Logo

Mathematical Sciences Research Institute

Home » Workshop » Schedules » Ehrhart Unimodality and Simplices for Numeral Systems

Ehrhart Unimodality and Simplices for Numeral Systems

Introductory Workshop: Geometric and Topological Combinatorics September 05, 2017 - September 08, 2017

September 06, 2017 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Liam Solus (Royal Institute of Technology (KTH))
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Abstract

In the field of Ehrhart theory, identification of lattice polytopes with unimodal Ehrhart h*-polynomials is a cornerstone investigation. The study of h*-unimodality is home to numerous long-standing conjectures within the field, and proofs thereof often reveal interesting algebra and combinatorics intrinsic to the associated lattice polytopes. Proof techniques for h*-unimodality are plentiful, and some are apparently more dependent on the lattice geometry of the polytope than others. In recent years, proving a polynomial has only real-roots has gained traction as a technique for verifying unimodality of h-polynomials in general. However, the geometric underpinnings of the real-rooted phenomena for h*-unimodality are not well-understood. As such, more examples of this property are always noteworthy. In this talk, we will discuss a family of lattice n-simplices that associate via their normalized volumes to the n^th-place values of positional numeral systems. The h*-polynomials for simplices associated to special systems such as the factoradics and the binary numerals recover ubiquitous h-polynomials, namely the Eulerian polynomials and binomial coefficients, respectively. Simplices associated to any base-r numeral system are also provably real-rooted. We will put the h*-real-rootedness of the simplices for numeral systems in context with that of their cousins, the s-lecture hall simplices, and discuss their admittance of this phenomena as it relates to other, more intrinsically geometric, reasons for h*-unimodality.

Supplements
29482?type=thumb Solus Notes 875 KB application/pdf Download
Video/Audio Files

9-Solus

H.264 Video 9-Solus.mp4 123 MB video/mp4 rtsp://videos.msri.org/data/000/029/357/original/9-Solus.mp4 Download
Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture. The videos are sold at cost for $20USD (shipping included). Please Click Here to send an email to MSRI to purchase the DVD.

See more of our Streaming videos on our main VMath - Streaming Video page.