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:20260404T110910Z UID:EmCats/1 DESCRIPTION:Title: The Universal Property of the Algebraic Path Problem\nby Jade Ma ster (University of California Riverside) as part of Em-Cats\n\n\nAbstract \nThe algebraic path problem generalizes the shortest path problem\, which studies graphs weighted in the positive real numbers\, and asks for the p ath between a given pair of vertices with the minimum total weight. This p ath may be computed using an expression built up from the "min" and "+" of positive real numbers. The algebraic path problem generalizes this from g raphs weighted in the positive reals to graphs weighted in an arbitrary co mmutative 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.\n\nIn this talk we will sh ow how solutions to the algebraic path problem are computed with a left ad joint\, and this opens the door to reasoning about the algebraic path prob lem 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 $R$-enric hed 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 prob lem efficiently on large graphs\, it helps to break it down into smaller s ub-problems. The universal property of the algebraic path problem gives in sight into the way that solutions to these sub-problems may be glued toget her to form a solution to the whole\, which may be regarded as a "practica l" application of abstract category theory.\n LOCATION:https://stable.researchseminars.org/talk/EmCats/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Minani Iragi (University of South Africa) DTSTART:20210929T150000Z DTEND:20210929T160000Z DTSTAMP:20260404T110910Z UID:EmCats/2 DESCRIPTION:Title: A categorical study of quasi-uniform structures\nby Minani Iragi (University of South Africa) as part of Em-Cats\n\n\nAbstract\nA 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\, a mong other things\, a definition of continuity. Uniform structures are top ological spaces with structure to support definitions such as uniform cont inuity and uniform convergence. Quasi-uniform structures then generalise t his idea in a similar way to how quasi-metrics generalise metrics\, that i s\, by dropping the condition of symmetry.\n\nIn this talk we will show ho w to view these as constructions on the category of topological spaces\, e nabling us to generalise the constructions to an arbitrary ambient categor y. 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 to pological 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 appl y topologically motivated ideas to categories of other branches of mathema tics\, such as groups\, rings\, or topological groups.\n LOCATION:https://stable.researchseminars.org/talk/EmCats/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Nuiok Dicaire (University of Edinburgh) DTSTART:20211117T160000Z DTEND:20211117T170000Z DTSTAMP:20260404T110910Z UID:EmCats/4 DESCRIPTION:Title: Localisable monads\, from global to local\nby Nuiok Dicaire (Uni versity of Edinburgh) as part of Em-Cats\n\n\nAbstract\nMonads have many u seful applications. In mathematics they are used to study algebras at the level of theories rather than specific structures. In programming language s\, monads 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 questi on is then how to handle several instances of such side-effects or a grade d collection of them. The general approach consists in defining many “sm all” monads and combining them together using distributive laws.\n\nIn t his talk\, we take a different approach and look for a pre-existing intern al structure to a monoidal category that allows us to develop a fine-grain ing of monads. This uses techniques from tensor topology and provides an i ntrinsic 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-c ategory. To motivate the talk\, we will consider two concrete applications in concurrency and quantum theory. This is all covered in our recent pape r: https://arxiv.org/abs/2108.01756 .\n LOCATION:https://stable.researchseminars.org/talk/EmCats/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Jennifer Brown DTSTART:20220413T150000Z DTEND:20220413T160000Z DTSTAMP:20260404T110910Z UID:EmCats/6 DESCRIPTION:Title: Skein Categories and Quantization\nby Jennifer Brown as part of Em-Cats\n\n\nAbstract\nThe beautiful AJ conjecture predicts that a (yet-un defined) quantization of one knot invariant — the A-polynomial — annih ilates another famous invariant\, the colored Jones polynomial. This conje cture was formulated independently by both mathematicians and physicists\, and is open but well supported.\n\nThe term "quantization" comes from phy sics\, where it describes the transition from a classical to a quantum des cription of a system. Mathematically\, it is a construction that deforms a commutative algebra into a non-commutative one.\n\nThe A-polynomial is co nstructed from the character variety of a knot's complement. We will descr ibe recent work on quantizing this construction using skein categories\, w ith the help of categorical actions\, monads\, and representable functors. This talk is based on joint work in progress with David Jordan and Tudor Dimofte.\n LOCATION:https://stable.researchseminars.org/talk/EmCats/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Juan F. Meleiro DTSTART:20220518T160000Z DTEND:20220518T170000Z DTSTAMP:20260404T110910Z UID:EmCats/8 DESCRIPTION:Title: Towards Modular Mathematics\nby Juan F. Meleiro as part of Em-Ca ts\n\n\nAbstract\nSynthetic Reasoning is a style of mathematics based on a xiomatic theories that aim to capture the fundamental and essential struct ures in a particular subject. Such theories are often type theories with i ntended interpretations inside structured categories such as toposes.\n\nB ut theorycrafting is currently an artisanal job\, that requires analysis a nd synthesis from scratch for every theory that will be created. A formal (and categorical) toolkit for manipulating these theories could aid the sy nthetic mathematician in their endeavors\, just as a toolbox can help any artisan in their craft.\n\nModular mathematics is mathematics based on the se formal theories that capture a way of Synthetic Reasoning in particular fields\, and can then be combined and compared. In this talk\, I will pre sent work in progress towards a framework for such modular mathematics. Un iversal Logic will be our guide for the capabilities that such a framework should provide\, including translation between\, and combinations of theo ries. I will present a formal theory called MMT (introduced by Florian Rab e) that follows such a guide. I will then present three formal approaches to the definition of the fundamental group\, each following a distinct sty le: a purely categorical\, a syntactical-categorical\, and a purely syntac tical one\; all in order to explore some possible ways to do Modular Mathe matics.\n LOCATION:https://stable.researchseminars.org/talk/EmCats/8/ END:VEVENT END:VCALENDAR