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:20260423T053138Z 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 END:VCALENDAR