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**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 properties

**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 [3], which motivates the work of Martinez et al. [4,5], Libermann [6], and the recent thesis by Fusca [7].

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.

[1] Poincaré H. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci. 1901;132:369-71.

[2] 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.

[3] Weinstein A. Lagrangian mechanics and groupoids. Fields Institute Proc. AMS. 1996;7:207-31.

[4] Martínez E. Lagrangian mechanics on Lie algebroids. Acta Applicandae Mathematica. 2001 Jul;67(3):295-320.

[5] 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.

[6] Libermann P. Lie algebroids and mechanics. Archivum mathematicum. 1996;32(3):147-62.

[7] 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.

**Title**: A-Homotopy Theory and Universal Covers

**Abstract**: A-homotopy theory is a homotopy theory developed for graphs. We would like to know if this homotopy relation gives the weak equivalences of a model structure on the category of graphs. We are trying to develop a weak factorization system on graphs as a stepping-stone to this goal. One requirement of a weak factorization system is the ability to factor a morphism into two morphisms from different classes. In order to do this, we are mimicking a strategy found in the homotopy theory of topological spaces that involves covering spaces and lifting properties. In this talk, I will give the construction of the universal cover of a graph.

**Title**: Building Cartesian Differential Categories as Kleisli Categories

**Abstract**: Cartesian differential categories (CDC) come equipped with a differential combinator, which provides a categorical axiomatization of the directional derivative from multivariable calculus, and so produces a derivative for every map. An important example of a CDC is the coKleisli category of a differential category. In 2018, BJORT constructed a CDC from abelian functor calculus. However, the BJORT example arises as a Kleisli category rather than a coKleisli category.

In this talk, I will generalize this story and explain how to construct a CDC as a Kleisli category. Since CDC are not self-dual, it is not as simple as taking the dual construction of the coKleisli category: in fact it is very different!

**Title**: Quillen-Barr-Beck cohomology for restricted Lie algebras

**Abstract**: The Hochschild cohomology for restricted Lie algebras classifies strongly abelian extensions of restricted Lie algebras. In this talk we define Quillen-Barr-Beck cohomology for the category of restricted Lie algebras and we prove that Quillen-Barr-Beck’s cohomology classifies general abelian extensions. Moreover, using Duskin-Glenn’s torsors cohomology theory, we prove a classification theorem for the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions of restricted Lie algebras. Finally, we give an interpretation of Cegarra-Aznar’s exact sequence for torsor cohomology. Thus,we obtain for a short exact sequence of restricted Lie algebras, an eight-term exact sequence for Quillen-Barr-Beck cohomology. This sequence replaces the five-term exact sequence proved by Eckmann-Stammbach in the context of Hochschild cohomology.

**Title**: Locally bounded enriched categories

**Abstract**: We define and study the notion of a locally bounded category enriched over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. Along with providing many new examples of locally bounded (closed) categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly’s reflectivity and local boundedness results for orthogonal subcategories and categories of models for sketches and theories. We also provide characterization results for locally bounded enriched categories in terms of enriched presheaf categories, and show that locally bounded enriched categories satisfy useful representability and adjoint functor theorems. We also define and study the notion of an alpha-bounded-small limit with respect to a locally alpha-bounded closed category, which parallels Kelly’s notion of alpha-small limit with respect to a locally alpha-presentable closed category. This is joint work with Rory Lucyshyn-Wright at Brandon University.

**Title**: TQFTs through Parallel Transport and Quantum Computing

**Abstract**: Quantum computing is captured in the formalism of the monoidal subcategory of Vect (category of complex vector spaces) generated by C^2 – in particular, quantum circuits are diagrams in Vect – while topological quantum field theories, in the sense of Atiyah, are diagrams in Vect indexed by cobordisms. We outline a program to formalize this connection. In doing so, we first equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a double category. Finite dimensional complex vector spaces and linear maps between them are given a suitable double categorical structure which we call FVect. Finally, we realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to FVect, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category. This talk reports on joint work with Steven Rayan.