What are the critical points of the master function of a nonrealizable matroid?
speaker: Graham Denham (University of Western Ontario, London)abstract: It is well-known that complex hyperplane arrangements can be conveniently resolved to normal crossing divisors with the help of the permutohedral toric variety. The cohomology algebras of the resulting wonderful compactifications are not only matroid invariants, but Adiprasito, Huh and Katz (2015) found that Hodge-theoretic constraints imposed on them by complex geometry persist for arbitrary matroids. The maximal likelihood variety of a complex arrangement captures the set of critical points of all rational functions with poles and zeros on the arrangement. Its bidegree (as a biprojective variety) agrees with the h-vector of the underlying matroid's broken circuit complex. I will describe work with Federico Ardila and June Huh in which we construct a combinatorial analogue of the maximal likelihood variety for arbitrary matroids, based on a construction that doubles the permutohedron. Although the concept in the title is fictitious, the analogy leads to a proof that the h-vector of the broken circuit complex is a log-concave sequence.
timetable:
Fri 9 Jun, 9:00 - 9:40, Aula Dini
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