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Title: More on the Categorical and Differential Algebra of Commutative Rigs
Abstract: In this talk we’ll continue our friendly introduction to the differential algebraic theory of commutative rings by moving towards answering the question: “In what sense are the Kahler differentials functorial in the first place?” We’ll start by introducing fibrations and their morphisms, move on to briefly discussing the Grothendieck Construction (which allows one to move between fibrations and pseudofunctors, depending on taste), and then explain the sense in which taking the module of relative Kahler differentials is functorial in CRig.
The scope and vibes of this talk are meant to be more relaxed and informal, so if you’ve never met rigs before and want to get to know how their differential algebra works this is a fantastic opportunity (which of course I’d say, for I am the speaker) to see. Alternatively, this makes precise many constructions involving Kahler differentials which are at best ad-hoc in traditional commutative ring theory.
Title: The Differential Algebra of Commutative Rigs: Derivations and Differentials
Abstract: In the last talk I gave on commutative rigs, we got to know a bit about how the categories of commutative rig algebras and their modules interact together. In this talk we’ll continue towards our ultimate goal of understanding the differential/algebraic geometry of rigs by getting to know how the theory of derivations and differentials interacts with rigs. I’ll start by recalling our main cast of characters, define what it means to be a derivation, define the rig module of Kahler differentials, and then start to show some of the properties these modules have. Depending on time we’ll discuss how to encode derivations as solutions to specific lifting problems and/or in what sense the Kahler differentials construction is functorial. This talk will be relatively gentle in delivery, so all are encouraged to come and see how cool rigs are!
Title: Generalising the Coecke-Pavlovic-Vicary Correspondence
Abstract:
In their paper [1], Coecke-Palvovic-Vicary (CPV) gives a correspondence between orthogonal bases in finite-dimensional Hilbert Spaces (FdHilb) and Commutative-Dagger Frobenius algebras in FdHilb. In the first part of my talk, I will go over this correspondence and the associated categorical statements.
In the second part of my talk, I will give an outline of the programme about how this correspondence can be generalised to arbitrary dimensions. I will first introduce the ingredients involved- Finiteness spaces, Lefschetz Spaces and time-permitting linear monoids and then will give a few results connecting these mathematical objects.
Reference:
[1] Coecke, B., Pavlovic, D., & Vicary, J. (2012). A new description of orthogonal bases. Mathematical Structures in Computer Science, 23(3).
Title: A Planned Outline for the Abstract Machine for the Message Passing Language’s Memory Management System
Abstract: CaMPL is a functional programming language designed by Cockett and Pastro which uses message-passing semantics for concurrency. In order to run a CaMPL program, the code is first compiled down into code for MPL, which is run by an interpreter. In this talk, I will discuss the design decisions I plan to take when implementing said interpreter in C.
When implementing a programming language, memory management is very important. The way a system stores, allocates, and garbage collects data has a high impact on its performance. Furthermore, there’s no ‘one size fits all’ solution. Every language has unqiue propeties which lend themselves to different backend memory mangement systems.
There are two aspects of CaMPL in particular that are relevant to the system I am implementing:
-functional sequntial semantics
-message passing semantics on the concurrent side
These traits are shared by the programming language Erlang, and many of my solutions will be adapted from those used by that programming language.
I will start by discussing the overall memory architecture of the system, which uses private heaps for individual processes and a shared global heap for messages. Then I’ll talk about the implications for garbage collection that this scheme has, and outline the local and global heap garbage collection algorithms. Next, I’ll discuss how memory allocation will work on both sides, and I’ll conclude by discussing the algorithms I plan to use to reduce memory fragmentation.
Title: The Categorical Algebra of Rigs and Kähler Differential Functor for Rigs
Abstract: The theory of rigs (rings without negatives), also known as semirings in the literature, is an interesting and important field of algebra with applications to theoretical computer science, logic, economics (through the use of tropical geometry and the tropical semiring), and also to arithmetic geometry (in the styles of Deitmar, or of Toën and Vasquie, or of Lorscheid — all of these are built to discuss a theory of schemes over the “field with one element”).
In this talk, based on joint work in progress with Robin Cockett, I will introduce the category of commutative rigs and indicate a careful and precise construction of the ways in which the categories of commutative rig algebras and modules over commutative rigs interact. More precisely, I will show that there are fibrations associated to both the commutative algebra and module constructions and that the underlying module/symmetric algebra functors sit as fibre-wise adjoints between the categories of said fibrations. Afterwards, depending on time, I will discuss some combination and/or permuation of the following topics: what localizations of rigs are, what the module of Kähler differentials are, how they arise as a functor into the module fibration, and also how the module of Kähler differentials interacts with localizations and tensor products.
Title: Drazin Inverses in Categories
Abstract: In this talk, I will introduce Drazin inverses from a categorical perspective. Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory and ring theory. Drazin inverses can also be defined for endomorphisms in any category. In this talk, I will introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, and explore various properties of these inverses.
Title: Semantics of Higher-Order Processes in Categorical
Message Passing Language (CaMPL)
Abstract: Categorical Message Passing Language (CaMPL) is a concurrent programming language based on a categorical semantic given by a linear actegory. The sequential side of CaMPL is a functional-style programming language, while the concurrent side supports message passing between processes along channels with concurrent types called protocols. A notable feature that could be added to CaMPL, is the support for higher-order processes on the concurrent side, allowing processes to be passed to other processes. While passing concurrent processes between processes is feasible, supporting recursive process definitions requires the ability to reuse the passed process multiple times. However, since concurrent resources cannot be duplicated, processes must be represented as sequential data. Consequently, the concurrent side must be enriched into the sequential side. In this presentation we talk about the categorical semantics that lets us store a concurrent processes as sequential data and yet use them.
Title: Functor calculus for multi-persistent homology
Abstract: Functor calculus is a general framework for studying functors, and comes in many flavors. We construct a variant of functor calculus, called poset cocalculus, for studying functors out of posets. Our motivation is to better understand multi-persistence modules, as these modules can be viewed as functors from R^k into a category of chain complexes. We demonstrate some properties of poset cocalculus, and apply these in concrete examples with multi-persistence modules. We finish by giving examples of poset cocalculus in other contexts, such as filtrations of simplicial complexes.
Title: Moduli Spaces of Arrows
Abstract: Many moduli spaces parametrize isomorphism classes in some category, and we can wonder if the morphisms of said category can be similarly parametrized. The relevant moduli stack encodes families of morphisms, but its points are still objects and not morphisms. The idea of this talk is to consider a moduli stack whose points are the morphisms involved in the original moduli problem. We will develop this idea for the specific case of the moduli problem of vector bundles over a fixed base and observe that this has a generalization: if we can parametrize arrows of vector bundles, we can parametrize diagrams thereof of more general shapes. We will then discuss some implications of this line of thought for the moduli problem of Higgs bundles and non-Abelian Hodge theory. Time permitting, we will discuss abstract moduli theory and homotopy theory in this light. This talk reports on joint work with Steven Rayan.
Title: Weighted pullbacks in V-graded categories
Abstract: Introduced by Richard Wood in 1976, categories graded by a monoidal category V generalize both V-actegories and V-enriched categories. In this talk, we review some basics of V-graded categories, and then we introduce a notion of weighted pullback in V-graded categories. Weighted pullbacks are certain weighted limits that generalize the usual (conical) pullbacks, yet they also specialize to certain notions of universal quantification and certain dependent products. Indeed, weighted pullbacks generalize simple products in the codomain fibration of a cartesian closed category with finite limits and, in particular, simple universal quantification in the subobject fibration of such a category. Generalizing the latter example, we introduce notions of simple product and simple universal quantification in V-actegories as special cases of the notion of weighted pullback. In particular, weighted pullbacks thus give rise to a notion of simple universal quantification in monoidal categories.
