Speakers
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- Jade Master (University of California Riverside)
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The Universal Property of the Algebraic Path Problem
- 17:00 UTC, 25th of August 2021
- Slides | YouTube
The Universal Property of the Algebraic Path Problem
The algebraic path problem generalizes the shortest path problem, which studies graphs weighted in the positive real numbers, and asks for the path between a given pair of vertices with the minimum total weight. This path may be computed using an expression built up from the "min" and "+" of positive real numbers. The algebraic path problem generalizes this from graphs weighted in the positive reals to graphs weighted in an arbitrary commutative semiring R. With appropriate choices of R, many well known problems in optimization, computer science, probability, and computing become instances of the algebraic path problem.
In this talk we will show how solutions to the algebraic path problem are computed with a left adjoint, and this opens the door to reasoning about the algebraic path problem using the techniques of modern category theory. When R is "nice", a graph weighted in R may be regarded as an R-enriched graph, and the solution to its algebraic path problem is then given by the free R-enriched category on it. The algebraic path problem suffers from combinatorial explosion so that solutions can take a very long time to compute when the size of the graph is large. Therefore, to compute the algebraic path problem efficiently on large graphs, it helps to break it down into smaller sub-problems. The universal property of the algebraic path problem gives insight into the way that solutions to these sub-problems may be glued together to form a solution to the whole, which may be regarded as a "practical" application of abstract category theory.
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- Minani Iragi (University of South Africa)
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A categorical study of quasi-uniform structures
- 15:00 UTC, 29th of September 2021
- Slides | YouTube
A categorical study of quasi-uniform structures
A topology on a set is usually defined in terms of neighbourhoods, or equivalently in terms of open sets or closed sets. Each of these frameworks allows, among other things, a definition of continuity. Uniform structures are topological spaces with structure to support definitions such as uniform continuity and uniform convergence. Quasi-uniform structures then generalise this idea in a similar way to how quasi-metrics generalise metrics, that is, by dropping the condition of symmetry.
In this talk we will show how to view these as constructions on the category of topological spaces, enabling us to generalise the constructions to an arbitrary ambient category. We will show how to relate quasi-uniform structures on a category with closure operators. Closure operators generalise the concept of topological closure operator, which can be viewed as structure on the category of topological spaces obtained by closing subspaces of topological spaces. This method of moving from Top to an arbitrary category is often called "doing topology in categories", and is a powerful tool which permits us to apply topologically motivated ideas to categories of other branches of mathematics, such as groups, rings, or topological groups.
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- Nuiok Dicaire (University of Edinburgh)
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Localisable monads, from global to local
- 16:00 UTC, 17th of November 2021
- Slides | YouTube
Localisable monads, from global to local
Monads have many useful applications. In mathematics they are used to study algebras at the level of theories rather than specific structures. In programming languages, monads provide a convenient way to handle computational side-effects which include, roughly speaking, things like interacting with external code or altering the state of the program's variables. An important question is then how to handle several instances of such side-effects or a graded collection of them. The general approach consists in defining many “small” monads and combining them together using distributive laws.
In this talk, we take a different approach and look for a pre-existing internal structure to a monoidal category that allows us to develop a fine-graining of monads. This uses techniques from tensor topology and provides an intrinsic theory of local computational effects without needing to know how the constituent effects interact beforehand. We call the monads obtained "localisable" and show how they are equivalent to monads in a specific 2-category. To motivate the talk, we will consider two concrete applications in concurrency and quantum theory. This is all covered in our recent paper: arxiv.org/abs/2108.01756.
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- Jennifer Brown (University of California Davis)
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Skein Categories and Quantization
- 15:00 UTC, 13th of April 2022
- Slides | YouTube
Skein Categories and Quantization
The beautiful AJ conjecture predicts that a (yet-undefined) quantization of one knot invariant — the A-polynomial — annihilates another famous invariant, the colored Jones polynomial. This conjecture was formulated independently by both mathematicians and physicists, and is open but well supported.
The term "quantization" comes from physics, where it describes the transition from a classical to a quantum description of a system. Mathematically, it is a construction that deforms a commutative algebra into a non-commutative one.
The A-polynomial is constructed from the character variety of a knot's complement. We will describe recent work on quantizing this construction using skein categories, with the help of categorical actions, monads, and representable functors. This talk is based on joint work in progress with David Jordan and Tudor Dimofte.
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- Juan F. Meleiro (Universidade de São Paulo)
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Towards Modular Mathematics
- 16:00 UTC, 18th of May 2022
- Slides | YouTube
Towards Modular Mathematics
Synthetic Reasoning is a style of mathematics based on axiomatic theories that aim to capture the fundamental and essential structures in a particular subject. Such theories are often type theories with intended interpretations inside structured categories such as toposes.
But theorycrafting is currently an artisanal job, that requires analysis and synthesis from scratch for every theory that will be created. A formal (and categorical) toolkit for manipulating these theories could aid the synthetic mathematician in their endeavors, just as a toolbox can help any artisan in their craft.
Modular mathematics is mathematics based on these formal theories that capture a way of Synthetic Reasoning in particular fields, and can then be combined and compared. In this talk, I will present work in progress towards a framework for such modular mathematics. Universal Logic will be our guide for the capabilities that such a framework should provide, including translation between, and combinations of theories. I will present a formal theory called MMT (introduced by Florian Rabe) that follows such a guide. I will then present three formal approaches to the definition of the fundamental group, each following a distinct style: a purely categorical, a syntactical-categorical, and a purely syntactical one; all in order to explore some possible ways to do Modular Mathematics.