Operadic Methods in Geometry II

This meeting will take place in Paris at the École Normale Supérieure de Paris from October 8th (3pm) to October 10th (1pm)
The goal of this meeting is to discuss some recent developments in the use of operadic methods in various flavours of geometry (complex, symplectic, algebraic, derived algebraic,...). The 1st edition of OMG took place in Barcelona in 2023.

Organizers

Registration is closed.

Speakers

Schedule

8 October 15:00-15:30 Welcome, coffee
15:30-16:30 Jakob Ulmer
16:45-17:45 Hugo Pourcelot
9 October 10:00-11:00 Samuel Muñoz
11:00-11:30 Coffee break
11:30-12:30 Marie-Camille Delarue
Lunch
14:30-15:30 Adela Zhang
15:30-16:00 Coffee break
16:00-17:00 Bruno Vallette
10 October 10:00 - 11:00 Connor Malin
11:00 - 11:30 Coffee break
11:30 - 12:30 Pedro Boavida de Brito

Abstracts

  • Pedro Boavida de Brito: Configuration spaces of manifolds with vanishing diagonal class
    The homology of configuration spaces of points in Euclidean space has well-known geometric representatives given by certain "planetary systems”. In the talk, I will describe what planetary systems may be for a general manifold. These classes provide homology generators and, for manifolds with vanishing diagonal class, the relations can be fully determined. The message is that under the vanishing diagonal class assumption (and with field coefficients) the answer is as simple as it can be. This overlaps results by Arabia and Petersen. The approach I’ll report on is rather low-tech (e.g. no spectral sequences will appear) and gives the ring structure in cohomology, which seems to be new. This is joint work with Geoffroy Horel and Danica Kosanovic.
  • Marie-Camille Delarue: Stable homology of Higman-Thompson groups
    The Higman-Thompson groups are groups of certain self-homeomorphisms of Cantor sets. We use a description provided by Skipper and Wu of the Higman-Thompson groups as groups of trees. We compute their homology in a stable range by constructing a scanning map following the work of Madsen, Weiss, Galatius and others. This map allows us to express the stable homology of the groups as the homology of the base point component of the infinite loop space of a certain spectrum.
  • Connor Malin: A simple construction of the self duality of E_n
    The interaction of the little disks operad E_n and Koszul duality has been, and continues to be, studied throughout the past 30 years. In particular, what is the relation between $E_n$ and its Koszul dual? We survey the history of this problem, and then describe our recent construction of an explicit, simple, and geometrically defined map which witness the equivalence of the $E_n$ operad and its Koszul dual, in the category of spectra.
  • Samuel Muñoz: On the homotopy type of spaces of long knots
    For given p less than d, the space of "long knots" is the embedding space rel boundary of the p-disc into the d-disc. By work of Boavida de Brito—Weiss et al., when d-p is at least 3, its homotopy type is closely related to that of the space of operad maps from E_p to E_d, where E_n denotes the little n-discs operad. In this talk, I will explain a computation of this homotopy type away from the prime 2 and in the so-called "concordance embedding stable range" which, by developments of Goodwillie—Krannich—Kupers, is at least 2d-p-5. The description features objects internal to surgery theory as well as relative algebraic K-theory/topological cyclic homology, appearing as part of an analogue for embeddings of a theorem of Weiss—Williams. I will also explain how this computation recovers rationally the 0- and 1-loop order parts of the hairy graph complex of Fresse—Turchin—Willwacher.
  • Hugo Pourcelot : Integration along the fibers for gebras over dioperads and shifted Poisson geometry
    Given a monoidal adjunction between rigid categories and a certain orientation datum on the right adjoint F, I will explain how to transport gebras over dioperads along F, via endowing this functor with a shifted Frobenius monoidal structure. This procedure should be thought of as a kind of integration along the fiber, which would be the case where F is the pushforward of a projection X x M --> X, with M a closed oriented manifold. Our motivation comes from derived Poisson geometry: in their seminal article introducing shifted symplectic structures, Pantev-Toën-Vaquié-Vezzosi proved an AKSZ theorem via an integration along the fibers procedure, which is conjectured to generalize to shifted Poisson structures. I will explain how our construction, when applied to the dioperad of shifted Lie bialgebras, can be regarded as a toy version of such a Poisson AKSZ construction. This is joint work in progress with Valerio Melani.
  • Jakob Ulmer : Extracting Invariants from Calabi-Yau Categories
    In the context of mirror symmetry, Kontsevich proposed to study categories instead of certain numerical invariants. For instance, Fukaya categories vs Gromov-Witten invariants of a symplectic manifold. How do such categories (CY categories) in turn encode these numbers? I will first survey an answer proposed by Costello, Caldararu-Tu, including successful applications. In the second half I will report on work in progress generalizing this. Physically, their procedure can be called closed string field theory (SFT), whereas I consider open-closed SFT. A key part is played by certain dg-BV algebras, for the open case it being the domain of the Loday-Quillen-Tsygan map. Finally I will present ideas about when and how to obtain a MCE in this dg-BV algebra. The MCE under the LQT map relates to a matrix models, such being known to encode numerical invariants. Time permitting I will explain how other structures appear, such as periodic cyclic homology of above categories, (curved) L-infinity structures on cyclic cochairs and non-commutative Chern characters.
  • Bruno Vallette: The higher algebras of topological recursion
    In 2018, Kontsevich and Soibelman introduced a new algebraic notion called Quantum Airy structure providing the minimal but general framework in which the Eynard—Oration topological recursion makes sense. Quantum Airy structures are systems of linear PDEs satisfying certain axioms, which admit a unique solution, the so-called partition function, and this solution is computed by the topological recursion. Several classes of examples of quantum Airy structures have been constructed from (non-commutative) Frobenius algebras appearing in the geometry of spectral curves and in vertex operator algebras. The corresponding partition functions provide the solution to numerous problems in the enumerative geometry of complex curves and in low-imensional quantum field theories. In this talk, we will show that quantum Airy structures are gebras over a Koszul properad and that their associated partition function is given by the properadic deformation gauge group action with explicit formulas. This is a joint work with Gaëtan Borot.
  • Adela Zhang: Operations on mod p TAQ cohomology and spectral partition Lie algebras.
    The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. We apply this spectral sequence to the Koszul dual pair given by the (non-unital) E_∞ operad and the spectral Lie operad. When the input are trivial E_∞-HF_p algebras, we obtain the structure of operations on mod p TAQ cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner–Mathew. In the colimit, the unary operations are Koszul dual to the Dyer–Lashof algebra in the sense of Priddy. There is also a shifted restricted Lie structure that can be detected by the homotopy fixed points spectral sequence, reflecting the Koszul duality between the commutative operad and the shifted Lie operad over F_p