Title: Abadi and Plotkin’s language in terms of differential structure
Abstract: Abadi and Plotkin defined the Simple Differential Programming Language (SDPL) to be a functional programming language with a first order type system where every program can be reverse differentiated. Importantly this language has conditionals and recursive functions. In this talk we will develop the denotational semantics and source code transformation semantics from a categorical point of view using the technology of reverse differential join restriction categories. In particular, using an interpretation into such a category we will show that source code transformations are modelled correctly, that the trace-differentiation technique is modelled, and we will express a denotational semantics for SDPL into any such category with enough points. Finally, we will use the theory of these categories to derive a modification to the operational semantics that yields an exponential speedup on the differentiation of loops. Also, as a bonus, we will do recursion in a restriction category!
Process calculus is a family of related approaches to formally model concurrent systems, e.g. Pi-calculus which pass channels as data along other channels; Ambient calculus which models distributed and mobile computations. Rho Calculus (Reflective Higher Order Calculus) is based on Pi-calculus; it is a closed theory in the form of an asynchronous message-passing calculus built on a notion of quoting. Names are quoted processes and unquoting is reification of processes. Names are subject to algebraic rules and reasoning such as substitutions and bisimulation.
Title: Microlinear Lawvere theories
In this talk we introduce Microlinear Lawvere theories, and show that differential objects in cartesian tangent categories are an example. One of the more striking applications of this theory is that every tangent category embeds into the Eilenberg-Moore category of a tensor differential category.
(Joint work with Jonathan Gallagher and Rory Lucyshyn-Wright)
Title: The symmetric algebra’s other universal properties (joint work with Richard Garner)
The symmetric algebra is known for being the free commutative algebra over a vector space. Therefore, it is universal amongst commutative algebras. It turns out, the symmetric algebra also has another universal property which can be used to construct the initial monoidal differential modality.
Title: String Diagrams for Regular Restriction Categories
Abstract: I will discuss how cartesian restriction categories can be reasoned about using string diagrams for monoidal categories, including how this extends to regular restriction categories. Specifically, we will see that every cartesian bicategory of relations (in the sense of Carboni and Walters) has a regular restriction category of partial maps as a subcategory, and that from a regular restriction category we can construct the category of relations of which it is the partial map subcategory. This all works whether or not our restriction categories are split.
Title: Dagger Frobenius relations for dagger linearly distributive categories
Abstract: Commutative dagger Frobenius algebras play a central role in categorical quantum mechanics due to their correspondence to orthogonal basis in the category of finite-dimensional Hilbert spaces. Subsequently, such algebras represent quantum observables. Measurement in a dagger monoidal category is an isometry, m: A \to X (i.e, m^\dagger m = 1_X) where A is any object, and X is a special commutative dagger Frobenius Algebra.
In this talk, I will generalize dagger Frobenius Algebras from dagger monoidal categories to dagger linearly distributive categories. We refer to the generalization as dagger linear monoids. I will provide the conditions under which a dagger linear monoid gives a dagger Frobenius algebra in a unitary category. We show the correspondence between dagger linear duals and dagger linear monoids. We find that in a complete and cocomplete category, limit of dagger linear monoids is a dagger linear monoid. Thus one can represent, possibly, infinite dimensional quantum observables using dagger linear monoids.
If time permits, I will discuss measurements for dagger linear monoids. A measurement for a dagger linear monoid is a retract from the linear monoid to a special commutative dagger Frobenius Algebra within the unitary core.
Title: The ZX& calculus
Abstract: Consider ZX&, the fragment of the ZX calculus generated by the copying/addition Frobenius algebras, the not gate and the and gate. I prove that this fragment is complete and universal for a prop of spans of sets, by freely adding units and counits to the inverse products of TOF (the category generated by the Toffoli gate and computational basis states/effects). To prove completeness, I first show show that adding a counit to TOF is the same as computing the classical channels, which is the the same as the discrete Cartesian completion. The completeness of ZX& is obtained via a two way translation between this extension of TOF with units and counits, and ZX&.
Title:Enriched limit doctrines, Lawvere theories.
Abstract: Following Lack and Rosicky’s “Notions of Lawvere Theories”, we will look at the theory of strongly finitely presentable objects. This leads to a notion of enriched Lawvere theory that mirrors the classical case, particularly commutative theories and morphisms of theories.
Title: The fast Weil to go Faa (with Ben MacAdam)
This talk will introduce the notion of a tangent complex in an arbitrary category. We will then use tangent complexes to show that tangent categories are precisely coalgebras of a comonad. We will then show how to reconstruct the Faa di Bruno construction as a subconstruction of the cofree tangent category, and we will not require any combinatorics to do this.
Title: Persisten Homology and it Generalization
Abstract: Persistent homology has seen more widespread use as a tool for analyzing data over the last decade. Generalizations of the theory to multiple parameter filtrations would have even broader applications. In this talk we’ll quickly see how (single parameter) persistent homology works and look at McCleary and Patel’s generalization of the notion of a persistence diagram by exploring a counter-example to their first attempt.