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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
Tags/Keywords
  • Ehrhart

  • weighted projective space

  • real-rooted polynomials

  • unimodal

  • symmetric

  • Eulerian polynomial

  • binomial coefficients

  • numeral system

  • factoradics

  • binary numbers

Primary Mathematics Subject Classification
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
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