BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jade Master (University of California Riverside) DTSTART:20210825T170000Z DTEND:20210825T180000Z DTSTAMP:20260422T225821Z UID:EmCats/1 DESCRIPTION:Title: T he Universal Property of the Algebraic Path Problem\nby Jade Master (U niversity of California Riverside) as part of Em-Cats\n\n\nAbstract\nThe a lgebraic path problem generalizes the shortest path problem\, which studie s graphs weighted in the positive real numbers\, and asks for the path bet ween 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 positi ve real numbers. The algebraic path problem generalizes this from graphs w eighted in the positive reals to graphs weighted in an arbitrary commutati ve semiring $R$. With appropriate choices of $R$\, many well known problem s in optimization\, computer science\, probability\, and computing become instances of the algebraic path problem.\n\nIn 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 usi ng 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 solutio n to its algebraic path problem is then given by the free $R$-enriched cat egory on it. The algebraic path problem suffers from combinatorial explosi on 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 eff iciently on large graphs\, it helps to break it down into smaller sub-prob lems. The universal property of the algebraic path problem gives insight i nto 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" appl ication of abstract category theory.\n LOCATION:https://researchseminars.org/talk/EmCats/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Minani Iragi (University of South Africa) DTSTART:20210929T150000Z DTEND:20210929T160000Z DTSTAMP:20260422T225821Z UID:EmCats/2 DESCRIPTION:Title: A categorical study of quasi-uniform structures\nby Minani Iragi (Unive rsity of South Africa) as part of Em-Cats\n\n\nAbstract\nA topology on a s et is usually defined in terms of neighbourhoods\, or equivalently in ter ms of open sets or closed sets. Each of these frameworks allows\, among ot her things\, a definition of continuity. Uniform structures are topologica l spaces with structure to support definitions such as uniform continuity and uniform convergence. Quasi-uniform structures then generalise this ide a in a similar way to how quasi-metrics generalise metrics\, that is\, by dropping the condition of symmetry.\n\nIn this talk we will show how to vi ew these as constructions on the category of topological spaces\, enabling us to generalise the constructions to an arbitrary ambient category. We w ill show how to relate quasi-uniform structures on a category with closure operators. Closure operators generalise the concept of topological closur e operator\, which can be viewed as structure on the category of topologic al spaces obtained by closing subspaces of topological spaces. This method of moving from Top to an arbitrary category is often called "doing topolo gy in categories"\, and is a powerful tool which permits us to apply topol ogically motivated ideas to categories of other branches of mathematics\, such as groups\, rings\, or topological groups.\n LOCATION:https://researchseminars.org/talk/EmCats/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Nuiok Dicaire (University of Edinburgh) DTSTART:20211117T160000Z DTEND:20211117T170000Z DTSTAMP:20260422T225821Z UID:EmCats/4 DESCRIPTION:Title: L ocalisable monads\, from global to local\nby Nuiok Dicaire (University of Edinburgh) as part of Em-Cats\n\n\nAbstract\nMonads have many useful a pplications. In mathematics they are used to study algebras at the level o f theories rather than specific structures. In programming languages\, mon ads provide a convenient way to \nhandle 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 t hen how to handle several instances of such side-effects or a graded colle ction of them. The general approach consists in defining many “small” monads and combining them together using distributive laws.\n\nIn this tal k\, we take a different approach and look for a pre-existing internal stru cture to a monoidal category that allows us to develop a fine-graining of monads. This uses techniques from tensor topology and provides an intrinsi c theory of local computational effects without needing to know how the co nstituent effects interact beforehand. We call the monads obtained "locali sable" and show how they are equivalent to monads in a specific 2-category . To motivate the talk\, we will consider two concrete applications in con currency and quantum theory. This is all covered in our recent paper: http s://arxiv.org/abs/2108.01756 .\n LOCATION:https://researchseminars.org/talk/EmCats/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Jennifer Brown DTSTART:20220413T150000Z DTEND:20220413T160000Z DTSTAMP:20260422T225821Z UID:EmCats/6 DESCRIPTION:Title: S kein Categories and Quantization\nby Jennifer Brown as part of Em-Cats \n\n\nAbstract\nThe 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 w as formulated independently by both mathematicians and physicists\, and is open but well supported.\n\nThe term "quantization" comes from physics\, where it describes the transition from a classical to a quantum descriptio n of a system. Mathematically\, it is a construction that deforms a commut ative algebra into a non-commutative one.\n\nThe A-polynomial is construct ed from the character variety of a knot's complement. We will describe rec ent work on quantizing this construction using skein categories\, with the help of categorical actions\, monads\, and representable functors. This t alk is based on joint work in progress with David Jordan and Tudor Dimofte .\n LOCATION:https://researchseminars.org/talk/EmCats/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Juan F. Meleiro DTSTART:20220518T160000Z DTEND:20220518T170000Z DTSTAMP:20260422T225821Z UID:EmCats/8 DESCRIPTION:Title: T owards Modular Mathematics\nby Juan F. Meleiro as part of Em-Cats\n\n\ nAbstract\nSynthetic Reasoning is a style of mathematics based on axiomati c 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.\n\nBut theo rycrafting is currently an artisanal job\, that requires analysis and synt hesis from scratch for every theory that will be created. A formal (and ca tegorical) toolkit for manipulating these theories could aid the synthetic mathematician in their endeavors\, just as a toolbox can help any artisan in their craft.\n\nModular mathematics is mathematics based on these form al theories that capture a way of Synthetic Reasoning in particular fields \, and can then be combined and compared. In this talk\, I will present wo rk 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 p urely categorical\, a syntactical-categorical\, and a purely syntactical o ne\; all in order to explore some possible ways to do Modular Mathematics. \n LOCATION:https://researchseminars.org/talk/EmCats/8/ END:VEVENT END:VCALENDAR