Gordon Williams: Prisms aren't boring
The organizers of the Algebraic Graph Theory International Webinar would like to invite you to another presentation, this time on January 18, 2022, at 7pm Central European Time (= 6pm UTC), delivered by Gordon Williams.
Title: Prisms Aren't Boring
From a geometric standpoint, prisms are relatively easy to conceptualize, even when formed over polytopes of high rank. The construction of a prism over an abstract polytope, map, or maniplex also has a rather natural description (Gleason and Hubard's discussion in  is quite nice, where it is an instructive example of a type of product). Since the prism operation seems comparatively easy to describe, one might be tempted to describe the prism operation as being somewhat boring. However, once one tries to understand the relationship between prisms and other polytopes, life gets a lot more interesting.
In this talk we are going to explore prisms over abstract polytopes from the perspective of regular covering theory, where the goal is to identify the minimal regular abstract polytope that covers a given polytope. In , the connection groups of prisms over polygons were described in some detail, and a consequence of a result in  is that prisms over polygons always have a unique minimal regular cover. In this talk, we'll prove a very general result about a family of prisms of higher rank that always possess a unique minimal regular cover, describe a conjecture about when even more do, and discuss some examples of some prisms over polytopes that likely don't. Along the way we'll discuss connections to the theory of maps and maniplexes, the importance of the connection group of a polytope in thinking about this kind of question, and some of the new computational tools we've been working on.
This is joint work with Gabriel Cunningham and Mark Mixer, and incorporates some results Cunningham and I developed with Daniel Pellicer.
 I. Gleason and I. Hubard. Products of abstract polytopes. J. Combin. Theory Ser. A, 157, 287-320, 2018.
 M. I. Hartley, D. Pellicer, and G. I. Williams. Minimal covers of the prisms and antiprisms. Discr. Math., 312(20), 3046-3058, October 2012.
 B. Monson, D. Pellicer, and G. I. Williams. Mixing and monodromy of abstract polytopes. Trans. of the AMS, 366, 2651-2681, 2014.
Further details may be found at http://euler.doa.fmph.uniba.sk/AGTIW.html where you can also find the slides and the recordings of our previous presentations. Also, if you wish to advertise an AGT friendly conference on this page, please send us the link.
Hoping to see you at the webinar, and wishing you all the best.
Isabel Hubard, Robert Jajcay and Primoz Potočnik