安定∞圏とそこから派生するHochschildホモロジーやHochschildコホモロジーについての研究をいくつかの観点から研究した。背景としてはホモロジー的ミラー対称性や$E_n$代数のfactorization層などがある。２０２０年度中に発見したfactorizationホモロジーの写像スタックへの自然変形（拡大）を応用してきわめて一般的な形で定式化し一般論を整備した。そのことを用いて、安定∞圏の族から周期的巡回ホモロジー（periodic cyclic homology）の族にHodge構造の類似の構造が入ることに応用した。その構造の構成についてはもう一つの構成法を見つけている。それは、HochshcildホモロジーとHochschildコホモロジーの対に入る代数構造とそのモジュライ理論的な解釈を見つけたうえでそれを応用するもので２０２０年度に見つけていた。２０２１年度はその技術的詳細をかなり簡略化し見通しをよくすることに成功した。この結果はOn D-modules of categories I、``Moduli Theory associated to Hochschild pairs''として私のホームページ、arXivのversion2として公開中である。さらに私自身で見つけた二つの構成法は各々長所を持っており、比較することが重要となる。これら一見異なる構成たちは同じものを構成することを、Koszul双対性などを応用することによって証明した。その結果はOn D-modules of categories IIとして公開中である。さらにこれらの理論を用いてGriffiths横断性を示した。またこれらから思いついてCY安定∞圏に対するBogomolov-Tian-Todorov定理を示した。これらの結果は、論文準備中である。

In this research project, I have studied the Hpdge-theoretic invariants arising from a family of stable infinity-categorties. I would like to summerize the resluts which I obtained. Let us conisder the Hochschild pair. I gave a conceptual and simple constructuon of the algberaic structure on the Hochschild pair associated to a stable infinity-category. Moreover, I discovered the moduli-theoretic interpretation of this algebraic structure on the Hochschild pair. I have constructed D-module structure on the periodic cyclic homology arising from a family of stable infinity-categories. I discovered two methods. The first method is an application of the Hochshicla pair. The second method is to use the canonical extension of factorization homology.

The research representative and Go Yamashita have constructed families of Wach modules of rank two and applied them to the study of crystalline deformation rings of dimension two. He and Satoshi Kondo have constructed lifts of the zeta elements in motivic cohomologies of Drinfeld modular varieties to their integral models satisfying norm relations, and have constructed a theory of topoi related to monoids. He and Yusuke Sugiyama have introduced a new notion of pseudo-tameness and, by using them, have proved that any algebraic curve over an algebraically closed field has a tame morphism to the projective line. He has introduced the derived double shuffle spaces and has applied them to show a double shuffle analogue of Broadhurst-Kreimer conjecture in depth four. He has found that a suitable quotient of a Hilber modular surface related to the L-function of a certain curve of genus is a Kummer surface.

The principal purpose of this program is to study a duality of tannakian type for higher categories and to apply it to various theory such as mixed motives. I proved a tannakian characterization theorem for symmetric monoidal stable infinity-categories that satisfy a certain simple condition (so-called fine tannakian infinity-categories). I applied this theory to mixed motives to obtain motivic Galois stacks and associated motivic Galois group. I also define a motivic rational homotopy type and its relation with motivic Galois actions and related notions. I applied the tannaka duality theory to motivic rational homotopy types.

In 2011-2012, we obtained several results concerning tannaka duality type theorems towards applications to mixed motives. Some results are purely categorical and derived algebraid theoretic, and others are about motives. One algebraic powerful machinery I constructed is tannakization in the realm of derived algebraic geometry. Applying it to motivic situations we constructed derived and underived motivic Galois groups. Moreover, I found a refined tannaka duality type theorem which are well-suited to motivic applications and studied the structure of motivic Galois groups for the cases where one cannot use techniques in mixed Tate motives .