Speakers

Title: Involution algebroids and their homotopy theory II

Abstract: We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. We first sketch the relationship between our new definition and the classical one. Then as an illustrative example of how to work with the new definition we develop some of the homotopy theory of involution algebroids. This is joint work with Ben MacAdam.

Title: Involution algebroids and their homotopy theory

Abstract: We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. We first sketch the relationship between our new definition and the classical one. Then as an illustrative example of how to work with the new definition we develop some of the homotopy theory of involution algebroids. This is joint work with Ben MacAdam.

Title: Dagger linear logic for categorical quantum mechanics

Abstract: Categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of this approach is that, while it supports finite dimensional processes, it does not generalize well to infinite dimensional processes. A natural categorical generalization of compact closed categories, in which to seek a description of categorical quantum mechanics without this limitation, is in ∗-autonomous categories or, more generally, linearly distributive categories (LDCs). 

Last winter, Robin, Cole and myself started development of this direction of generalization. An important first step is to establish the behavior of the dagger in this more general setting. Thus, we simultaneously developed the categorical semantics of dagger linear logic.  The goal of my talk is to give the definition of Mixed Unitary Categories [1] which is a key structure in the development of the quantum mechanics in the setting of LDCs. I will also show a running example of this category during my talk.

Reference:
[1] Robin Cockett, Cole Comfort, and Priyaa Srinivasan “Dagger linear logic for categorical quantum mechanics”, arXiv:1809.00275 [math.CT], Sep. 2018.

This week we:-

– recap the definition of dagger-LDC, dagger mix and dagger isomix categories
– define unitary structure for dagger isomix categories
– show that compact LDCs are linearly equivalent to monoidal categories
– define a mixed unitary category 
– (If time permits) Finiteness spaces, an example of a Mixed Unitary Category

Title: Dagger linear logic for categorical quantum mechanics

Abstract: Categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of this approach is that, while it supports finite dimensional processes, it does not generalize well to infinite dimensional processes. A natural categorical generalization of compact closed categories, in which to seek a description of categorical quantum mechanics without this limitation, is in ∗-autonomous categories or, more generally, linearly distributive categories (LDCs). 

Last winter, Robin, Cole and myself started development of this direction of generalization. An important first step is to establish the behavior of the dagger in this more general setting. Thus, we simultaneously developed the categorical semantics of dagger linear logic.  The goal of my talk is to give the definition of Mixed Unitary Categories [1] which is a key structure in the development of the quantum mechanics in the setting of LDCs. I will also show a running example of this category during my talk.

Reference:
[1] Robin Cockett, Cole Comfort, and Priyaa Srinivasan “Dagger linear logic for categorical quantum mechanics”, arXiv:1809.00275 [math.CT], Sep. 2018.

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.