日時 2025年1月11日（土）ー 12日（日）
会場 九州大学西新プラザ（アクセス）
講演者
荒川研資，下村磨生将，野坂武史，Yichen Tong，南範彦，山形颯
スケジュール
1月11日（土）
13:20 -- 14:20 南範彦（大和大学, OCAMI）
高次余次元有理不変量とそれらがホモトピー論に現れる例たち
(Higher codimensional birational invariants arising in homotopy theory)
14:30 -- 15:30 山形颯（福岡大学）
Categorification of polynomial invariants of matroids
15:40 -- 16:40 荒川研資（京都大学）
Axiomatic approach to symmetric sequences and infinity operads
16:50 -- 17:50 野坂武史（東京科学大学）
Continuous cocycles of some gauge groups and diffeomorphisms groups from Bott-maps
1月12日（日）
9:30 -- 10:30 下村 磨生将（高知大学）
Notes on Greek letter elements
10:45 -- 11:45 Yichen Tong（Westlake University）
On the fundamental group and the magnitude-path spectral sequence of a directed graph
アブストラクト
南範彦
高次余次元有理不変量とそれらがホモトピー論に現れる例たち
(Higher codimensional birational invariants arising in homotopy theory)
In this talk, I shall present a theorem, which immediately leads to a pedagocial student-friendly problem for classical homotopy theory major students, allowing them to apply classical homotopy theory to algebraic geometry, and, thereby, facilitating them to understand algebraic geometry.
This pedagocial student-friendly problem is just the computation of the cokernel of the Thom reduction map from the complex cobordism to the integral cohomology for complex projective smooth varieties, for which this theorem endows with some higher codimensional birational invariant interpretations.
For the Godeaux-Serre varieties, such computations are reduced to the case of more familiar (for homotopy theorists) classifying spaces of finite groups. This theorem is a tip of an iceberg of my more general results on higher codimensional birational invariants.
(This talk completely supersedes and upgrades all of my recent talks, including the one at Kochi Autumn Workshop in September 2024, on similar subjects.)
山形颯
Categorification of polynomial invariants of matroids
Khovanov introduced a bigraded cohomology theory of links, whose graded Euler characteristic is the Jones polynomial. Analogously, several constructions of the Khovanov-type (co)homology theories have been provided beyond the knot theory, such as the chromatic cohomology of graphs and the characteristic homology of hyperplane arrangements. A matroid is a structure that reflects the notion of abstract dependency, including cycles in graphs and linear dependency of vectors. In particular, we can obtain matroids from both graphs and arrangements of hyperplanes. In this talk, we provide (co)homology groups of the characteristic polynomial of matroids as a generalization of the chromatic cohomology and characteristic homology of hyperplane arrangements.
This talk is based on the joint work with Takuya Saito.
荒川研資
Axiomatic approach to symmetric sequences and infinity operads
A symmetric sequence in a category C is a functor from the category of finite sets and bijections into C. When C is closed symmetric monoidal and cocomplete, the category of symmetric sequences inherits a monoidal structure with the composition product as its tensor product. Operads in C then correspond to monoids under this monoidal structure.
One way to define operads in the infinity categorical setting is to generalize the composition product monoidal structure and define infinity operads as monoids in it. However, there are several, equally appealing ways to generalize the monoidal structure, and their comparison has proven difficult. It is therefore unclear whether these different generalizations lead to equivalent theories of infinity operads.
In this talk, we show that the specific encoding of the composition product monoidal structure does not affect the resulting theory of infinity operads. More precisely, we show that any choice of the monoidal structure satisfying natural axioms leads to an equivalent theory of infinity operads.
野坂武史
Continuous cocycles of some gauge groups and diffeomorphisms groups from Bott-maps
The topology of gauge groups and of diffeomorphism groups of manifolds has been studied from many approaches with several motivations and applications. In this talk, we focus on the continuous group cohomology of the groups, and suggest new procedure of constructing continuous cocycles of the groups in some strong conditions. The idea of the construction is to regard the higher transgressions of the Chern-Simons forms via a Bott map as a cocycle condition using the bordism groups. I will discuss some relations to Gelfand-Fucks cohomology and Bott-Thurston cocycles, and present some non-trivial cocycles of some groups.
下村 磨生将
Notes on Greek letter elements
The stable homotopy groups of spheres have important generators, which are called Greek letter elements. These generators play an important role in stable homotopy theory, and we have many open problems around them. In recent works, we propose some approaches to these problems. We talk about these approaches.
Yichen Tong
On the fundamental group and the magnitude-path spectral sequence of a directed graph
Directed graphs are crucial combinatorial objects and are of great interest in both pure and applied mathematics. In particular, methods from algebraic topology, especially homotopy theory, have been introduced to study (directed) graphs. The fundamental group and the path homology of a directed graph were introduced by Grigor’yan, Lin, Muranov, and Yau, and are related through the Hurewicz theorem. Later, Di, Ivanov, Mukoseev, and Zhang showed that the fundamental group admits a natural sequence of quotient groups called r-fundamental groups, which captures the quasi-metric structure of a directed graph that the fundamental group cannot capture. On the other hand, the magnitude-path spectral sequence, due to Asao, connects magnitude homology and path homology of a directed graph, and it may be thought as a sequence of homology of a directed graph. In this talk, we study relations of the r-fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem.
This is joint work with Daisuke Kishimoto.
世話人 浅尾泰彦（福岡大学） 石黒賢士（福岡大学） 岸本大祐（九州大学） 蔦谷充伸（九州大学） 宮内敏行（福岡大学）
資金 基盤研究 (C) 「ポリヘドラルプロダクトのトポロジーと組み合わせ構造の研究」(研究代表者：岸本大祐，研究課題番号：22K03284)
基盤研究 (C) 「J-準同型による球面の非安定ホモトピー群の大域構造の研究」(研究代表者：宮内敏行，研究課題番号：22K03326)