BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anton Dochtermann (Texas State University) DTSTART:20201021T140000Z DTEND:20201021T150000Z DTSTAMP:20260423T021425Z UID:CATGT/13 DESCRIPTION:Title: S hellings\, chordality\, and Simon's conjecture\nby Anton Dochtermann ( Texas State University) as part of Applications of Combinatorics in Algebr a\, Topology and Graph Theory\n\n\nAbstract\nA simplicial complex X is "sh ellable" if there exists an ordering of its facets that satisfies nice int ersection properties. Shellability imposes strong topological and algebrai c conditions on X and its Stanley-Reisner ring\, and has been an important tool in geometric and algebraic combinatorics. Examples of shellable com plexes include boundaries of simplicial polytopes and the independence com plex of matroids. In general it is difficult (NP hard) to determine if a given complex is shellable\, and X is said to be "extendably shellable" if a greedy algorithm always succeeds. A conjecture of Simon posits that th e k-skeleton of a simplex on vertex set [n] is extendably shellable. \n\n Simon's conjecture has been established for k=2 but until recently all oth er nontrivial cases were open. We show how the case k=n-3 follows from an application of chordal graphs and the notion of "exposed edges"\, and in f act prove that any shellable d-dimensional complex on at most d+3 vertices is extendably shellable. This leads to a notion of higher-dimensional ch ordality which connects Simon's conjecture to tools in commutative algebra and simple homotopy theory. We also explore other cases of Simon's conjec ture and for instance prove that any vertex decomposable complex can be co mpleted to a shelling of a simplex skeleton. Parts of this are joint work with Culertson\, Guralnik\, Stiller\, and Oh.\n LOCATION:https://researchseminars.org/talk/CATGT/13/ END:VEVENT END:VCALENDAR