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SUMMARY:Ayah Almousa (University of Minnesota - Twin Cities)
DTSTART:20221201T151500Z
DTEND:20221201T170000Z
DTSTAMP:20260422T065352Z
UID:CJCS/93
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CJCS/93/">Ro
 ot Polytopes\, Tropical Types\, and Toric Edge Ideals</a>\nby Ayah Almousa
  (University of Minnesota - Twin Cities) as part of Copenhagen-Jerusalem C
 ombinatorics Seminar\n\n\nAbstract\nThis is joint work with Anton Dochterm
 ann (Texas State) and Ben Smith (Manchester). We consider arrangements of 
 tropical hyperplanes where the apices of the hyperplanes are taken to infi
 nity in certain directions. Such an arrangement defines a decomposition of
  Euclidean space where a cell is determined by its `type' data\, analogous
  to the covectors of an oriented matroid. By work of Develin-Sturmfels and
  Fink-Rincón\, these `tropical complexes' are dual to (regular) subdivisi
 ons of root polytopes\, which in turn are in bijection with mixed subdivis
 ions of certain generalized permutohedra. Extending previous work with Jos
 wig-Sanyal\, we show how a natural monomial labeling of these complexes de
 scribes polynomial relations (syzygies) among `type ideals' which arise na
 turally from the combinatorial data of the arrangement. In particular\, we
  show that the cotype ideal is Alexander dual to a corresponding initial i
 deal of the lattice ideal of the underlying root polytope. This leads to n
 ovel ways of studying algebraic properties of various monomial and toric i
 deals\, as well as relating them to combinatorial and geometric properties
 . In particular\, our methods of studying the dimension of the tropical co
 mplex leads to new formulas for homological invariants of toric edge ideal
 s of bipartite graphs\, which have been extensively studied in the commuta
 tive algebra community.\n
LOCATION:https://researchseminars.org/talk/CJCS/93/
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