Speakers

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.

Title: Introduction to univalence in Coq IV
Abstract: This week we conclude our sequence of talks on homotopy type theory (HoTT). We revisit the univalence axiom and use a simple example to illustrate its use.
The audience is encouraged to follow the development of the theory on their own computers so please bring a laptop if you want to do this! (You will need about 300mb of space to install the Coq proof assistant.) There will be short exercises that the audience can complete at their own pace which will not be vital to the theory but rather are intended to familiarise the audience with the proof assistant and tactics.

Title: Introduction to univalence in Coq III
Abstract: This week we continue our sequence of talks on homotopy type theory (HoTT) for which we are approaching the denouement. First we define h-propositions and what it means to be a contractible type. Then we distinguish between a couple of different types of equivalence and show that a particular choice satisfies all the conditions for being an h-proposition. If we have time we will formulate the univalence axiom and make some straightforward deductions that illustrate its use.
The audience is encouraged to follow the development of the theory on their own computers so please bring a laptop if you want to do this! (You will need about 300mb of space to install the Coq proof assistant.) There will be short exercises that the audience can complete at their own pace which will not be vital to the theory but rather are intended to familiarise the audience with the proof assistant and tactics.

Title: Introduction to univalence in Coq II
Abstract: This week we continue our sequence of talks on homotopy type theory (HoTT). We will introduce some more tactics and work towards the univalence axiom.
The audience is encouraged to follow the development of the theory on their own computers so please bring a laptop if you want to do this! (You will need about 300mb of space to install the Coq proof assistant.) There will be short exercises that the audience can complete at their own pace which will not be vital to the theory but rather are intended to familiarise the audience with the proof assistant and tactics.