2019 Workshop on Deformation Theory and Homotopy Algebra, Southwest Jiaotong UniversityEmei Campus,May 611,2019.
OnSite Registration: 2:00pm6:00pm, 5 May(Sunday), Hushan Hotel Openning Ceremony: 8:30am9:00am, 6 May(Monday), No.5 conference room.

May 6 Monday 
May 7 Tuesday 
May 8 Wednesday 
May 9 Thursday 
May 10 Friday 
May 11 Saturday 
9:0010:00 
Kai Behrend 
Free Discussion 
Ralph Kaufmann 
Alberto Cattaneo 
Si Li 
Free discussion 
10:0010:30 
Tea break 
Tea break 
Tea break 
Tea break 
10:3011:30 
Zhuo Chen 
Chengming Bai 
Guodong Zhou 
Yuri Berest 
15:0016:00 
Andrey Lazarev 
Raphael Ponge 
Francesco Bonechi 
Free Discussion 
16:0016:30 
Tea break 
Tea break 
Tea break 
16：3017:30 
Mathieu Stienon 
Yu Ye 
Junwu Tu 
Title and Abstract
Chengming Bai (Chern institute of Mathematics)
Title: Deformations and their controlling cohomologies of $\mathcal O$operators
Abstract: We establish a deformation theory of a kind of linear operators, namely, $\mathcal O$operators in consistence with the general principles of deformation theories. On one hand, there is a suitable differential graded Lie algebra whose MaurerCartan elements characterize $\mathcal O$operators and their deformations. On the other hand, there is an analogue of the Andr\'eQuillen cohomology which controls the deformations of $\mathcal O$operators. Infinitesimal deformations of $\mathcal O$operators are studied and applications are given to deformations of skewsymmetric $r$matrices for the classical YangBaxter equation. This is a joint work with Li Guo, Yunhe Sheng and Rong Tang.

Kai Behrend(University of British Columbia )
Title：dgmanifolds form a category of fibrant objects
Abstract：This is a report on very recent work in progress (joint with HY Liao and P Xu) proving the theorem in the title. The purpose is to embed differentiable manifolds in a context where homotopy theory is possible: for example, the fibre products and intersections of differentiable manifolds, which do not exist in the category of differentiable manifolds, do exist as homotopy fibered products in the category of fibrant objects I will describe. Furthermore, deformation theory, by which I mean the homotopy theory of differential graded Lie algebras, is embedded in this context as well. The hope is that this will lead to a simpler context for derived differentiable topology than other more involved constructions. The proof is an application of the transfer theorem for curved Linfinity algebras.

Yuri Berest (Cornell University)
Title: Derived algebraic groups
Abstract: Classically, an affine algebraic group (or affine group scheme) G over
a field k is defined by its functor of points, which is a representable groupvalued functor on the category of commutative kalgebras. It is known that the natural (degreewise) extension of G to the category of simplicial commutative algebras has, unfortunately, poor homotopical properties: in particular, it is not a homotopy invariant functor. In the literature, going back to the classical work of R.Swan and F.Waldhausen, one can find different ways to remedy this problem by giving different homotopical approximations of G. In this talk, we address the question of how to construct and compare such approximations. This question is motivated by
the recent work of S. Galatius and A. Venkatesh on derived Galois deformation rings
and my joint work with A.C.Ramadoss and W.k.Yeung on representation homology of simplicial groups and topological spaces.

Francesco Bonechi (INFN)
Title: Poisson structures on differentiable stacks
Abstract: Quasi Poisson groupoids are bivector fields on Lie groupoids that are compatible with the groupoid multiplication and satisfy the Jacobi identity in a homotopical sense. We discuss their properties with respect to Morita equivalence. More generally, we introduce the graded 2Lie algebra of polyvector fields and its Morita invariance. This is a joint work with N.Ciccoli, C.LaurentGengoux and P.Xu.

Alberto Cattaneo(Universität Zürich)
Title: An Introduction to the BVBFV Formalism
Abstract: The BVBFV formalism unifies the BV formalism (which deals with the problem of fixing the gauge of field theories on closed manifolds) with the BFV formalism (which yields a cohomological resolution of the reduced phase space of a classical field theory). I will explain how this formalism arises and how it can be quantized.

Zhuo Chen (Tsinghua University)
Title: SH Leibniz algebra from homotopic embedding tensors
Abstract: The embedding tensor appears in the gauging procedure of supergravity theories. Kotov and Strobl prove that there exists a correspondence between embedding tensors and Leibniz algebras, and show that the associated tensor hierarchy only depends on the corresponding Leibniz algebra. In this talk, we consider a homotopic version of embedding tensors in the context of dg geometry and derive SH(strongly homotopy) Leibniz algebras from homotopic embedding tensors. Several interesting examples will be discussed. This is a joint work with M.Xiang and T.Zhang.

Ralph Kaufmann(Purdue University)
Title: Deformations, Hopf algebras and Feynman categories.
Abstract: There are several Hopf algebras arising from in physics, number theory and geometry whose existence can be traced back to simplicial structures, cooperads and Feynman categories. We review these constructions which yield the Hopf algebras governing multizeta values, chains on double loop spaces and renomalization. We then discuss deformations in this context. They first appear in a noncommutative setting. Here the Hopf algebras can be seen as a limit point of a deformation of bialgebras in a formal parameter q. A second deformation appears, when considering more general multiplications in the bialgebra. This leads to considerations of filtration and associated graded objects. The result here is that the under given conditions the general Hopf/bialgebras are deformations of a quotient of
a free graded construction.

Andrey Lazarev (Lancaster University )
Title: Homotopy theory of monoids
Abstract: I will explain how the category of discrete monoids models the homotopy category of connected spaces. This correspondence is based on derived localization of associative algebras and could be viewed as an algebraization result, somewhat similar to rational homotopy theory (although not as structured). Time permitting, I will describe one application of this circle of ideas, namely a simple proof of a generalization of Adams’s cobar construction to general nonsimply connected spaces obtained recently by HessTonks and RiveraZeinalian using different methods. This is a joint with J. Chuang and J. Holstein.

Si Li (Tsinghua University )
Title: LodayQuillenTsygan Theorem and Large N gauge theory
Abstract: We describe the coupling of holomorphic ChernSimons theory at large N with Kodaira Spencer gravity. The 1st order deformation is realized by the LodayQuillenTsygan Theorem on the Lie algebra cohomology of large N matrices. We show that the dynamics of KodairaSpencer gravity is fully recovered from this large N holomorphic ChernSimons theory. This gives a construction of quantum openclosed Bmodel. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for openclosed topological Bmodel. At one loop this anomaly cancellation is analogous to the GreenSchwarz mechanism. As an application, we introduce a type I version of KodairaSpencer theory in complex dimension 5 and show that it can only be coupled to holomorphic ChernSimons theory with gauge group SO(32) at quantum level. This coupled system is conjectured to be a supersymmetric localization of type I string theory.

Mathieu Stienon(Pennsylvania State University)
Title: Atiyah class and Todd class of dg manifolds
Abstract: Exponential maps arise naturally in the contexts of Lie theory and of smooth manifolds. The infinite jets of these exponential maps are related to the PoincaréBirkhoffWitt isomorphism and the complete symbols of differential operators. We will discuss how these exponential maps can be extend to the context of dg manifolds. As an application, we will describe a natural Linfinity structure associated with the Atiyah class of a dg manifold.

Raphael Ponge (Sichuan University)
Title: Cyclic homology of crossedproduct algebras
Abstract: There is a great amount of work on the cyclic homology of crossedproduct algebras, by BaumConnes, BrylinskiNistor, Connes, Crainic, ElliottNatsumeNest, FeiginTsygan, GetzlerJones, Nest, Nistor, among others. However, at the exception of the characteristic map of Connes from the early 80s, we don’t have explicit chain maps that produce isomorphisms at the level of homology and provide us with geometric constructions of cyclic cycles in the case of group actions on manifolds or varieties.
The aim of this talk is to present the construction of explicit quasiisomorphisms for crossed products associated with actions of discrete groups. Along the way we recover and clarify various earlier results (in the sense that we obtain explicit chain maps that yield quasiisomorphisms). In particular, we recover the spectral sequences of FeiginTsygan and GetzlerJones, and derive an additional spectral sequence.
In the case of group actions on manifolds we have an explicit description of cyclic homology and periodic cyclic homology. In the finite order case, the results are expressed in terms of what we call ``mixed equivariant homology”, which interpolates group homology and de Rham cohomology. This is actually the natural receptacle for a cap product of group homology with equivariant cohomology. As a result taking cap products of group cycles with equivariant characteristic classes naturally gives to a geometric construction of cyclic cycles. For the periodic cyclic homology we recover earlier results of Connes and BrylinskiNistor via a Poincar\'e duality argument. For the nonperiodic cyclic homology the results seem to be new. In the infinite order case, we fix and simplify the misidentification of cyclic homology by Crainic.
In the case of group actions on smooth varieties we obtain the exact analogues of the results for group actions on manifolds. In particular, in the special case of finite group actions on smooth varieties we recover recent results of BrodzkiDaveNistor via the construction of an explicit quasiisomorphism.

Junwu Tu(ShanghaiTech University)
Title: The BogomolovTianTodorov Theorem of cyclic Ainfinity algebras
Abstract: Let A be a finitedimensional smooth unital cyclic Ainfinity algebra. Assume furthermore that A satisfies the HodgetodeRham degeneration property. We prove the noncommutative analogue of the BogomolovTianTodorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of A is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of A is also smooth.

Yu Ye(University of Science and Technology of China )
Title: Clifford deformation and noncommutative quadrics
Abstract: In this talk, we will introduce Clifford deformations for Koszul Frobenius algebras, which correspond to quadric hypersurface algebras via Koszul duality. It turns out that a quadric ypersurface is a noncommutative isolated singularity if and only if the corresponding Clifford deformation algebra is semisimple as a $Z_2$graded algebra. We also recover Knorrer's Periodicity theorem for quadric hypersurfaces without using matrix factorizations. This is a joint work with Jiwei He.

Guodong Zhou(East China Normal University )
Title: Cohomology of lower dimensional Poisson algebras with applications to Happel's question
Abstract: We will talk about two results about computation of Poisson cohomology. In the first part, following P. Monnier, we compute the Gerstenhaber algebra structure over the Poisson cohomology of certain isolated singularities in dimension two. In the second part, we show how to use the interplay between Poisson cohomology and Hochschild cohomology to construct counterexamples to Happel's question. This talk is based on ongoing joint work with ZiHao Qi.