スケジュール
1月7日（日）
13:30 -- 14:00 山縣真（福岡大学）
Some remarks on invariant rings under the actions of reflection groups associated to the Weyl groups of symplectic groups
14:15 -- 15:15 荒川研資（京都大学）
Classification diagram of marked simplicial sets
15:30 -- 16:30 箕輪悠希（京都大学）
On the parametrized topological complexity of sphere bundles over spheres
16:45 -- 17:45 岸本大祐（九州大学）
Tight complexes are Golod
1月8日（月）
9:30 -- 10:30 Yichen Tong（京都大学）
Homotopy commutativity in Riemannian symmetric spaces
10:45 -- 11:45 加藤諒（高知工科大学）
An analogue of the dichotomy conjecture on monoidally distributive posets
アブストラクト
荒川研資
Classification diagram of marked simplicial sets
The process of freely adjoining inverses, known as localization of categories, is one of the most mysterious and curious aspects of category theory. On the one hand, a good understanding of localizations is indispensable in many areas of math; the theory of derived categories is a good example. On the other hand, computing localization is notoriously hard. In this talk, we will see that things get even more interesting in higher category theory by looking at examples of localizations of ∞-categories. After that, we will explain Mazel-Gee's theorem on localizations and classification diagrams, which allows us to access localizations of ∞-categories in some cases, and give its new proof found by the author.
加藤諒
An analogue of the dichotomy conjecture on monoidally distributive posets
Hovey proposed the dichotomy conjecture on the stable homotopy category of spectra. Hovey-Palmieri proved many interesting facts around the dichotomy conjecture from the viewpoint of the Bousfield lattice. Kato-Shimomura-Tatehara defined the notion of monoidally distributive posets as a generalization of the Bousfield lattice. In this talk, we consider an analogue of the dichotomy conjecture on monoidally distributive posets, and prove several results around the analogue.
岸本大祐
Tight complexes are Golod
Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into a Euclidean space, studied in differential geometry. Golodness is a property of a noetherian ring, defined in terms of the Poincare series of its Tor algebra, and Golodness of a simplicial complex is defined by that of the Stanley-Reisner ring. Recent results on polyhedral products suggest connection between these two notions for manifold triangulations, and in 2023, Iriye and I proved that they are equivalent for 3-dimensional manifold triangulations. In this talk, I will present that tight complexes are always Golod, which implies Golodness and tightness are equivalent for all manifold triangulations. The main technical ingredient is the higher prism operator.
This is a joint work with Kouyemon Iriye.
Yichen Tong
Homotopy commutativity in Riemannian symmetric spaces
A fundamental problem on H-spaces is to find whether or not a given H-space is homotopy commutative. In particular, there are few answers for loop spaces of simply-connected finite complexes, including those of homogeneous spaces. Ganea considered complex and quaternion projective spaces in 1967, and Kishimoto, Takeda and the speaker considered all irreducible Hermitian symmetric spaces in 2022. In this talk, I would like to introduce a recent result on all irreducible Riemannian symmetric spaces, which extends the previous ones. In particular, we developed a tool to deal with AII=SU(2n)/Sp(n) and EIV=E_6/F_4, to which the techniques for other cases fail to apply. As a corollary, this tool also determines whether a complex Steifel manifold possesses an H-structure or not. If time permits, I will also share my further plan on measuring the non-commutativity of loop spaces of symmetric spaces by Samelson products.
This is a joint work with Daisuke Kishimoto, Yuki Minowa, and Toshiyuki Miyauchi.
箕輪悠希
On the parametrized topological complexity of sphere bundles over spheres
Let p:E-->B$ be a fibration, and let E^I_B be a space of paths γ:I-->E such that p\circ γ is a constant path. The parametrized topological complexity of p is defined as the sectional category of the evaluation fibration E^I_B-->E \times_B E. It was first introduced by Cohen, Farber and Weinberger and they gave a cohomological lower bound. In this talk, I will talk about the parametrized topological complexity of sphere bundles over spheres. More precisely, I will give a lower bound using the weak category, and then I will show that the lower bound can be greater than the cohomological one. I will also talk about the determination of the parametrized topological complexity in some cases.
山縣真
Some remarks on invariant rings under the actions of reflection groups associated to the Weyl groups of symplectic groups
The modular representations of some Weyl groups are considered. If two compact connected Lie groups are locally isomorphic, the complex representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod 2 reductions, we consider the structure of invariant rings related to symplectic groups.