Title: Realization Spaces of Polytopes
Abstract:In this talk I will present a model for the realization space of a polytope which represents a polytope by its slack matrix. This model provides a natural algebraic relaxation for the realization space, and comes with a defining ideal which can be used as a computational engine to answer questions about the realization space. We will see how this model is related to more classical realization space models (representing realizations by Gale diagrams or points of the Grassmannian). I will also show how the relationship of the slack model to the classical models can be used to improve computational efficiency of the slack model.
Title: The shapes of level curves of real polynomials near strict local minima
Abstract: We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.
The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity\; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.
Title: Gorenstein polytopes
Abstract: A Gorenstein polytope is a lattice polytope one of whose dilated polytopes is a reflexive polytope. In my talk, after reviewing Gorenstein polytopes from a viewpoint of enumeration of lattice points, the conjecture that every (0, 1)-polytope is a face of a Gorenstein (0, 1)-polytope will be discussed. No special knowledge will be required to understand my talk.