講演者 赤坂奎茉，前川拓海，箕輪悠希，柳田幸輝，Ye Liu
スケジュール
12月20日（土）
14:00 -- 14:30 赤坂奎茉（千葉大学）
(Op)lax Twisted Arrow (∞,n)-categories and its Applications to Condensations
14:50 -- 15:50 箕輪悠希（京都大学）
On the topological complexity of spaces with nontrivial fundamental groups
16:10 -- 17:10 Ye Liu（Xi'an Jiaotong-Liverpool University）
Holonomy Lie algebra of a geometric lattice
12月21日（日）
9:30 -- 10:30 前川拓海（東京大学）
Seiberg-Witten stable homotopy invariants via (∞,2)-functorial six-operations
10:45 -- 11:45 柳田幸輝（佐賀大学）
Dijkgraaf-Witten invariant in topological K-theory
アブストラクト
赤坂奎茉（千葉大学）
The Twisted Arrow (∞,n)-category and the Cospan (∞,n)-category
For an (∞-)category C, one can construct its twisted arrow category TwAr(C). For a 2-category, one may further relax commutativity by allowing (op)lax squares, Combining these two ideas, we are led to the notion of (op)lax twisted arrow (2-)categories TwAr(C)^{(op)lax}. In this work, we define (op)lax twisted arrow constructions for general (∞,n)-categories.
As an application, we show that these constructions allow one to reconstruct condensations, which in turn provide a convenient framework for describing idempotent completions of (∞,n)-categories.
前川拓海（東京大学）
Seiberg-Witten stable homotopy invariants via (∞,2)-functorial six-operations
Based on the 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) oo-categorical adjunction. After Gaitsgory-Rozenblyum, these six-operations fit into a single (oo,2)-functor out of the 2-category of correspondences. In this talk, we will present 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 closed spin c four-manifold, introduced by Furuta and Bauer, does correspond to a 2-morphism in that (oo,2)-functoriality.
箕輪悠希（京都大学）
On the topological complexity of spaces with nontrivial fundamental groups
アブストラクト
柳田幸輝（佐賀大学）
Dijkgraaf-Witten invariant in topological K-theory
The Dijkgraaf–Witten (DW) invariant is a topological invariant of oriented closed 3-manifolds, defined by using the ordinary homology of the classifying space of a finite group. As its K-homological analogue, we consider KDW invariants. In this talk, I will explain a representation-theoretic method to compute KDW invariants and present explicit computations for several Brieskorn homology spheres with PSL_2 (F_p). These are the first explicit computations of DW invariants in cases where both the finite group and the fundamental group of the manifolds are non-nilpotent.
Ye Liu（Xi'an Jiaotong-Liverpool University）
Holonomy Lie algebra of a geometric lattice
Motivated by T. Kohno’s result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice and a free Lie subalgebra. This yields the structure of the holonomy Lie algebra of a finite hypersolvable (including supersolvable) lattice. As applications, we obtain the structure of the holonomy Lie algebra of (the Salvetti complex of) a supersolvable oriented matroid, and that of a hypersolvable arrangement.
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) 「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)