## LMS YaMCATS Category Theory and its Applications Lecture Series.

Welcome to the LMS YaMCATS Category Theory and its Applications Lecture Series webpage.

This lecture series ‘category theory and its applications’ is linked to the LMS funded Yorshire and Midlands Category Seminar (YaMCATS) network.

This lecture series consists of two topics which illustrate the power of categorical methods to other areas of pure mathematics, mathematical physics and computer science.

The first topic is on the theory of differential categories, to be presented by Jean-Simon Lemay (University of Oxford). The theory of differential categories uses category theory to study the foundations of differentiation. Differential categories have been able to formalize various aspects of differentiation, from the very basic foundational aspects of differentiation to the more complex notions of differential geometry.

The second topic is mapping class groupoids and motion groupoids, to be presented by Fiona Torzewska (University of Leeds).The lectures will explore the use of these categorical structures in the context of topological phase of matter, that is a physical system whose behaviour is effectively described by a topological quantum field theory.

**Starting on Wednesday 28 October 2020 11am (GMT) to 12pm (GMT) for 7 weeks**

Please register with the ICMS here, to receive future announcements and the zoom link.

**Call details will be sent out 30mins before the start of the seminar**

This event is kindly funded by the London Mathematical Society.

### Programme of Lectures:

### All lectures are recorded and archived by the ICMS, please view them here.

**Jean-Simon Lemay **(University of Oxford) : *The Theory of Differential Categories. *

Abstract: The theory of differential categories uses category theory to study the foundations of differentiation. Differential categories have been able to formalize various aspects of differentiation, from the very basic foundational aspects of differentiation to the more complex notions of differential geometry. In these lectures, we propose to introduce and summarize the theory of differential categories, as well as discuss interesting examples and applications. There will be three (possibly four) lectures:

**Lecture 1)** We will provide an overview introduction to the theory of differential categories. In particular, we will take a look at the "map of differential categories" and discuss how each stage of the theory of differential categories are connected. We will also discuss the first stage differential categories, which are axiomatized by the basic algebraic foundations of differentiation and provide the categorical semantics of differential linear logic.

Please find the slides for this lecture here.

**Lecture 2) **We will discuss Cartesian differential categories, which formalize the directional derivative and the theory of multivariable differential calculus over Euclidean spaces, and provide the categorical semantics of the differential lambda-calculus.

Please find the slides for this lecture here.

**Lecture 3)** We will discuss tangent categories, which formalize tangent bundle structure and the theory of smooth manifolds.

**Bonus: Lecture 4)** Reverse differentiation is used in programming for efficient computations. We will discuss Cartesian reverse differential categories, a recent introduction to the theory of differential categories, which axiomatizes reverse differentiation, and their relationship to Cartesian differential categories, which axiomatize the forward derivative.

**Fiona Torzewska **(Univeristy of Leeds) : *Mapping class groupoids and motion groupoids*

A topological phase of matter is a physical system whose behaviour may be effectively described via a topological quantum field theory i.e. functor from cob to vect. The study of topological quantum field theories has applications in quantum computing but also involves a lot of beautiful mathematics which is interesting in its own right. A central role in the description of topological phases of matter in 2 (spatial) dimensions is played by the representations of braid groups. A natural generalisation to study the statistics of higher (spatial) dimensional phases of matter is then to look for generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the motions groups of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce generalisations of these two definitions. In each case we will show first that these give us groupoids and then that we can get back to the classical definitions by considering the endomorphisms of a single object. The mapping class groupoid is a simpler construction but is not in general the right notion to take when considering particles moving through space. We will construct a functor from the motion groupoid to the mapping class groupoid and hence see which cases we can study only the mapping class groupoid. We will use lots of examples to aid intuition and intend this talk to be accessible to those with minimal knowledge of topology.

Please find the slides for the first lecture here.

The provisional schedule for the lectures:

Oct 28: **Jean-Simon Lemay **(University of Oxford)

Nov 4: **Fiona Torzewska **(Univeristy of Leeds)

Nov 11: **Jean-Simon Lemay **(University of Oxford)

Nov 18: **Fiona Torzewska **(Univeristy of Leeds)

Nov 25: **Jean-Simon Lemay **(University of Oxford)

Dec 2: **Fiona Torzewska **(Univeristy of Leeds)

Dec 9: **Jean-Simon Lemay **(University of Oxford) or **Fiona Torzewska **(Univeristy of Leeds)