consider 3-braids coming from the periodic motion of 3 points on Lissajous curves. We classify Lissajous 3-braids and present a parametrization in terms of natural numbers together with slopes. We also discuss some properties of pseudo-Anosov stretch factors for Lissajous 3-braids. The main tool is the shape sphere --- the configuration space of the oriented similarity classes of triangles. This is a joint work with Hiroaki Nakamura and Hiroyuki Ogawa.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 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