#### Statement on Black Lives Matter

**Title**: TBA

**Abstract**: TBA

**Title**: TBA

**Abstract**: TBA

**Title**: Introduction to the de Rham Cohomology

**Abstract**: In this talk I will cover the details behind the construction of the de Rham cohomology. In particular, I will focus on its boundary map – the exterior derivative – and understanding what this cohomology tells us about closed and exact differential forms.

**Title**: Tangent infinity categories revisited (Part 2)

**Abstract**: In this talk I will discuss recent work with Michael Ching and Florian Schwarz on Tangent Infinity Categories. Over the summer, an error was discovered in our previous work. In this talk I will explain the error and how we propose to fix it.

**Title**: Tangent infinity categories revisited (Part 1)

**Abstract**: In this talk I will discuss recent work with Michael Ching and Florian Schwarz on Tangent Infinity Categories. Over the summer, an error was discovered in our previous work. In this talk I will explain the error and how we propose to fix it.

**Title**: Morita equivalence and bicategories of fractions: a simple geometric idea gets out of hand.

**Abstract**: We start by considering the geometry of spaces with singularities: in these spaces, the local geometry is not homogeneous as in a manifold, but varies from point to point. Lie groupoids provide a way of encoding this information. However, this encoding is not unique: different Lie groupoids may represent the same singular space. This leads to the idea of ‘Morita equivalence’ of Lie groupoids: two groupoids are Morita equivalent if they encode the same underlying geometric structure. To formalize this equivalence we consider the ‘localization’ of Lie groupoids, which creates a ‘bicategory of fractions’ in which the Morita equivalences become isomorphisms.

The goal of this talk is to describe this bicategory of fractions construction in all its diagrammatic glory, and convince you that it is geometrically motivated and meaningful. I will start with the geometric intuition of how Lie groupoids are used to encode singular spaces, and how Morita equivalence arises as a natural consequence, with examples and pictures. I will then explain the localization construction, starting from the familiar idea of localizing a prime ideal in a commutative ring and extending to more complicated contexts, culminating in the bicategory of fractions we are interested in. At the end I will mention applications and current work in this area. No background with Lie groupoids or bicategories is assumed and all scary diagrams will be thoroughly explained and motivated.

*This event will be in a hybrid format. Besides the option to attend in person the presentation will be streamed on zoom. For the zoom link and password, please contact florian.schwarz@ucalgary.ca*

**Title**: Turing Categories

**Abstract**:

This talk is based on the following papers/notes:

(1) “Introduction to Turing categories” with Pieter Hofstra

(2) “Timed set, functional complexity, and computability” with Boils, Gallagher, Hrubes

(3) “Total maps of Turing categories” with Pieter Hofstra and Pavel Hrubes

(4) “Estonia notes” on my website

Turing categories are the theory of “abstract computability”. Their development followed my meeting Pieter Hofstra. He was in Ottawa at the time and he subsequently joined me as a postdoc. The core theory was developed in Calgary before he returned to Ottawa as a faculty member. Tragically he died earlier this year when there was still so much to do and, indeed, that he had done, but had not published.

Turing categories are important because they characterize computability in a minimal traditional context. These ideas are not original to Pieter and I: De Paola, Heller, Longo, Moggi, and others had all travelled in this terrain before we did. Pieter and I simply took the ideas polished them a bit and moved them a step further on a road which still stretches ahead.

So, the purpose of the talk is to try and explain what all this was about … and what we were striving to accomplish. To do this I have to introduce restriction categories and Turing categories in that context. Then I will describe a family of models which are fundamental to computer science. Finally, I will take a quick look along the road at some open issues.

**Title**: Kan extensions as a framework to extend resource monotones

**Abstract**: A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monontone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into ([0,∞],≤), and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.

**Title:** Tangent categories, Cartesian differential categories and how they are related

**Abstract:** Tangent categories are a categorical generalization of the category of manifolds by having maps like the projection and the zero-section of the tangent bundle that fulfill certain relations. With a similar strategy, Cartesian differential categories generalize the category of finite-dimensional R-vector spaces and smooth maps.Unsurprisingly the construction generalizing manifolds and the construction generalizing vector-spaces are related. More precisely there is an adjunction between the category of Cartesian tangent categories and the category of Cartesian differential categories. Explaining this adjunction is the main goal of this talk.

**Title:** Categories of Kirchoff Relations

**Abstract:** It is known that the category of affine Lagrangian relations over a field F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of affine Lagrangian relations. The purpose of this paper is to provide a characterization of the relations in this subcategory — and in important subcategories thereof — in terms of parity-check and generator matrices as used in error detection.

In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff’s current law. We characterize these Kirchhoff relations in terms of parity-check matrices and, study two important subcategories: the deterministic Kirchhoff relations and the lossless relations.