Title : Graded (co-)Differential Categories
Abstract : Co-differential categories are additive symmetric monoidal categories where you can differentiate certain kind of morphisms with the help of a combination of a monad, monoids and a deriving transformation.
Recently, J.-S. Pacaud Lemay proved that the category of finite-dimensional vector spaces cannot be endowed with a nontrivial structure of co-differential category. And this is very unlikely that it will work better with some familiar categories such as the category of finite sets and relations or the category of all Hilbert spaces.
In this talk, I will introduce a generalisation of co-differential categories: graded co-differential categories, which basically consist in putting the word “graded” everywhere in the axioms of a co-differential category. I will explain how the preceding categories can be endowed with the structure of a graded co-differential category as example of a general categorical construction which use symmetric powers.
This is joint work in progress with Jean-Simon Pacaud-Lemay.
Title: An (∞,2)-categorical pasting theorem
Abstract: Power’s 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (∞,2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (∞,2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus.
Title: Describing principal bundles and pushing TQFTs forward
Abstract: In the first part of this talk we will discuss several ways, how principal bundles over a manifold can be described. The main two of them are maps from the base manifold into the group’s classifying space and assignments of group elements to the codimension one structures of a special decomposition, called fine stratification. Both of them provide an equivalence of categories to the category of principal bundles. Having all these equivalent descriptions, one can translate geometric constructions with principal bundles into discrete combinatorial constructions. In the second part of this talk I will outline, starting from a pushforward construction for equivariant Topological Quantum Field Theories, how this can be used to define a pushforward construction for defect Topoplogical Quantum Field Theories.
This is joint work with Gregor Schaumann, as part of my Master’s project.
Titre : From trees to infinity-operads
Abstract : The first goal of this talk will be to introduce the dendroidal world, which generalises the simplicial world. Instead of working with the category Delta, we will work with the category Omega, a category of trees, introduced by Moerdijk and Weiss. Presheaves on Delta are called dendroidal sets and generalise simplicial sets. I will explain how operads and infinity-operads appear in this setting. In a second part, I will introduce a notion of homology for infinity-operads, using a bar construction.
This is a joint work with Ieke Moerdijk.
Title: Measurement in the symplectic setting.
Abstract: In previous work, we have given generators for affine Lagrangian relations over an arbitrary field; exploiting the graphical calculus for affine relations. In the case of a finite field of odd prime order d, we have shown that this is isomorphic to qudit stabilizer circuits. In this talk, the question of measurement in this symplectic setting will be addressed. We show that the CPM construction applied to affine Lagrangian relations yields affine coisotropic relations; which can be obtained by adding the discard relation to affine Lagrangian relations. By splitting the decoherence maps in affine coisotropic relations, we obtain a two-sorted presentation for classical/stabilizer quantum circuits. We show that this is equivalent to adding injection and coinjection relations to affine Lagrangian relations. Time permitting, we will discuss the connection of this work to classical and quantum additive codes.
Recording (Passcode: 58n&dKr$)
Title: RO(C_2)-graded coefficients of C_2-Eilenberg-MacLane spectra
Abstract: In non-equivariant topology, the ordinary homology of a point is described by the dimension axiom and is quite simple – namely, it is concentrated in degree zero. The situation in G-equivariant topology is different. This is due to the fact that Bredon homology – the equivariant counterpart of the ordinary homology – is naturally graded over RO(G), the ring of G-representations. Whereas the equivariant dimension axiom describes the part of the Bredon homology of a point which is graded over trivial representations, it does not put any requirements on the rest of the grading – in which the homology may be quite complicated.
The RO(G)-graded Bredon homology theories are represented by G-Eilenberg-MacLane spectra, and thus the Bredon homology of a point is the same thing as coefficients of these spectra. During the talk I will present the method of computing the RO(C_2)-graded coefficients of C_2-Eilenberg-MacLane spectra based on the Tate square. As demonstrated by Greenlees, the Tate square gives an algorithmic approach to computing the coefficients of equivariant spectra. In the talk we will discuss how to use this method to obtain the RO(C_2)-graded coefficients of a C_2-Eilenberg-MacLane spectrum as a RO(C_2)-graded abelian group. We will also present the multiplicative structure of the C_2-Eilenberg-MacLane spectrum associated to the Burnside Mackey functor. This allows us to further describe the RO(C_2)-graded coefficients of any C_2-Eilenberg-MacLane spectrum as a module over the coefficients of the C_2-Eilenberg-MacLane spectrum of the Burnside Mackey functor. Finally, we will discuss the RO(C_2)-graded ring structure of coefficients of spectra associated to ring Mackey functors.
Title: Automorphisms of seemed surfaces, modular operads and Galois actions
Abstract: The idea behind Grothendieck-Teichmüller theory is to study the absolute Galois group via its actions on (the collection of all) moduli spaces of genus g curves. In practice, this is often done by studying an intermediate object: The Grothendieck-Teichmüler group, GT.
In this talk, I’ll describe an algebraic gadget built from simple decomposition data of Riemann surfaces. This gadget, called an infinity modular operad, provides a model for the collection of all moduli spaces of genus g curves with n boundaries, which we justify by showing that the automorphisms of this algebraic object is isomorphic to a subgroup of Grothendieck-Teichmüller group.
Title: The diagonal of the operahedra
Abstract: The set-theoretic diagonal of a polytope has the crippling defect of not being cellular: its image is not a union of cells. Our goal here is to develop a general theory, based on the method introduced by N. Masuda, H. Thomas, A. Tonks and B. Vallette, in order to understand and manipulate the cellular approximations of the diagonal of any polytope. This theory will allow us to tackle the problem of the cellular approximation of the diagonal of the operahedra, a family of polytopes ranging from the associahedra to the permutohedra, and which encodes homotopy operads. In this way, we obtain an explicit formula for the tensor product of two such operads, with interesting combinatorial propertiesReference: https://arxiv.org/abs/2110.14062
Title: New tangent structure on Lie algebroids and Lie groupoids
Abstract: The tangent bundle on a smooth manifold is, in a sense, sufficient structure to develop Lagrangian mechanics. In a famous note from 1901, Poincare reformulated Lagrangian mechanics by replacing the tangent bundle with a Lie algebra acting on a smooth manifold [1, 2]. Poincare’s formalism leads to the Euler-Poincare equations, which capture the usual Euler-Lagrange equations as a specific example. In 1996, Weinstein sketched out a general program building on Poincare’s ideas to formulate mechanics on Lie groupoids using Lie algebroids , which motivates the work of Martinez et al. [4,5], Libermann , and the recent thesis by Fusca .
In this talk, we will look at Weinstein’s program through the lens of tangent categories, which are a categorical abstract of the Weil functor formalism. We show that Lie algebroids can be reformulated as a certain category of tangent functors from Weil algebras into smooth manifolds. This tangent structure on the category of Lie algebroids agrees with Martinez’s presentation of Lie algebroids as generalized tangent bundles.
 Poincaré H. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci. 1901;132:369-71.
 Marle CM. On Henri Poincaré’s note “Sur une forme nouvelle des équations de la Mécanique”. Journal of geometry and symmetry in physics. 2013;29:1-38.
 Weinstein A. Lagrangian mechanics and groupoids. Fields Institute Proc. AMS. 1996;7:207-31.
 Martínez E. Lagrangian mechanics on Lie algebroids. Acta Applicandae Mathematica. 2001 Jul;67(3):295-320.
 de León M, Marrero JC, Martínez E. Lagrangian submanifolds and dynamics on Lie algebroids. Journal of Physics A: Mathematical and General. 2005 Jun 1;38(24):R241.
 Libermann P. Lie algebroids and mechanics. Archivum mathematicum. 1996;32(3):147-62.
 Fusca D. A groupoid approach to geometric mechanics (Doctoral dissertation, University of Toronto).
Title: Wheeled PROP structure on stable cohomology
Abstract: Wheeled PROPs, introduced by Markl, Merkulov and Shadrin are PROPs equipped with extra structures which can treat traces.
In this talk, after explaining the notions of PROPs and wheeled PROPs, I will describe a wheeled PROP structure on stable cohomology of automorphism groups of free groups with some particular coefficients. I will explain how cohomology classes constructed previously by Kawazumi can be interpreted using this wheeled PROP structure and I will construct a morphism of wheeled PROPs from a PROP given in terms of functor homology and the wheeled PROP evoked previously.
This is joint work with Nariya Kawazumi.