## 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://faculty.ms.u-tokyo.ac.jp/~topology/index.html Tea: 16:30 - 17:00 コモンルーム

### 2018年05月15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

[ 講演概要 ]
Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.

### 2018年05月08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
Dan Cristofaro-Gardiner 氏 (University of California, Santa Cruz)
Beyond the Weinstein conjecture (ENGLISH)
[ 講演概要 ]
The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

### 2018年04月24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

Singular Fibers of smooth maps and Cobordism groups (JAPANESE)
[ 講演概要 ]
Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.

### 2018年04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
Tamás Kálmán 氏 (東京工業大学)
Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)
[ 講演概要 ]
In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

### 2018年04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

２次元結び目のモース・ノビコフ数について (JAPANESE)
[ 講演概要 ]
2001年にPajitnov, Rudolph, Weberは，古典的絡み目に対してMorse-Novikov数を定義し，その基本的な性質を研究した．この不変量は，Alexander多項式やNovikovホモロジーおよびそれらの「ねじれ版」との関連から盛んに研究されている．本講演では，２次元結び目に対してMorse-Novikov数を定義し，いくつかの性質を述べる．特に，２次元結び目のモーション・ピクチャーやスピン構成法との関係について解説する．尚，本講演の内容はAndrei Pajitnov氏（ナント大学）との共同研究にもとづいている．

### 2018年04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

Chain level loop bracket and pseudo-holomorphic disks (JAPANESE)
[ 講演概要 ]
Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.

### 2018年03月30日(金)

15:00-16:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Matteo Felder 氏 (University of Geneva)
Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)
[ 講演概要 ]
The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

### 2018年03月30日(金)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Florian Naef 氏 (Massachusetts Institute of Technology)
Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)
[ 講演概要 ]
Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.

### 2018年02月21日(水)

17:00-18:30   数理科学研究科棟(駒場) 122号室
Tea: Common Room 16:30-17:00
Gwénaël Massuyeau 氏 (Université de Bourgogne)
The category of bottom tangles in handlebodies, and the Kontsevich integral (ENGLISH)
[ 講演概要 ]
Habiro introduced the category B of « bottom tangles in handlebodies », which encapsulates the set of knots in the 3-sphere as well as the mapping class groups of 3-dimensional handlebodies. There is a natural filtration on the category B defined using an appropriate generalization of Vassiliev invariants. In this talk, we will show that the completion of B with respect to the Vassiliev filtration is isomorphic to a certain category A which can be defined either in a combinatorial way using « Jacobi diagrams », or by a universal property via the notion of « Casimir Hopf algebra ». Such an isomorphism will be obtained by extending the Kontsevich integral (originally defined as a knot invariant) to a functor Z from B to A. This functor Z can be regarded as a refinement of the TQFT-like functor derived from the LMO invariant and, if time allows, we will evoke the topological interpretation of the « tree-level » of Z. (This is based on joint works with Kazuo Habiro.)

### 2018年01月30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

Persistence-like distance on Tamarkin's category and symplectic displacement energy (JAPANESE)
[ 講演概要 ]
The microlocal sheaf theory due to Kashiwara and Schapira can be regarded as Morse theory with sheaf coefficients. Recently it has been applied to symplectic geometry, after the pioneering work of Tamarkin. In this talk, I will propose a new sheaf-theoretic method to estimate the displacement energy of compact subsets in cotangent bundles. In the course of the proof, we introduce a persistence-like pseudo-distance on Tamarkin's sheaf category. This is a joint work with Tomohiro Asano.

### 2018年01月23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

An invariant of 3-manifolds via homology cobordisms (JAPANESE)
[ 講演概要 ]
For a closed 3-manifold X, we consider the topological invariant defined as the minimal integer g such that X is obtained as the closure of a homology cobordism over a surface of genus g. We prove that the invariant equals one for every lens space, which is contrast to the fact that some lens spaces do not admit any open book decomposition whose page is a surface of genus one. The proof is based on the Chebotarev density theorem and binary quadratic forms in number theory.

### 2018年01月23日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室

Wrapping projections and decompositions of Keinian groups (JAPANESE)
[ 講演概要 ]
Let $S$ be a closed surface of genus $g ¥geq 2$. The deformation space $AH(S)$ consists of (conjugacy classes of) discrete faithful representations $\rho:\pi_{1}(S) \to PSL_{2}(\mathbb{C})$.
McMullen, and Bromberg and Holt showed that $AH(S)$ can self-bump, that is, the interior of $AH(S)$ has the self-intersecting closure.
Both of them demonstrated the existence of self-bumping under the exisetence of a non-trivial wrapping projections from an algebraic limits to a geometric limits which wraps an annulus cusp into a torus cusp.
In this talk, given a representation $\rho$ at the boundary of $AH(S)$, we characterize a wrapping projection to a geometric limit associated to $\rho$, by the information of the actions of decomposed Kleinian groups of the image of $\rho$.

### 2018年01月16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
On adequacy and the crossing number of satellite knots (JAPANESE)
[ 講演概要 ]
It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.

### 2018年01月16日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室

A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)
[ 講演概要 ]
A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

### 2017年12月19日(火)

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

Deformation of holomorphic quadratic differentials and its applications (JAPANESE)
[ 講演概要 ]
Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

### 2017年12月12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

On the self-intersection of singular sets of maps and the signature defect (JAPANESE)
[ 講演概要 ]

### 2017年12月05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

Derivations and cohomologies of Lipschitz algebras (JAPANESE)
[ 講演概要 ]
For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

### 2017年11月28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
Sang-hyun Kim 氏 (Seoul National University)
Diffeomorphism Groups of One-Manifolds (ENGLISH)
[ 講演概要 ]
Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

### 2017年11月21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

The space of short ropes and the classifying space of the space of long knots (JAPANESE)
[ 講演概要 ]
We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

### 2017年11月07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

On an explicit example of topologically protected corner states (JAPANESE)
[ 講演概要 ]
In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a `product’’ of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

### 2017年10月31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
Yash Lodha 氏 (École Polytechnique Fédérale de Lausanne)
Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)
[ 講演概要 ]
Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

### 2017年10月24日(火)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30, Lie群論・表現論セミナーと合同

ラグランジュ交叉のフレアホモロジーに対する部分多様体論からのアプローチ (JAPANESE)
[ 講演概要 ]

### 2017年10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

Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)
[ 講演概要 ]
We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

### 2017年10月10日(火)

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

Poset-stratified spaces and some applications (JAPANESE)
[ 講演概要 ]
A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

### 2017年10月03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
Athanase Papadopoulos 氏 (IRMA, Université de Strasbourg)
Transitional geometry (ENGLISH)
[ 講演概要 ]
I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert A’Campo and recent joint work with A’Campo and Yi Huang.