Tuesday Seminar on Topology

Seminar information archive ~10/03Next seminarFuture seminars 10/04~

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

Seminar information archive


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Christine Vespa (Université de Strasbourg)
Higher Hochschild homology as a functor (ENGLISH)
[ Abstract ]
Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Ken’ichi Ohshika (Gakushuin University)
Thurston’s bounded image theorem (ENGLISH)
[ Abstract ]
The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurston’s original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurston’s original version making use of more recent technology. This is joint work with Cyril Lecuire.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiraku Nakajima (Kavli IPMU, The University of Tokyo)
Coulomb branches of 3d SUSY gauge theories (JAPANESE)
[ Abstract ]
I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Jongil Park (Seoul National University)
A topological interpretation of symplectic fillings of a normal surface singularity (ENGLISH)
[ Abstract ]
One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure.
Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, I’d like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, I’ll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.


17:00-18:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Florian Naef (Université de Genève)
On a moduli space interpretation of the Turaev cobracket (ENGLISH)
[ Abstract ]
Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator on the moduli space of flat connection with values in a super Lie algebra equipped with an odd pairing. This is joint work with A. Alekseev, J. Pulmann and P. Ševera.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Michael Hutchings (University of California, Berkeley)
Cube capacities (ENGLISH)
[ Abstract ]
We define a new series of symplectic capacities using equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but can be computed in many more examples. In particular, we use these capacities to find many examples of symplectic embeddings of cubes where the cube is as large as possible. This is joint work with Jean Gutt.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Anastasiia Tsvietkova (Okinawa Institute of Science and Technology, Rutgers University)
Representations of knot groups (ENGLISH)
[ Abstract ]
We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods, this does not involve any polyhedral decomposition or triangulation of the link complement, and uses only a link diagram satisfying a few mild restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is a joint work with Kathleen Peterson (Florida State University).


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yusuke Kuno (Tsuda University)
Generalized Dehn twists on surfaces and homology cylinders (JAPANESE)
[ Abstract ]
This is a joint work with Gwénaël Massuyeau (University of Burgundy). Lickorish's trick describes Dehn twists along simple closed curves on an oriented surface in terms of surgery of 3-manifolds. We discuss one possible generalization of this description to the situation where the curve under consideration may have self-intersections. Our result generalizes previously known computations related to the Johnson homomorphisms for the mapping class groups and for homology cylinders. In particular, we obtain an alternative and direct proof of the surjectivity of the Johnson homomorphisms for homology cylinders, which was proved by Garoufalidis-Levine and Habegger.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Marek Kaluba (Adam Mickiewicz Univeristy)
On property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb{Z})$ (ENGLISH)
[ Abstract ]
We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \ge 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n \ge 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \ge 6$) and of $\mathrm{SL}_n(\mathbb{Z})$ (with $n \ge 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n >6$.


13:00-14:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Anderson Vera (Université de Strasbourg)
Johnson-type homomorphisms and the LMO functor (ENGLISH)
[ Abstract ]
One of the main objects associated to a surface S is the mapping class group MCG(S). This group plays an important role in the study of 3-manifolds. Reciprocally, the topological invariants of 3-manifolds can be used to obtain interesting representations of MCG(S).

One possible approach to the study of MCG(S) is to consider its action on the fundamental group P of the surface or on some subgroups of P. This way, we can obtain some kind of filtrations of MCG(S) and homomorphisms, called Johnson type homomorphisms, which take values in certain spaces of diagrams. These spaces of diagrams are quotients of the target space of the LMO functor. Hence it is natural to ask what is the relation between the Johnson type homomorphisms and the LMO functor. The answer is well known in the case of the Torelli group and the usual Johnson homomorphisms. In this talk we consider two other different filtrations of MCG(S) introduced by Levine and Habiro-Massuyeau. We show that the respective Johnson homomorphisms can also be deduced from the LMO functor.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takeshi Torii (Okayama University)
Discrete G-spectra and a model for the K(n)-local stable homotopy category (JAPANESE)
[ Abstract ]
The K(n)-local stable homotopy categories are building blocks for the stable homotopy category of spectra. In this talk I will construct a model for the K(n)-local stable homotopy category, which explicitly shows the relationship with the Morava E-theory E_n and the stabilizer group G_n. We consider discrete symmetric G-spectra studied by Behrens-Davis for a profinite group G. I will show that the K(n)-local stable homotopy category is realized in the homotopy category of modules in discrete symmetric G_n-spectra over a discrete model of E_n.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masashi Ishida (Osaka University)
On non-singular solutions to the normalized Ricci flow on four-manifolds (JAPANESE)
[ Abstract ]
A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton introduced it as an important special class of solutions and proved that the underlying 3-manifold is geometrizable in the sense of Thurston. In this talk, we will discuss properties of 4-dimensional non-singular solutions from a gauge theoretical point of view. In particular, we would like to explain gauge theoretical invariants give rise to obstructions to the existence of 4-dimensional non-singular solutions.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Vincent Florens (Université de Pau et des Pays de l'Adour)
Slopes and concordance of links (ENGLISH)
[ Abstract ]
We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Motoko Kato (The University of Tokyo)
Fixed points for group actions on non-positively curved spaces (JAPANESE)
[ Abstract ]
In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takuya Sakasai (The University of Tokyo)
Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)
[ Abstract ]
There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Hidetoshi Masai (Tokyo Institute of Technology)
On continuity of drifts of the mapping class group (JAPANESE)
[ Abstract ]
When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.


10:30-12:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Michael Heusener (Université Clermont Auvergne)
Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)
[ Abstract ]
This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Oguni (Ehime University)
Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)
[ Abstract ]
A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroshige Shiga (Tokyo Institute of Technology)
The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)
[ Abstract ]
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
François Fillastre (Université de Cergy-Pontoise)
Co-Minkowski space and hyperbolic surfaces (ENGLISH)
[ Abstract ]
There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Daniel Matei (IMAR Bucharest)
Resonance varieties and matrix tree theorems (ENGLISH)
[ Abstract ]
We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Boris Hasselblatt (Tufts University)
Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)
[ Abstract ]
A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuanyuan Bao (The University of Tokyo)
An Alexander polynomial for MOY graphs (JAPANESE)
[ Abstract ]
An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masaharu Ishikawa (Keio University)
Positive flow-spines and contact 3-manifolds (JAPANESE)
[ Abstract ]
A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).


17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Emmy Murphy (Northwestern University)
Loose Legendrians and arboreal singularities (ENGLISH)
[ Abstract ]
Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.

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