On the realization space of the cube

Seminar
Speaker
Daniel Kalmanovich (Hebrew University)
Date
26/01/2020 - 15:30 - 14:00Add to Calendar 2020-01-26 14:00:00 2020-01-26 15:30:00 On the realization space of the cube Given a d-dimensional polytope P, its realization space consists of all d-polytopes that have the same combinatorial type as P. For instance, consider the realization space of the d-simplex: since the d-simplex is projectively unique (i.e., any d-simplex can be transformed into any other d-simplex by a projective transformation) the realization space of the d-simplex is connected (and in fact, contractible). As a consequence, any two simplicial d-polytopes, after applying an appropriate projective transformation to one of them, can be glued along a facet to produce a convex polytope. This realizes the connected sum operation for simplicial polytopes geometrically.  We consider the realization space of the d-cube. The d-cube is not projectively unique for , and we cannot realize the connected sum operation geometrically for cubical d-polytopes, . We enlarge the set of transformations, and show that any two realizations of the d-cube are connected by a finite sequence of transformations from this larger set, and that the realization space of the d-cube is contractible. Furthermore, we use this sequence to define an analog of the connected sum operation for cubical d-polytopes. I will mention generalizations to other families of polytopes, and an application of the cubical connected sum. Based on joint work with Karim Adiprasito and Eran Nevo. Room 201, Math and CS Building (Bldg. 216) אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Room 201, Math and CS Building (Bldg. 216)
Abstract

Given a d-dimensional polytope P, its realization space consists of all d-polytopes that have the same combinatorial type as P. For instance, consider the realization space of the d-simplex: since the d-simplex is projectively unique (i.e., any d-simplex can be transformed into any other d-simplex by a projective transformation) the realization space of the d-simplex is connected (and in fact, contractible). As a consequence, any two simplicial d-polytopes, after applying an appropriate projective transformation to one of them, can be glued along a facet to produce a convex polytope. This realizes the connected sum operation for simplicial polytopes geometrically. 

We consider the realization space of the d-cube. The d-cube is not projectively unique for d\ge 3, and we cannot realize the connected sum operation geometrically for cubical d-polytopes, d\ge 4. We enlarge the set of transformations, and show that any two realizations of the d-cube are connected by a finite sequence of transformations from this larger set, and that the realization space of the d-cube is contractible. Furthermore, we use this sequence to define an analog of the connected sum operation for cubical d-polytopes. I will mention generalizations to other families of polytopes, and an application of the cubical connected sum.

Based on joint work with Karim Adiprasito and Eran Nevo.

תאריך עדכון אחרון : 21/01/2020