Conference Timetable

Conference Abstracts

A. Khoroshkin: Contractads and Generating Series of Cohomology of Generalized Configuration Spaces

I will introduce a new operadic-type algebraic structure called a "contractad," where the space of operations is indexed by connected graphs, and compositions correspond to the contraction of connected subgraphs. A key example of a contractad arises from generalized configuration spaces on $R^n$, where the edges of a graph dictate which points cannot coincide. I will show how this algebraic framework can be utilized to extract information about the Hilbert series of cohomology rings.

In addition, Hamiltonian paths and acyclic orientations provide combinatorial examples of contractads, and I will present surprising formulas for the appropriate generating series for complete multipartite graphs.

L. Positselski: Exact and Abelian DG-categories

Abelian or exact DG-categories are a setting in homological algebra generalizing both (1) complexes in abelian/exact categories and (2) DG-modules, sheaves of DG-modules, curved DG-modules, etc. An exact DG-category has two underlying exact categories; for the abelian DG-category of DG-modules, these are the abelian category of DG-modules and closed morphisms of degree 0 between them, and the abelian category of graded modules and morphisms of degree 0.

I will explain the construction of an "almost involution" on DG-categories, which switches the two underlying additive categories and allows to assign, e.g., to the DG-category of DG-modules, the additive/abelian category of graded modules.

D. Bashkirov: Operadic Structures in Some 2-Dimensional Lattice Models

The talk concerns an approach to studying the symmetry, self-similarity, and renormalization properties of some lattice models, such as the 2d Ising model, from an operadic perspective. We will discuss how a variant of the framed little 2-disks operad can be used to describe the combinatorics of spin domain distributions and explore an operad governing a certain bootstrap percolation process on a 2d grid. The latter leads to an operad structure on bistochastic matrices.

D. Blanc: The Filtered Model for Rational Differential Graded Lie Algebras, Revisited

In the late 70s, Yves Felix and Halperin-Stasheff showed how one could deform a certain bigraded model for a given graded-commutative algebra H over the rationals to yield a (filtered) cofibrant model for any differential graded-commutative algebra A having H as its cohomology. Not surprisingly, a similar procedure is possible for a differential graded Lie algebra L. We show how a simplicial version of this construction can be used to produce a complete sequence of higher-order invariants for the homotopy type of L.

This is joint work with Samik Basu and Debasis Sen.

D. White: A Unified Approach to the Homotopy Theory of Operads and Algebras

I will report on joint work with Michael Batanin and Florian De Leger studying the homotopy theory of the Grothendieck construction, given a category B and a functor F from B^op to CAT. We introduce powerful new techniques for constructing transferred model structures on Grothendieck constructions, such as the category of pairs (P, A) where P is a (symmetric or non-symmetric) colored operad and A is a P-algebra or a left P-module.

From such a (semi-)model structure on pairs, we produce a "horizontal" (semi-)model structure on the base B and "vertical" (semi-)model structures on the fibers F(P). This recovers in a unified framework all known (semi-)model structures on categories of operads and their algebras and produces new model structures, e.g., on the category of twisted modular operads.

Additionally, we study when these model structures are left proper and when a weak equivalence in the base B gives rise to a Quillen equivalence of fibers.

A. Lazarev: Hochschild cohomology of the second kind (joint with A. Guan and J. Holstein)

Let A be a differential graded algebra. Its cohomological Hochschild complex C(A,A) is a double complex, one differential being induced by the differential in A and the other one is the Hochschild differential. Any double complex admits two totalizations, one given by direct sums and the other – by direct products. The ordinary Hochschild complex is formed by taking the direct product totalization. The Hochschild complex obtained by the direct sum totalization was considered by Positselski and Polischchuk and they called it the Hochschild complex of the second kind. Their goal was to prove that the Hochschild cohomology of A of the second kind is isomorphic to the ordinary Hochschild cohomology (of the first kind) of the category of cofibrant perfect A-modules; they proved that this is true under certain technical assumptions on A.

In this talk I will consider another version of the Hochschild cohomology of the second kind, which is lies between the Positselski-Polischchuk Hochschild cohomology and the ordinary Hochschild cohomology. Its definition is not as elementary as that of Positselski-Polishchuk but it has better formal properties.

We consider two examples giving geometrically meaningful results: one when A is the Dolbeault algebra of a smooth complex projective variety and the other when A is the de Rham algebra of a smooth manifold. In the first case the Hochschild cohomology is isomorphic to the Hodge cohomology of the variety, and in the second it leads to the string homology of the manifold.

Ping Xu: Derived differentiable manifolds

One of the main motivations behind derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both cases can be considered as fiber products of manifolds which may not be smooth in classical differential geometry. Thus we need to extend the category of differentiable manifolds to a larger category in which one can talk about "homotopy fiber products". In this talk, we will discuss a solution to this problém in terms of bundles of $L_\infty$-algebras. This is mainly based on a joint work with Kai Behrend and Hsuan-Yi Liao.

M. Stienon: Noncommutative calculi for differential graded manifolds

Differential graded manifolds are a useful concept that provides a common framework for many important classes of algebro-geometric structures, including homotopy Lie algebras, foliations, and complex manifolds. In this talk, we will describe noncommutative calculi for differential graded manifolds and then outline a Duflo-Kontsevich type theorem. The Duflo theorem of classical Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds are special instances of this theorem.

I. Galvez-Carrilo: Decomposition spaces: convexity and the Crapo complementation formula.

We define the notion of convex subspace $K$ of a decomposition space $X$ and establish a Crapo formula for the Möbius function, $\mu_X=\mu_{X- K}+\mu_x*\zeta_K*\mu_X$. At the objective level this is an explicit homotopy equivalence of $\infty$-groupoids. Taking homotopy cardinality recovers the classical algebraic identities: the Björner-Walker formula if $X$ is just a locally finite poset, and the original Crapo formula in the case of a finite lattice.

A. Tonks: On the origins of the Loday-Ronco B-infinity structures

Thanks to a helpful remark of Martin after a different conference talk, we can show that B-infinty structure discovered by Loday and Ronco in their work on the structure theory of (non-cocommutative) cofree Hopf algbras, is a twisting of the trivial B-infinity algebra by a specific automorphism. Of course, this gives an astonishingly simple proof of their properties.

D. Trnka: Operadic Grothendieck construction

I will talk about the Grothendieck construction in operadic context. That is, a construction that presents an operad as a category, equipped with some extra structure - the structure of an operadic category. For set-valued operads (and other operad-like structures), this was described by Batanin and Markl, together with an equivalence of operads and so-called discrete operadic fibrations. The goal of my talk is to extend this result to operads valued in small categories, for which we need to introduce operadic 2-categories. A trivial example of a categorical operad-like structure is a monoidal category. We describe its operadic Grothendieck construction and we relate it to the 'Para' construction, which appears in machine learning.

J. M. Moreno: Higher order Massey products for algebras over operads

We will recall the classical Massey products for associative algebras. Then, explain how Fernando Muro understood triple products from an operadic perspective. Finally, I will explain how Oisín Flynn-Conolly and myself have built on top of Fernando's framework to define higher-order Massey products for algebras over operads in chain complexes. This is joint work with Oisín Flynn-Conolly.

F. De Leger: The lattice path operad and fibration sequences

In their paper "The lattice path operad and Hochschild cochains", Batanin and Berger introduced a coloured operad in Set called lattice path operad L, with a filtration L_m of it, for any non-negative integer m. They also introduced a construction called condensation of a coloured operad and proved that the condensation of L_m yields an E_m operad. For m=2, they deduced from it a new proof of Deligne's conjecture, L_2 being the operad for multiplicative non- symmetric operads. Closely related to this is Turchin/Dwyer-Hess double delooping formula for the totalization of a multiplicative non-symmetric operad. A natural question to ask is whether there is an m-fold delooping formula associated to the m-th filtration stage of L. In this talk we will recall the notion of lattice path operad and its filtration and we will conjecture a fibration sequence associated to any stage of the filtration. Such fibration sequences would extend the results of Turchin/Dwyer-Hess.

D. Dunina: Wreath product of operadic categories and Boardman-Vogt product of operads

Operadic categories were introduced in 2015 by M. Batanin and M. Markl as a generalization of various "operad-like" structures, including classical operads, their variants, versions of PROPs, and other similar structures. Later, motivated by the Boardman-Vogt tensor product of operads, M. Batanin introduced the notion of a wreath product of operadic categories, aiming to establish a correspondence between the two mentioned constructions. In this talk, we define the wreath product of operadic categories, briefly examine some of its properties, and establish a relationship between the wreath product of operadic categories and the Boardman-Vogt product of operads. This is a work in progress, and we hope to extend the presented results to achieve the desired correspondence.

O. Kravchenko: Poisson algebra bundles on configuration spaces

In a joint work with Alessandra Frabetti and Leonid Ryvkin https://arxiv.org/abs/2407.15287) we consider a category of vector bundles over a base manifold which is a space of unordered configurations of points. We see such a space as a manifold of non constant dimension. It turns out that there are two monoidal structures on this category: the usual point-wise tensor product called Hadamard, and a new one similar to the external tensor product, called Cauchy. We also define a Poisson bracket coming from physics. The Leibniz rules with respect to both products are not totally obvious. While the motivation for this work comes from physics the algebraic part of the structure is interesting in its own right. I would love to add something up-to homotopy in the picture but not sure yet of the result.

Ralph Kaufmann: Bimodules and operad type theories

We discuss a way to think about operads as monoidal bi-module monoids in a monoidal functor category. This allows us to describe Feynman categories, Unique Factorization Categories and their plus constructions in a succinct way. Bar and co-bar constructions become very natural in this framework. It also allows comparison to other constructions such as moment categories and operadic categories, which we will discuss if time allows.