Tuesday Seminar on Topology (April -- July, 2022)

[Japanese] [Past Programs]

This is an online seminar on Zoom.

Last updated September 16, 2022
Information :　
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

April 19, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom. Pre-registration is not required.

Toshihisa Kubo (Ryukoku University)

On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces

Abstract: Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a \emph{differential symmetry breaking operator} (differential SBO for short) ([T. Kobayashi, Differential Geom. Appl. (2014)]). In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$. In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

April 26, 17:00-18:00 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom. Pre-registration is not required.

Yoshiki Oshima (The Univesity of Tokyo)

On the existence of discrete series for homogeneous spaces

Abstract: When a Lie group G acts transitively on a manifold X, an irreducible subrepresentation of L²(X) is called a discrete series representation of X. One may ask which homogeneous space X has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

May 10, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Hokuto Konno (The University of Tokyo)

Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory

Abstract: I will discuss two different kinds of applications of Seiberg-Witten (Floer) homotopy theory involving involutions. The first application is about the Nielsen realization problem, which asks whether a given finite subgroup of the mapping class group of a manifold lifts to a subgroup of the diffeomorphism group. Although every finite subgroup is known to lift in dimension 2, there are manifolds of dimension greater than 2 for which the Nielsen realization fails. However, only few examples have been known in dimension 4. I will show that "4-dimensional Dehn twists" yield a large class of new examples. The second application is about 4-dimensional invariants of knots. I will introduce a version of "Floer K-theory for knots", and will explain that this framework gives the first comparison result for the smooth and topological versions of a certain knot invariant, called stabilizing number. Although the above two topics (Nielsen realization and knots) may seem to have different flavors, they are derived from a common idea. The first one is proved using a constraint on smooth involutions on a closed 4-manifold from Seiberg-Witten homotopy theory by Yuya Kato, and the second one is derived from a generalization of Kato's result to 4-manifolds with boundary using Seiberg-Witten Floer homotopy theory. This talk is partially based on joint work with Jin Miyazawa and Masaki Taniguchi.

May 17, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Tatsuro Shimizu (Tokyo Denki University)

Contribution of simple loops to the configuration space integral

Abstract: For a manifold with a representation of the fundamental group, the configuration space integral associated with a graph (Feynman diagram) gives a real number. An appropriate linear combination of graphs gives an invariant of the manifold with the representation. In this talk, we discuss the contribution of simple loops to the configuration space integral. Hutchings, Lee and Kitayama give geometric descriptions of the Reidemeister torsion by using circle valued Morse functions. By using these descriptions and a Morse theoretical description of the configuration space integral, we have equations among the Reidemeister torsion and the contributions of simple loops in some cases. In this talk, we extend the equations for some other cases and give a computational example of the configuration space integrals by using Morse function.

May 24, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Christine Vespa (IRMA, Université de Strasbourg / JSPS)

Polynomial functors associated with beaded open Jacobi diagrams

Abstract: The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by gr^op. By restriction to this subcategory, morphisms in the linear category A give rise to interesting contravariant functors on the category gr, encoding part of the composition structure of the category A.
In recent papers, Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on gr^op which are outer functors, in the sense that inner automorphisms act trivially.
In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on gr^op which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

May 31, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Kazushi Ueda (The Univesity of Tokyo)

Stable Fukaya categories of Milnor fibers

Abstract: We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

June 7, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Yoshikazu Yamaguchi (Waseda University)

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

Abstract: We discuss a relation between a dynamical zeta function defined by the geodesic flow on a 2-dimensional hyperbolic orbifold and the asymptotic behavior of the Reidemeister torsion for the unit tangent bundle over the orbifold. The unit tangent bundle over a hyperbolic orbifold is a Seifert fibered space with a geometric structure given by the universal cover of PSL(2, R). This geometric structure induces an SL(2, R)-representation of the fundamental group. Here the asymptotic behavior of the Reidemeister torsion means the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle over a hyperbolic orbifold and the SL(n, R)-representations induced by the SL(2, R)-one of its fundamental group. For a hyperbolic 3-manifold, we can derive the hyperbolic volume from the limit of the leading coefficient in the Reidemeister torsion with a dynamical zeta function according to previous works. For the unit tangent bundle over a 2-dimensional hyperbolic orbifold, which is not a hyperbolic 3-manifold, we can find the orbifold Euler characteristic of the orbifold in the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle from the relation with the dynamical zeta function defined by the geodesic flow on the orbifold.

June 14, 17:30-18:30 -- Online on Zoom. Pre-registration required.

Katsuhiko Kuribayashi (Shinshu University)

Cartan calculi on the free loop spaces

Abstract: A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham complex. In this talk, a `second stage' of the Cartan calculus is investigated. In a more general setting, the stage is formulated with a Lie algebra representation of the Andre-Quillen cohomology of a commutative differential graded algebra A on the endomorphism ring of the Hochschild homology of A in terms of the homotopy Cartan calculi in the sense of Fiorenza and Kowalzig. Moreover, the Lie algebra representation in the Cartan calculus is interpreted geometrically as a map from the rational homotopy group of the monoid of self-homotopy equivalences on a simply-connected space M to the derivation ring on the loop cohomology of M. An extension of the representation to the string cohomology and its geometric counterpart are also discussed together with the BV exactness which is a new rational homotopy invariant introduced in our work. This talk is based on joint work in progress with T. Naito, S. Wakatsuki and T. Yamaguchi.

June 21, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Kazuhiro Ichihara (Nihon University)

Cosmetic surgeries on knots in the 3-sphere

Abstract: A pair of Dehn surgeries on a knot is called purely (resp. chirally) cosmetic if the obtained manifolds are orientation-preservingly (resp. -reversingly) homeomorphic. It is conjectured that if a knot in the 3-sphere admits purely (resp. chirally) cosmetic surgeries, then the knot is a trivial knot (resp. a torus knot or an amphicheiral knot). In this talk, after giving a brief survey on the studies on these conjectures, I will explain recent progresses on the conjectures. This is based on joint works with Tetsuya Ito (Kyoto University), In Dae Jong (Kindai University), and Toshio Saito (Joetsu University of Education).

July 5, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Yushi Nakano (Tokai University)

Non-existence of Lyapunov exponents for homoclinic bifurcations of surface diffeomorphisms

Abstract: Lyapunov exponent is widely used in natural science including mathematics, such as a tool to find chaotic signal or a foundation of non-uniformly hyperbolic systems theory. However, its existence (outside of the supports of invariant probability measures) is seldom discussed. In this talk, I consider the problem of whether the Lyapunov irregular set, i.e. the set of points at which Lyapunov exponent fails to exist, has positive Lebesgue measure. I will show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas, as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. This is a joint work with S. Kiriki, X. Li and T. Soma.

July 12, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Sungkyung Kang (Center for Geometry and Physics, Institute of Basic Science)

Cable knots and involutive Heegaard Floer homology

Abstract: Heegaard Floer homology (and its variants) carries an intrinsic symmetry, which conjecturally corresponds to the Pin(2)-equivariance in Seiberg-Witten Floer homology. By exploiting the symmetry, we prove that (odd,1)-cables of the figure-eight knots are linearly independent in the concordance group of rationally slice knots, and present a first example of rationally slice knots of complexity 1 which are not slice. Furthermore, we establish an explicit connection between involutive knot Floer theory and involutive bordered Floer theory of knot complements, and use it to prove a similar result for iterated cables of figure-eight knots. A part of this talk is based on a joint work with J. Hom, M. Stoffregen, and J. Park.