AlgebraiC Topology NETWORK

Upcoming Events

30 April 16:30--18:00

Shinshu Topology Seminar (Homepage)

浅尾泰彦（九州大学） 

Magnitude homology and Anick resolution

Hepworth-Willerton, Leinster-Shulmanによって与えられたマグニチュードホモロジーの定義は、マグニチュードの圏化となるようにデザインされた具体的な鎖複体のホモロジーであった。講演者の問題意識の一つとして、このホモロジーが既存の文脈の中でどう解釈されるべきか、というものがある。この講演ではAsao-IvanovによるマグニチュードホモロジーのTor関手としての記述に基づいて、「有限次元代数の表現論」的な解釈を与える試みについて紹介する。またAsao-Wakatsukiによる極小射影分解の構成を用いたマグニチュードホモロジーの計算を、Anick resolutionという表現論的な枠組みに沿って紹介する。この講演は一部アーロン・チャン氏 (名大)と若月駿氏（名大）とのディスカッションに基づく。

20 - 21 June

Homotopy Okinawa

Venue

Okinawa Prefectural Gender Equality Center "Tiruru" (Homepage)

Celebration

We will celebrate Prof. Toshiyuki Akita's 60th birthday.

Confirmed Speakers

Toshiyuki Akita (Hokkaido Univrsity)

Yasuhiko Asao (Kyushu University)

Yoh Katoh (Tokyo University of Science)

Ye Liu (Xi’an Jiaotong-Liverpool University)

Takahiro Matsushita (Shinshu University)

Yuki Minowa (Kyushu University)

Takefumi Nosaka (Institute of Science Tokyo)

Takao Sato (Tokyo University of Science)

Ryokichi Tanaka (Kyoto University)

Financial Support

JSPS KAKENHI 23K03113 (Takeshi Torii)

Organizers

Daisuke Kishimoto (Kyushu University)

Shuichi Tsukuda (University of the Ryukyus)

Abstract

Ye Liu

Magnitude homology of real hyperplane arrangements

Magnitude is a cardinality-like invariant of metric spaces or enriched categories measuring the effective size. Its categorification, the magnitude homology, is a more powerful invariant. For a real hyperplane arrangement, or more generally, an oriented matroid, the tope graph encapsulates considerable amount of information. Since tope graphs are equipped with the shortest path metric, we feed them to the magnitude and magnitude homology machinery to derive new invariants of real hyperplane arrangements. We prove some structural results of the magnitude of arrangements, including reciprocity, palindromic numerator and denominator. For magnitude homology of arrangements, we give combinatorial descriptions in small length and prove that tope graphs are diagonal if and only if the arrangement is Boolean. We present a face decomposition of magnitude homology, using which we obtain a combinatorial formula of diagonal magnitude Betti numbers. Many open problems are posted for future study. In particular, we conjecture that magnitude and magnitude homology of arrangements are determined by the intersection lattice.

Ryokichi Tanaka

Rough similarity rigidity via ergodic theory of topological flows

For every non-elementary hyperbolic group, we give a necessary and sufficient condition for two given word metrics to be roughly similar, i.e., they are within bounded distance after multiplying by a positive constant, in terms of mean distortion. The proof is based on the ergodic theory of topological flows associated with general hyperbolic groups. We will also mention explicit illustrative examples and applications, including the real analyticity of intersection numbers for families of dominated (Anosov) representations. This talk is based on joint work with Stephen Cantrell (St Andrews).