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Two problems in Geometric Combinatorics
Faculty Research Seminar| Speaker: | Fu Liu, UC Davis |
| Location: | Zoom |
| Start time: | Tue, Mar 9 2021, 2:10PM |
A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset (with a reasonable structure), determine whether there exists a polytope whose face lattice is the poset. In my recent joint work with Federico Castillo, we provided a realization of a poset constructed by Kapranov as a hybrid between the face poset of the permutohedron and the associahedron. In the first part of the talk, I will introduce the general realization problem and briefly discuss our construction.
In the second part of the talk, I will discuss the Ehrhart positivity question on polytopes. Ehrhart polynomials count the number of lattice points in dilations of a fixed polytope, and I am interested in studying the question of when the coefficients of this polynomial are positive using geometric tools.
I will try to make the talk self-contained, and thus no previous knowledge of polytopes is required.