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Tuesday Seminar on Topology
Seminar information archive ～01/07｜Next seminar｜Future seminars 01/08～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
Future seminars
2026/01/13
17:00-18:00 Room #hybrid/056 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Sogo Murakami (The University of Tokyo)
On the shadowing property of differentiable dynamical systems beyond structural stability (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Sogo Murakami (The University of Tokyo)
On the shadowing property of differentiable dynamical systems beyond structural stability (JAPANESE)
[ Abstract ]
The shadowing property, which has been extensively studied in connection with hyperbolic differentiable dynamical systems, is a dynamical concept ensuring that approximate orbits with small errors (commonly referred to as pseudo-orbits) can be traced by a true orbit. This property is one of the fundamental notions closely related to structural stability. In this talk, I will present the conditions under which the shadowing property holds for differentiable dynamical systems that are not structurally stable, in both discrete-time and continuous-time settings. In the first part of the talk, conditions guaranteeing the shadowing property for Axiom A diffeomorphisms will be discussed. In particular, I will explain the T^{s,u}-condition, and its relationship with the C^0-transversality condition introduced by PetrovPilyugin. I will then give a sufficient condition for having the shadowing property for Axiom A diffeomorphisms. In the second part, results concerning the shadowing property on chain recurrent sets for flows will be presented. While Robinson (1977) showed that every hyperbolic set exhibits the shadowing property, it is known that no singular hyperbolic set with non-isolated hyperbolic singularity, such as the Lorenz attractor, admits the shadowing property (Wen-Wen, 2020). Motivated by this, Arbieto et al. conjectured that any chain recurrent set with attached (non-isolated) hyperbolic singularities cannot possess the shadowing property. In this talk, a counterexample to this conjecture will be constructed.
[ Reference URL ]The shadowing property, which has been extensively studied in connection with hyperbolic differentiable dynamical systems, is a dynamical concept ensuring that approximate orbits with small errors (commonly referred to as pseudo-orbits) can be traced by a true orbit. This property is one of the fundamental notions closely related to structural stability. In this talk, I will present the conditions under which the shadowing property holds for differentiable dynamical systems that are not structurally stable, in both discrete-time and continuous-time settings. In the first part of the talk, conditions guaranteeing the shadowing property for Axiom A diffeomorphisms will be discussed. In particular, I will explain the T^{s,u}-condition, and its relationship with the C^0-transversality condition introduced by PetrovPilyugin. I will then give a sufficient condition for having the shadowing property for Axiom A diffeomorphisms. In the second part, results concerning the shadowing property on chain recurrent sets for flows will be presented. While Robinson (1977) showed that every hyperbolic set exhibits the shadowing property, it is known that no singular hyperbolic set with non-isolated hyperbolic singularity, such as the Lorenz attractor, admits the shadowing property (Wen-Wen, 2020). Motivated by this, Arbieto et al. conjectured that any chain recurrent set with attached (non-isolated) hyperbolic singularities cannot possess the shadowing property. In this talk, a counterexample to this conjecture will be constructed.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2026/01/20
17:00-18:00 Room #hybrid/056 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Takumi Maegawa (The University of Tokyo)
A six-functor construction of the Bauer-Furuta invariant (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Takumi Maegawa (The University of Tokyo)
A six-functor construction of the Bauer-Furuta invariant (JAPANESE)
[ Abstract ]
Building on the pioneering works of Verdier and Grothendieck, and later developed by Kashiwara-Schapira, the six-functor formalism for sheaves enables us to understand cohomological duality theorems and transfer maps in terms of certain (stable) ∞-categorical adjunction. Following Gaitsgory-Rozenblyum, these six operations fit into a single (∞,2)-functor out of the 2-category of correspondences. In this talk, we will recall these modern points of view on the six-functor formalism, and as an application, we will see that the stable homotopy theoretic refinement of the Seiberg-Witten invariant defined for a closed spin c four-manifold, introduced by Furuta and Bauer, does correspond to a 2-morphism in that (∞,2)-functoriality.
[ Reference URL ]Building on the pioneering works of Verdier and Grothendieck, and later developed by Kashiwara-Schapira, the six-functor formalism for sheaves enables us to understand cohomological duality theorems and transfer maps in terms of certain (stable) ∞-categorical adjunction. Following Gaitsgory-Rozenblyum, these six operations fit into a single (∞,2)-functor out of the 2-category of correspondences. In this talk, we will recall these modern points of view on the six-functor formalism, and as an application, we will see that the stable homotopy theoretic refinement of the Seiberg-Witten invariant defined for a closed spin c four-manifold, introduced by Furuta and Bauer, does correspond to a 2-morphism in that (∞,2)-functoriality.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2026/01/21
16:00-17:00 Room #hybrid/118 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Ingrid Irmer (Southern University of Science and Technology)
Understanding the well-rounded deformation retraction of Teichmüller space (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Ingrid Irmer (Southern University of Science and Technology)
Understanding the well-rounded deformation retraction of Teichmüller space (ENGLISH)
[ Abstract ]
The term "well-rounded deformation retraction" goes back to a paper of Ash in which equivariant deformation retractions of the space of $n\times n$ positive-definite real symmetric matrices acted on by $SL(n,\mathbb{Z})$ were studied. An informal analogy between families of groups, such as $SL(n,\mathbb{Z})$, $Out(F_{n})$ and mapping class groups, suggests the existence of a similar equivariant deformation retractions of the actions of $Out(F_{n})$ and mapping class groups on well-chosen spaces. In all these examples, there are spaces on which the respective groups act with known equivariant deformation retractions onto cell complexes of the smallest possible dimension --- the virtual cohomological dimension of the group. The purpose of this talk is to explain that the equivariant deformation retraction of the action of the mapping class group on Teichmüller space can be understood to be a piecewise-smooth analogue of Ash's well rounded deformation retraction. The key idea is to understand the role of duality in correctly drawing this analogy.
[ Reference URL ]The term "well-rounded deformation retraction" goes back to a paper of Ash in which equivariant deformation retractions of the space of $n\times n$ positive-definite real symmetric matrices acted on by $SL(n,\mathbb{Z})$ were studied. An informal analogy between families of groups, such as $SL(n,\mathbb{Z})$, $Out(F_{n})$ and mapping class groups, suggests the existence of a similar equivariant deformation retractions of the actions of $Out(F_{n})$ and mapping class groups on well-chosen spaces. In all these examples, there are spaces on which the respective groups act with known equivariant deformation retractions onto cell complexes of the smallest possible dimension --- the virtual cohomological dimension of the group. The purpose of this talk is to explain that the equivariant deformation retraction of the action of the mapping class group on Teichmüller space can be understood to be a piecewise-smooth analogue of Ash's well rounded deformation retraction. The key idea is to understand the role of duality in correctly drawing this analogy.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2026/01/21
17:30-18:30 Room #hybrid/118 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Stavros Garoufalidis (Southern University of Science and Technology)
What are Lie superalgebras good for? (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Stavros Garoufalidis (Southern University of Science and Technology)
What are Lie superalgebras good for? (ENGLISH)
[ Abstract ]
I will try to answer, as honestly as I can, this question. Lie superalgebras are important in mathematical physics (supersymmetry), in representation theory, in categorification, in quantum topology, but also in classical topology. Namely, they may detect the genus of a smallest spanning surface of a knot. Come and listen about some theorems and experimental evidence, and decide for yourself if this is an accident, a conspiracy theory, or a manifestation of the truth!
[ Reference URL ]I will try to answer, as honestly as I can, this question. Lie superalgebras are important in mathematical physics (supersymmetry), in representation theory, in categorification, in quantum topology, but also in classical topology. Namely, they may detect the genus of a smallest spanning surface of a knot. Come and listen about some theorems and experimental evidence, and decide for yourself if this is an accident, a conspiracy theory, or a manifestation of the truth!
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
