Combinatorial obstructions to quasi-isomorphism

This talk surveys recent advances in the study of Massey products, a classical algebraic invariant of homotopy types introduced in the 1960s by Massey, and subsequently built upon by May, Retkah and Allday. Recently, thanks to the work of Buijs, Moreno-Fernández and Murillo, these objects have once again emerged as a significant focus of interest in homotopical algebra. Over the past five years, several generalisations have been introduced: 1) operadic Massey products for Koszul operads, 2) cotriple products in positive characteristic, and 3) determinant Massey products capturing more complex relations in CDGAs. In this talk, we survey these recent developments and explain how these obstructions offer a toolkit for treating the associative and commutative quasi-isomorphism problem in various contexts. We conclude with some open problems. Part of this talk is based on joint work with Moreno-Fernández and Muro.

Stable homology of braided Higman--Thompson groups

Thompson's groups are groups of piecewise linear self-homeomorphisms of an interval where the cut points are dyadic rationals. Braided versions were introduced independently by Brin and Dehornoy. We build a topological model for these groups in the form of categories of 1-"cobordisms" in R^3 where objects are configurations of points and morphisms are paths of configurations which are allowed to collide in certain ways. We then construct a scanning map on these topological models to compute the stable homology of the braided variants of the Thompson groups.

Postnikov invariants in rational homotopy theory

Algebraic models for rational homotopy types were proposed by Quillen in 1969 and later by Sullivan in 1977. Quillen’s model, based on Lie algebras, makes it possible to describe simply connected rational homotopy types, whereas Sullivan’s model, based on commutative algebras, describes finite and nilpotent homotopy types. More recently, Buijs, Félix, Murillo and Tanré on the one hand, and Robert-Nicoud and Vallette on the other, extended Quillen’s model to the non-simply connected case.

In this talk, we will present a theory of Postnikov invariants for complete (homotopy) Lie algebras that is compatible with the integration functor of Robert-Nicoud and Vallette. We will then deduce an integration of Chevalley–Eilenberg cohomology into the cohomology of spaces with local coefficients.

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C*-categorical prefactorization algebras of superselection sectors

A major result by Doplicher, Haag, and Roberts in algebraic quantum field theory shows that one can derive particle statistics from local quantum observables using first principles. While this may sound like a string of technical jargon, the core mathematical idea is neat: there exists a braided (or symmetric) monoidal category of superselection sectors—interpreted as charges or direct summands of the Hilbert space of states—that can be constructed entirely from the net of algebras of local observables. The statistics (i.e. what distinguishes bosons from fermions) of an irreducible sector can then be extracted from the action of the braid (or symmetric) group on tensor powers of that sector within the aforementioned category.

In this talk, we aim to introduce these ideas to the audience, assuming no prior background. We will also present a new construction of the braided monoidal category of superselection sectors, employing tools from higher algebra, which makes transparent the geometry/topology underlying this theory. This is based on joint work with Benini, Naaijkens, and Schenkel [arXiv:2505.07960] and with Benini, and Schenkel [arXiv:2604.24865].

Homotopy category of homotopy algebras (after B. Vallette)

I will explain that the category of strongly homotopy algebras over quadratic Koszul operads (which includes the Three Graces: A-infinity, L-infinity, and C-infinity) possesses an "almost" model structure, such that the localization obtained by inverting weak equivalences is equivalent to the category of suitable homotopy classes. The central reference for the material of my talk is B. Vallette: Homotopy theory of homotopy algebras, Ann. Inst. Fourier 70(2), 2020, 683-738. 

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PD operads and partition Lie algebras

Partition Lie algebras have been introduced by Brantner and Mathew as certain homotopy theoretic refinements of dg-Lie algebras that control deformation problems in positive characteristic. In this talk, I will try to explain how partition Lie algebras (and other types of Lie algebras) can be understood as certain types of algebras over a PD operad, a refinement of the usual notion of an operad. A version of Koszul duality for such PD operads gives rise to Lie-algebraic descriptions of various types of formal problems appearing in derived algebraic geometry. Based on joint work with Lukas Brantner and Ricardo Campos.