Properads are gadgets that are more expressive than operads, which are capable of modeling some types of bialgebras while still admitting a satisfactory theory of Koszul duality. The focus of this talk will be on a corresponding homotopy coherent notion, that of infinity properads. In particular, we will give a concrete model for these, and discuss methods for enrichment in monoidal categories and a model for (bi)algebras. This talk is based on joint work with Hongyi Chu, and with Marcy Robertson and Donald Yau.

The absolute Galois group of the field of rational numbers and the

Grothendieck-Teichmueller group introduced by V. Drinfeld in 1990 are among the most mysterious objects in mathematics.

My talk will be devoted to GT-shadows. These tantalizing objects may be thought of as “approximations”to elements of the mysterious Grothendieck-Teichmueller group. They form a groupoid and act on Grothendieck's child's drawings. Currently, the most amazing discovery related to GT-shadows is that the orbits of child's drawings with respect to the action of the absolute Galois group (when they can be computed) and the orbits of child's drawings with respect to the action of GT-shadows coincide! If time permits,I will show how to work with the software package for GT-shadows and their action on child's drawings.My talk is partially based on the joint paper https://arxiv.org/abs/2008.00066 with Khanh Q. Le and Aidan A. Lorenz.

In this talk, we investigate GK-dimension of nonsymmetric operads and prove that the GK dimension of a nonsymmetric operad falls into {0,1} ∪ [2,∞); this is a generalisation to nonsymmetric operads of the Bergman Gap Theorem for associative algebras. This is a joint work with Yongjun Xu, James Zhang and Xiangui Zhao.

This talk describes joint work with Suhyeon Lee (Berkeley), Brendan Murphy (University of Washington) and Luke Trujillo (Harvey Mudd College).I will explain the definition of a PROB, which generalizes the PROPs of Boardman and Vogt, and give several examples. The example of particular interest in this talk will be SB, the PROB of singular braids. To the authors' knowledge, the category SB has not appeared in the literature before. This PROB is the free braided monoidal category containing an object whose n-fold tensor powers admit an action of the singular braid monoid on n strands. If time allows, I will give a sketch of the associated braided operad of SB. A long term goal of this research is to use this technology to give a general framework for producing and studying invariants of singular knots and links, which may be obtained from singular braid invariants.

A Poisson algebra is a vector space $\mathcal{A}$ over a field $k$ endowed with two bilinear operations, a multiplication denoted by $x\otimes y\mapsto x\cdot y$ and a Poisson bracket denoted by $x\otimes y\mapsto (x,y)$, such that $(\mathcal{A},\cdot)$ is a commutative associative algebra, $(\mathcal{A},(-,-))$ is a Lie algebra, and $\mathcal{A}$ satisfies the Leibniz identity

$$(x, y \cdot z) = (x, y) \cdot z + y \cdot (x, z).$$

We develop a new theory of Gr\"{o}bner--Shirshov bases for Poisson algebras, which requires introducing several nontrivial new techniques. As an application, we solve the word problem for certain one relator Poisson algebras when the leading monomial of the single relation is a Lie monomial.

A Novikov algebra $(\mathcal{A},\circ)$ is a vector space with the identities

$x\circ(y\circ z)-(x\circ y)\circ z =y\circ(x\circ z)-(y\circ x)\circ z (\mbox{left symmetry}), (x\circ y)\circ z=(x\circ z)\circ y (\mbox{right commutativity}).$

A Novikov--Poisson algebra $(\mathcal{A}, \cdot,\circ)$ is a vector space with two bilinear operations $\cdot$ and $\circ$ such that $(\mathcal{A}, \circ)$ forms a Novikov algebra, $(\mathcal{A}, \cdot)$ forms a commutative associative algebra, and the following two identities holds:

$(x\cdot y)\circ z= x\cdot (y\circ z), (x\circ y)\cdot z-x\circ (y\cdot z) =(y\circ x)\cdot z-y\circ (x\cdot z), x, y, z\in \mathcal{A}.$

We construct a linear basis for a free Novikov--Poisson algebra when the algebra has a unit with respect to the commutative and associative bilinear operation. Finally, we show that every Novikov--Poisson algebra (with unit) can be embedded into its universal enveloping special Novikov--Poisson admissible algebra.

We determine the \emph{$L_\infty$-algebra} that controls deformatio- ns of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying {\bf \textmd{LieRep}} pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie$_\infty$-algebras}. This is a joint work with Andrey Lazarev and Yunhe Sheng.

Suppose $(A,d_A,\{\mu_k\})$ is an $A_\infty$-algebra over a commutative ring $R$, whose underlying chain complex $(A,d_A)$ is a deformation retract of a complex $(C,d_C)$. It is a result of Kontsevich and Soibelman that the retract data allows one to explicitly transfer the $A_\infty$-structure from $(A,d_A)$ to a homotopy equivalent $A_ \infty$-structure on $(C,d_C)$.

In this talk, I will answer a question recently posed by Dennis Sullivan concerning the characterization of transferred structures for algebras over a suitable class of operads. For the deformation retract scenario presented above, Sullivan's question can be stated as follows: Can one characterize, up to equivalence, the transferred structure as the unique $A_\infty$-structure on $C$ whose restriction to $A$ is the original $(A,d_A,\{\mu_k\})$?

This is based on joint work arXiv:2006.00072 with Martin Markl.

The embedding tensor formalism is a powerful tool in gauged supergravity theory. Each embedding tensor gives rise to a Leibniz algebra structure on the field content. We consider its homotopy counterpart which we call homotopy embedding tensor to a differential graded Lie algebra. It turns out that there associates functorially a homotopy Leibniz algebra to each homotopy class of homotopy embedding tensors. We will explain how to obtain a homotopy Leibniz algebra from a homotopy embedding tensor via a summation over labelled rooted forests.

This is a joint work Zhuo Chen and Huabin Ge.

Recently, Rota's Classification Problem on algebraic operator identities has been studied in the context of Gr\"obner-Shirshov bases and rewriting systems, giving an equivalence of the Classification Problems with the existence of the Gr\"obner-Shirshov bases and the convergence of the corresponding rewriting system, all for associative algebras with a linear operator. In this talk, we discuss the generalization of Rota's Classification Problem to general algebraic structures, namely operads, in particular nonsymmetric operads. This approach also allows us to study (disconnected) operads from the viewpoint of the Classification Problem. Applications to Koszulity of operads are obtained.

Baker-Campbell-Hausdorff (BCH) formula and several variants in Lie theory can be obtained through mould calculus which also gives easily the generalizations. I will give a brief introduction to mould theory and the proofs of BCH formula and some related formulas. The talk is based on joint work with Yong LI (Chern Institute, Tianjin) and David SAUZIN (IMCCE, Paris).

In this talk, we show that there is a twisted noncommutative symplectic structure on AS-regular algebras, where the twisting comes from the Nakayama automorphism of the algebra. This twisted symplectic structure gives an isomorphism between the derived tangent complex and the derived twisted cotangent complex of the AS-regular algebra, and when taking the commutator quotient space, we get Van den Bergh’s noncommutative Poincare duality. This talk is based on an ongoing research joint with Eshmatov and Liu.

Integrals over configuration spaces arise naturally from quantum field theories and provide links between algebra and geometry. For example, topological QFT on the circle leads to an algebraic analogue of index theorem; topological QFT on the disk leads to Kontsevich's Formality Theorem on deformation quantization. In this talk, we introduce the notion of regularized integral to formulate an analytic theory for integrals over configuration spaces of Riemann surfaces that come from 2d chiral QFT. An an application, we explain how such regularized integrals lead geometrically to modular forms and are related to certain chiral analogue of index theorem. This is joint work with Jie Zhou. Preprint available at arXiv:2008.07503.