Higher structures and deformations in representation theory

研究建议集中在Hochschild产生的更高代数结构，以及相关的上同调和更高范畴结构，如A-无穷范畴。我的目标是开发新的和先进的技术，适用于代数、表示论、拓扑学和几何学中的广泛问题。在过去的几年里，我一直在研究由奇异Hochschild上同调引起的更高代数结构。我已经证明了奇异Hochschild上同调具有与经典Hochschild上同调相同的丰富结构：上同调中的Gertenhaber代数和复水平上的B-无穷代数。由此，Keller证明了奇异Hochschild上同调与dG奇点范畴的Hochschild上同调是同构的。他还推测，这种同构提升为B-无穷大代数的拟同构。我的研究提案的第一个目标是证明上面提到的凯勒猜想。为了实现这一目标，与X-W。我们引入了奇点范畴的一个显式dg增强。在这个方案中，我们计划构造奇异Hochschild上同调上的B-无穷结构在dG增强上的自然作用，这将产生B-无穷大准同构提升Keller同构。第二个目标是进一步探索奇异Hochschild上同调在拓扑学和几何学中的应用。我们与M.Rivera一起证明了任意dg Frobenius代数的奇异Hochschild上同调都有丰富的高阶代数结构。在本提案中，我们计划展示这种更高的结构实际上来自Sullivan dg道具的自然作用，该道具最初用于在字符串拓扑中建模操作。这一建议还涉及(分次)代数的(A-无穷)变形及其在表示论中的应用。最近，我们和S.Barmeier一起发展了一种组合方法来研究任意有关系的有限箭图的路代数的形变理论，这使得我们可以很容易地计算特定类别的例子。第三个目标是在Keller猜想的推动下，理解奇点范畴在形变量化下的行为。特别地，我们期望循环商奇点的奇点范畴与自然泊松结构在奇点上的形变量化是等价的。第四个目标是用分次柔代数的曲面模型来刻画其导性等价的A-无穷变形。第五个目标是利用我们的组合方法研究和验证Stroppel关于扩展的Khovanov弧代数的猜想，从而得到这些代数的内在形式化.

The research proposal is focusing on higher algebraic structures arsing from Hochschild, and related, cohomology and on higher categorical structures such as A-infinity categories. I aim to develop new and advanced techniques applicable to a wide range of problems in algebra, representation theory, topology, and geometry. Over the past few years, I have been investigating higher algebraic structures arising from singular Hochschild cohomology. I have shown that singular Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber algebra in cohomology and a B-infinity algebra at the complex level. Suggested by this result, Keller proved that singular Hochschild cohomology is isomorphic to the Hochschild cohomology of the dg singularity category. He also conjectured that this isomorphism lifts to a quasi-isomorphism of B-infinity algebras. The first objective in my research proposal is to prove Keller's conjecture stated above. To achieve this, jointly with X. -W. Chen we introduced an explicit dg enhancement of singularities categories. In this proposal, we plan to construct a natural action of the B-infinity structure on singular Hochschild cohomology on this dg enhancement, which will yield a B-infinity quasi-isomorphism lifting Keller's isomorphism. The second objective is to explore further applications of singular Hochschild cohomology in topology and geometry. Jointly with M. Rivera we have shown that there is a rich higher algebraic structure on the singular Hochschild cohomology of any dg Frobenius algebra. In this proposal, we plan to show that this higher structure actually comes from a natural action of the Sullivan dg PROP, originally used to model operations in string topology. This proposal also concerns (A-infinity) deformations of (graded) algebras and their applications to representation theory. Recently jointly with S. Barmeier we have developed a combinatorial method to study the deformation theory of a path algebra of any finite quiver with relations, which allows us to easily compute particular classes of examples. The third objective aims to understand the behaviour of singularity categories under deformation quantisation, motivated by Keller's conjecture. Particularly, we expect an equivalence between singularity categories of a cyclic quotient singularity and of the deformation quantisation of a natural Poisson structure on the singularity. The fourth objective is to describe the A-infinity deformations of graded gentle algebras up to derived equivalence, in terms of their surface models. In particular, we expect that some A-infinity deformations correspond to doing surgery on the surface models.The fifth objective is to use our combinatorial method to study and verify Stroppel's conjecture for extended Khovanov arc algebras, which yields the intrinsic formality of these algebras.