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The Slack Realization Space of a Polytope

Connections for Women Workshop: Geometric and Topological Combinatorics August 31, 2017 - September 01, 2017

August 31, 2017 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Rekha Thomas (University of Washington)
Location: MSRI: Simons Auditorium
  • Polytopes

  • slack matrix

  • slack ideal

  • realization spaces

  • binomial ideal

  • toric ideal

  • projective uniqueness

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



We introduce a new model of a realization space of a polytope that arises as the positive part of a real variety. The variety is determined by the slack ideal of the polytope, a saturated determinantal ideal of a sparse generic matrix that encodes the combinatorics of the polytope. The slack ideal offers a uniform computational framework for several classical questions about polytopes such as rational realizability, projectively uniqueness, non-prescribability of faces, and realizability of combinatorial polytopes. The simplest slack ideals are toric. We identify the toric ideals that arise from projectively unique polytopes. New and classical examples illuminate the relationships between projective uniqueness and toric slack ideals.

29417?type=thumb Thomas Notes 3.28 MB application/pdf Download
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H.264 Video 1-Thomas.mp4 188 MB video/mp4 rtsp://videos.msri.org/data/000/029/306/original/1-Thomas.mp4 Download
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