Polytopes and Friends - Abstracts
Monday
Georg Loho: TBA
Irem Portakal: TBA
Tuesday
Alex Black: Shadows of Polytopes
Abstract: A shadow of a polytope is its image under a linear projection to two dimensions. Shadows appear in various applications including parametric optimization, algebraic complexity theory, and the theory of the simplex method. In each of these applications, the primary concern is the number of vertices of a shadow, which is used to bound the run-time of several algorithms. In this talk, I will survey known results about sizes of shadows and highlight my work on the topic. Throughout I will describe important open problems. This talk is partially based on joint work with Francisco Criado.
Eleni Tzanaki: Upper bounds on the number of faces of the Minkowski sum of polytopes
Given two convex polytopes $P, Q$, their Minkowski sum, which is again a polytope, is defined as $P+Q = \{p+q:\, p\in P, q\in Q\}$. In this talk I will present tight expressions for the maximum number of $k$-dimensional faces, $0 \leq {k} \leq {d-1}$, of the Minkowski sum $P_1 + \cdots + P_r$, of $r$ convex $d$-dimensional polytopes $P_1, \ldots, P_r$ in $\mathbb R^d$ where $d \geq {2}$ and $r<d$, as a (recursively defined) function of the number of vertices of the polytopes. These upper bounds are proved making use of basic notions such as $f$- and $h$-vector calculus, stellar-subdivisions and shellings, and generalize the steps used by McMullen to prove the Upper Bound Theorem for polytopes. The key idea behind the approach is to express the Minkowski sum $P_1 + \cdots + P_r$ as a section of the Cayley polytope $C$ of the summands; bounding the $k$-faces of $P_1 + \cdots + P_r$ reduces to bounding the subset of the $(k + r - 1)$-faces of $C$ that contain vertices from each of the $r$ polytopes. I will focus on the steps of the proof for the Minkowski sum of two polytopes. I will then explain how these steps should be adjusted when one considers the Minkowski sum of more polytopes.
Vincent Pilaud: Wigglyhedra
Abstract: The talk will present the construction of the wigglyhedron, a polytope whose vertices correspond
* to wiggly pseudotriangulations (some sort of degenerations of the pseudotriangulations of a point set in general position) defining the facets of the wiggly complex,
* and to wiggly permutations (some permutations defined by a pattern avoidance condition) defining the wiggly lattice.
We will see that the wigglyhedron contain copies of all type A Cambrian associahedra, and evoke the motivation of the wiggly complex from categorical representation theory.
This is based on joint work with Asilata Bapat (arXiv:2407.11632).
Wednesday
Jacinta Torres: Minkowski properties for string polytopes
Abstract: String polytopes were introduced by Littelmann as generalizations of Gelfand-Tsetlin patterns to arbitrary semi-simple complex algebraic groups. For the general linear group, the defining inequalities of these polytopes are given in terms of wiring diagrams. We show certain Minkowski sum properties of them in this context, inspired by recent work with Patimo on formulas for Kostka-Foulkes polynomials. We use an explicit construction by De Laporte-Genz-Koshevoy of the crystal structure on the set of lattice points. This is joint work with Lara Bossinger.