Paul Bressler (U. de los Andes in Bogota)

- Title: Atiyah class d'après M.Kapranov

- Abstract: I will describe a construction of a canonical L-infinity structure on the (shifted by -1) tangent bundle and its canonical representations. Time permitting, I will describe a generalization of above picture to pairs of Lie algebroids.

Andrei Caldararu (U. of Wisconsin)

- Title: Functoriality of HKR isomorphisms and orbifold Hochschild products

- Abstract: There are two generalizations to arbitrary embeddings of the classical HKR isomorphism (which refers to the diagonal embedding).  They are due to Arinkin-Caldararu in the first form, and to Arinkin-Caldararu-Hablicsek and Grivaux (independently) in the second form. N one of these works addressed the question of whether these isomorphisms are functorial, in the case of a flag of two consecutive embeddings. I will discuss some surprising results in the study of this question, and I will conclude with applications to the construction of an associative product on the polyvector field cohomology of a quotient [X/G] for X smooth and G abelian. This is joint work with Shengyuan Huang.

Zhuo Chen (Tsinghua U.)     [ Lecture Note ]

- Title: Hopf algebras arising from dg manifolds

- Abstract: Let (M,Q) be a dg manifold. The space of shifted vector fields (X(M)[−1],L_Q) is a Lie algebra object in the homology category H(dg−mod) of dg modules over (M,Q), the Atiyah class α_M being its Lie bracket. The triple (X(M)[−1],L_Q;α_M) is also a Lie algebra object in the Gabriel-Zisman homotopy category Π(dg−mod). In this talk, we describe the universal enveloping algebra of (X(M)[−1],L_Q;α_M) and prove that it is a Hopf algebra object in Π(dg−mod). As an application, we study Fedosov dg Lie algebroids and recover a result of Chen, Stiénon and Xu on the Hopf algebra arising from a Lie pair. This is a joint work with JH Cheng and DD Ni.

Bumsig Kim (KIAS)

- Title: Atiyah classes and Chern characters for global matrix factorizations

- Abstract: The talk is based on a joint work with Alexander Polishchuk. We define the Atiyah class for a global matrix factorization and use it to give a formula for the categorical Chern character and the boundary-bulk map for matrix factorizations, generalizing the formula due to Caldararu. Our approach is based on developing the Lie algebra analogies observed by Kapranov and Markarian.

Honglei Lang (China Agricultural U.)     [ Lecture Note ]

- Title: Atiyah classes for generalized holomorphic vector bundles

- Abstract: For a generalized holomorphic vector bundle, we construct the Atiyah class, which is the obstruction of the existence of generalized holomorphic connections. Similar to the holomorphic case, such Atiyah classes can be defined by three approaches: the Cech cohomology, the extension class of the first jet bundle as well as the Lie pair. This is a joint work with Xiao Jia and Zhangju Liu.

Hsuan-Yi Liao (KIAS)     [ Lecture Note ]

- Title: Kontsevich--Duflo type theorem for dg manifolds
- Abstract: In this talk, we describe a Kontsevich--Duflo type theorem for dg manifolds. The Duflo theorem of Lie theory and the Kontsevich theorem regarding the Hoschschild cohomology of complex manifolds can both be derived as special cases of this Kontsevich--Duflo type theorem for dg manifolds. This is a joint work with Mathieu Stienon and Ping Xu.

Daniel Murfet (U. of Melbourne)     [ Lecture Note ]

- Title: A-infinity categories of matrix factorisations via A-infinity idempotents

- Abstract: I'll present a new point of view on the process of constructing an A-infinity model of a particular differential graded category, the category of matrix factorisations, which emphasises the role of A-infinity idempotents and Atiyah classes. In contrast to earlier work in this context, which has focused on the standard generator and its endomorphism algebra, we care about all objects and potentials over arbitrary base rings.

Jae-Suk Park (POSTECH)

- Title: Homotopy Equivariant hbar-Superconnection and Quantum Field Theory

- Abstract: I will explain that the every quantum correlation function and its higher analogues of a quantum field theory are governed by the coalgebraic analogue of certain $L_infty$-homotopy equivariant hbar-Superconnection.

Alexey A. Sharapov (National Research Tomsk State U.)     [ Lecture Note ]

-Title: Cup Product and Deformations of A-infinity algebras

- Abstract: Starting from some motivating physical examples, I am going to discuss the problem of formal deformation of A-infinity algebras. More specifically, I will present a simple method for the deformation of multi-parameter families of A-infinity algebras. Central to the method is the concept of cup-product on A-infinity cohomology, which we define in terms of brace operations on the Hochschild complex. To exemplify the general method, I will consider formal deformations of dg-algebras in the category of minimal A-infinity algebras, including dg-algebras associated with certain symplectic reflection algebras.

Mathieu Stienon (Penn State U.)     [ Lecture Note ]

- Title: Atiyah class of a dg vector bundlerelative to a dg Lie algebroid

Boris Tsygan (Northwestern U.)

- Title: Topics in deformation quantization

Arkady Vaintrob (U. of Oregon)

- Title: Homological vector fields and Atiyah classes

- Abstract: Many geometric and algebraic objects can be described and studied in terms of homological vector fields (HVFs), integrable odd vector fields on supermanifolds. Examples include complex and generalized complex structures, Lie algebras and Lie bialgebras, Poisson manifolds, and homotopy algebroid actions.  Atiyah classes provide another family of constructions bridging geometry and Lie algebra type structures. I will talk about Atiyah classes in the context of HVFs. In particular I will define them for HVFs and their pairs and show how this helps to clarify and extend some earlier constructions of Atiyah classes.

Simon Willerton (U. of Sheffield)     [ Lecture Note ]

- Title: Hopf Monads, Hopf algebras and diagrammatics

- Abstract: [This is joint work with Christos Aravanis.]  The Atiyah class equips the shifted tangent sheaf of a complex manifold with the structure of a Lie algebra and the derived category is in a certain sense the representation category of this.  This Lie algebra has a universal enveloping algebra U in the derived category which can be defined in terms of a push forward and pull back of the structure sheaf along the diagonal embedding.  This is formally analogous to the group algebra of a finite group in its representation category.  I will show how this can be used to equip U with the structure of a Hopf algebra using the technology of Hopf monads (due to Bruguiere, Lack and Virelizier).  The key to the existence of the antipode is the projection formula.  I will show explicitly how to prove the projection formula holds in this case using a diagrammatic approach.

Maosong Xiang (Huazhong U. of Science & Technology)     [ Lecture Note ]

- Title: Atiyah classes and Hochschild cohomology of integrable distributions

- Abstract: Each integrable (or involutive) distribution of a smooth manifold gives rise to a dg manifold and a Lie algebroid pair. In this talk, we will first discuss about how to identify the Atiyah and Todd classes of this dg manifold to those of the corresponding Lie algebroid pair. Then we will show that the Hochschild cohomology of the previous dg manifold and Lie algebroid pair are isomorphic as Gerstenhaber algebras. In particular, we recover the Kontsevich-Duflo theorem for complex manifolds in terms of dg manifolds. This is a joint work in Zhuo Chen and Ping Xu.