Speakers

Title: A Topology on Categories of Locally Ringed Space and Applications to Arithmetic Geometry

Abstract: In this talk we present some relevant background on locally ringed spaces and consider a problem that appears in arithmetic geometry. We then use this problem to motivate a class of functors that arise in arithmetic geometry, provide examples of these functors, and show how these functors can induce a Grothendieck topology on their codomains. We then use this topology and its topos of sheaves to make progress towards our motivating problem.

Title: Cubical Sets and A-Homotopy Theory

Abstract: In this talk, I will describe the cubical category and cubical sets. Then I will show how A-homotopy theory, a discrete homotopy theory for graphs, gives us an example of a cubical set. Time permitting, I will discuss how this cubical set might lead us to a chain complex.

Title: Extending CNOT to real stabilizer quantum mechanics

Abstract: The stabilizer formalism for quantum mechanics is an important tool for implementing fault tolerant quantum circuits.  In this talk we first give a brief overview of the stabilizer formalism.  We also will discuss the angle-free fragment of the ZX calculus, which is complete for the real fragment of stabilizer quantum mechanics.  We use this fact to extend the category CNOT to be complete for this fragment of quantum mechanics.

Title: Abstract Symplectic Geometry

Abstract: In recent years, symplectic geometry has used increasingly sophisticated categorical and homotopical machinery. We will consider how tangent categories may simplify some of these constructions.

Title: Differential Algebras in Codifferential Categories

Abstract: Differential categories have lead to abstract formulations of several notions of differentiation such as, to list a few, the directional derivative, Kahler differentials, differential forms, smooth manifolds , and De Rham cohomology. Therefore, if the theory of differential categories wishes to champion itself as the axiomatization of the fundamentals of differentiation: differential algebras should fit naturally in this story.  In this talk, I’ll talk about differential algebras and how they fit in the theory of differential categories. 

Title: Additive Bundles and their Connection Theory

Abstract: In this talk we consider the generalization of connection theory from the second tangent bundle of a smooth manifold to double additive bundles in an arbitrary category. We then extend this generalization to higher ordered connections on n-fold additive bundles. 

Title: Two dimensional Lie theory

Abstract: This week I present an outline of a joint project with Ben MacAdam. The main aim is to generalise the theory of Lie groupoids and Lie algebroids by using 2-cubical sets. One advantage of this approach is that it avoids a certain quotient that is required in the classical theory and is therefore more amenable to generalisation in terms of tangent categories. An additional advantage of this approach is that when the tangent category is assumed representable the appropriate modification of the Lie approximation functor becomes representable also.

Title: The free Lie algebras in Tebbe’s calculation of the derivatives of atomic functors
Abstract: A discrete module is a functor from finite pointed sets to chain complexes of R-modules. There are two ways to do functor calculus for discrete modules. The first is to find the Taylor tower in a way analogous to the Taylor series of functions of a real variable. A second approach is to something more akin to Lagrangian approximation. For functions of a real variable, f, the n-th Lagrangian approximation is the degree n polynomial function which agrees with f on n+1 point. In the case of a discrete module, F, one uses a left Kan extension to produce the best Lagrangian approximation to F. The quotient of successive Lagrangian polynomial functors are called atomic functors.
In her PhD thesis, Amelia Tebbe showed that the n-th derivatives of atomic functors, in the sense of Goodwillie calculus, involve products of free Lie algebras and simple cross effects. The goal of this talk is to present this calculation, and to ask the audience if this looks familiar.

Title: Interpretations of algebraic theories, and the adjunctions they induce: Part II
Abstract: Many cases of free/forgetful adjunctions are special cases of a more general theorem: any interpretation of algebraic theories induces an adjunction on their categories of models. Free monoid, free groups, free modules, tensor algebras, and polynomial rings are all instances of this. In my talk I will prove this theorem.

Title: Interpretations of algebraic theories, and the adjunctions they induce
Abstract: Many cases of free/forgetful adjunctions are special cases of a more general theorem: any interpretation of algebraic theories induces an adjunction on their categories of models. Free monoid, free groups, free modules, tensor algebras, and polynomial rings are all instances of this. In my talk I will prove this theorem.