Tuesday Seminar on Topology

[Japanese]   [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo (with live streaming on Zoom)
or
17:00 -- 18:00 Online seminar on Zoom.

Last updated December 20, 2024
Information :@
Kazuo Habiro
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

Pre-registration is required.
Once you register, you can attend all our seminars until the end of March, 2024.
The zoom meeting will be opened 15 minutes before start.
Making audio and video recordings is prohibited.
October 8, 17:00-18:30 -- Room 123 with live streaming on Zoom

Hokuto Konno (The University of Tokyo)

Dehn twists on 4-manifolds

Abstract: Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.

October 17 (Thu), 17:00-18:30 -- Room 123 with live streaming on Zoom

Makoto Enokizono (The University of Tokyo)

Slope inequalities for fibered complex surfaces

Abstract: Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology. It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves. In this talk, after outlining the background of these studies, I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.

October 22, 17:00-18:00 -- Online on Zoom

Tatsuki Kuwagaki (Kyoto University)

On the generic existence of WKB spectral networks/Stokes graphs

Abstract: The foliation determined by a quadratic differential on a Riemann surface is a classical object of study. In particular, considering leaves through zero points has been of interest in connection with WKB analysis, Teichmüller theory, and quantum field theory. WKB spectral network (or Stokes graph) is a higher-order-differential version of this notion. In this talk, I will discuss the proof of existence of WKB spectral network for a large class of differentials. If time permits, I will explain its relationship with Lagrangian intersection Floer theory.

October 29, 17:00-18:30 -- Room 056 with live streaming on Zoom

Takahito Naito (Nippon Institute of Technology)

Cartan calculus in string topology

Abstract: The homology of the free loop space of a closed oriented manifold (called the loop homology) has rich algebraic structures. In the theory of string topology due to Chas and Sullivan, it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket. On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology, which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.
In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus. Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients. As an application, the nilpotency of some loop homology classes are determined.

November 5, 17:00-18:30 -- Room 056 with live streaming on Zoom

Jun Murakami (Waseda University)

On complexified tetrahedrons associated with the double twist knots and its application to the volume conjecture

Abstract: In this talk, I first would like to explain complexified tetrahedrons which subdivide the complement of a double twist knot, and then talk about their application to the volume conjecture. Complexified tetrahedrons are introduced by using the fundamental group, and they are deformations of the regular ideal octahedron which is a half of the complement of the Borromean rings. Such deformations correspond to the surgeries of the Borromean rings producing the double twist knots.
On the other hand, the colored Jones polynomials of the double twist knots are given by the quantum 6j symbol with some extra terms. We see the correspondence of the quantum 6j symbol and the volume of a complexified terahedron by using the Neumann-Zagier function, and we can apply such correspondence to prove the volume conjecture for the double twist knots. To do this, the ADO invariant is used instead of the colored Jones invariant. The l'Hopital's rule is applied to get the ADO invariant, and integral by parts solves the big cancellation problem. At the last, it is shown that the application of the saddle point method is not so hard for this case.

November 12, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar] -- Room 056 with live streaming on Zoom

Junko Inoue (Tottori University)

Holomorphically induced representations of some solvable Lie groups

Abstract: From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group G in an L2-space of some holomorphic sections of some line bundle over a G-homogeneous space associated with a polarization for a linear form of the Lie algebra of G. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.

In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations ρ in some general settings. We would like to discuss the following problems:
(1) conditions of non-vanishing of ρ,
(2) decomposition of ρ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.

November 19, 17:00-18:30 -- Room 056 with live streaming on Zoom

Bruno Scárdua (Federal University of Rio de Janeiro)

On real center singularities of complex vector fields on surfaces

Abstract: One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([2]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique dfun théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude dfun cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.

November 26, 17:00-18:30 -- Room 056 with live streaming on Zoom

Masaki Natori (The University of Tokyo)

A proof of Bott periodicity via Quot schemes and bulk-edge correspondence

Abstract: The bulk-edge correspondence refers to the phenomenon typically found in topological insulators, where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary). In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence. It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory. The proof of Bott periodicity has been approached from various perspectives. We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.

December 3, 17:30-18:30 -- Room 056 with live streaming on Zoom

Jun-ichi Inoguchi (Hokkaido University)

Surfaces in 3-dimensional spaces and Integrable systems

Abstract: Surfaces of constant mean curvature in hyperbolic 3-space have different aspects depending on the value of mean curvature. In particular, the class of surfaces of constant mean curvature H<1 has no Euclidean or spherical correspondents. I would explain how to construct surface of constant mean curvature H<1 in hyperbolic 3-space by the method of Integrable Systems (joint work with Josef F. Dorfmeister and Shinpei Kobayashi).

December 10, 17:00-18:30 -- Room 056 with live streaming on Zoom

Shun Wakatsuki (Nagoya University)BR>
Computation of the magnitude homology as a derived functor

Abstract: Asao-Ivanov showed that the magnitude homology of a finite metric space is isomorphic to the derived functor Tor over some ring. In this talk, I will explain an application of the theory of minimal projective resolution to this derived functor. Especially in the case of a geodetic graph, torsion-freeness and a criterion for diagonality of the magnitude homology are established. Moreover, I will give computational examples including cyclic graphs. This is a joint work with Yasuhiko Asao.

December 17, 17:00-18:30 -- Room 056 with live streaming on Zoom

Emmanuel Graff (The University of Tokyo)

Is there torsion in the homotopy braid group?

Abstract: In the 'Kourovka notebook,' V. Lin questions the existence of a non-trivial epimorphism from the braid group onto a non-abelian torsion-free group. The homotopy braid group, studied by Goldsmith in 1974, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsion-free for less than six strands. In this presentation, we will see a new approach based on the broader concept of welded braids, along with algebraic techniques, to determine whether the homotopy braid group provides a complete answer to Linfs question.

January 14, 17:00-18:00 -- Room 056 with live streaming on Zoom

Leo Yoshioka (The University of Tokyo)

Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop graphs

Abstract: In this talk, we give some non-trivial cocycles and cycles of the space of long embeddings R^j --> R^n modulo immersions. First, we construct a cocycle through configuration space integrals with the simplest 2-loop graph cocycle of the Bott-Cattaneo-Rossi graph complex for odd n and j. Then, we give a cycle from a chord diagram on oriented lines, which is associated with the simplest 2-loop hairy graph. We show the non-triviality of this (co)cycle by pairing argument, which is reduced to pairing of graphs with the chord diagram. This construction of cycles and the pairing argument to show the non-triviality is also applied to general 2-loop (co)cycles of top degree. If time permits, we introduce a modified graph complex and configuration space integrals to give more general cocycles. This new graph complex is quasi-isomorphic to both the hairy graph complex and the graph complex introduced in embedding calculus by Arone and Turchin. With these modified cocycles, our pairing argument provides an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long j-knots R^j -->R^{j+2}, which Budney-Gabai and Watanabe first established.