Nonpositive Curvature Property of Modular Semilattices and Orthoscheme Complexes

a nonpositive curvature property of modular n.w
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Explore the nonpositive curvature property in modular semilattices, orthoscheme complexes, and CAT(0) spaces. Discover the orthoschemes, orthoscheme complexes, geodesic metric spaces, distributive lattices, modular lattices, and noncrossing partition lattices in the context of mathematical theory and applications.

  • Modular Semilattices
  • Orthoscheme Complexes
  • CAT(0) Spaces
  • Geodesic Metric Spaces
  • Mathematical Theory
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  1. A Nonpositive Curvature Property of Modular Semilattices Hiroshi Hirai University of Tokyo hirai@mist.i.u-tokyo.ac.jp The 11-th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications Tokyo, May 29, 2019 1

  2. Orthoscheme (named by Coxeter 1991) Def. ?-dimensional orthoscheme ?-dim. simplex of vertices ? 0,0, ,0 , 1,0, ,0 , (1,1,0, ,0), ,(1,1,1, ,1) 111 3 2 11 110 10 00 000 100 2

  3. Orthoscheme Complex (Brady-McCammond 2012) : graded poset ? := the order complex of metrized by orthoschemes ?3 111 ( ?,?2) ?2 110 ?1 000 100 ?0 3

  4. ?() ? ? ? The length ?(?) of a path ?:[0,1] ?( ) is well-defined ? ?,? inf ? ? ?: ?,? path} (? ,?) is geodesic metric space 4

  5. What are posets to have CAT(0) orthoscheme complex ? ? Cartan, Alexandrov, Topogonov, curvature 0 Def: CAT(0)-space (Gromov 1987) geodesic metric space (?,?) s.t. every triangle is slimmer ? ? 2 (?,?) ? ?,? = ? ?,? = ? ?,? = ? ?2 ? ?2 ? ?2 ? ? ? ? ? ? ? ?,? ? ?2 FACT: CAT(0)-space is uniquely geodesic 5

  6. Boolean Lattice Lem: = 2{1,2, ,?} ? 0,1? CAT(0) 6

  7. Distributive Lattice ?( ) Birkhoff Lem: = The family of ideals in a poset (?, ) ? ? 0,1? ? ? ? ? (?,? ?:? ?)} CAT(0) c.f. order polytope [Stanley 86; Fujishige 84] 7

  8. Modular Lattice ... ? Thm. [Chalopin-Chepoi-H-Osajda 2014] : modular lattice ? : CAT(0) Thm. [H.18] ?: is submodular ?:?( ) is convex MVSP, nc-rank computation [Hamada-H. 17] 8

  9. Brady-McCammond Conjecture (Brady-McCammond 2012) 1 Noncrossing partition on 1,2, ,8 Ex. { 8,1 , 7,2,3 , 4,6 , 5 } 8 2 3 7 Noncrossing partition lattice := { noncrossing partitions } w.r.t. refinement relation 4 6 5 Conj. [BM12] : noncrossing partition lattice ? : CAT(0) Thm. [BM12] BM conjecture true braid group is CAT(0) group 9

  10. The Space of Phylogenetic Trees (Billera-Holmes-Vogtmann 2001) Def: tree-space ??(truncated version) { 0,1 weighed root trees having ? as leaves } = {? 0,12[?]| supp ? is laminar } 0.4 0.9 0.9 0.3 0.3 5 1 2 3 4 0.4 1 2 4 5 3 Thm. [BHV01]: ?? is CAT(0) Owen11,Owen-Provan11: Polytime algorithm for geodesics 10

  11. Gromovs Characterization (Gromov 1987) ? 2?: abstract simplicial complex on ? ? ? ? ? ? ? ? {? 0,1? | supp ? ?} = ? ? 0,1? Thm. [G87] ? ? is CAT(0) ?is flag ? is the family of stable sets of some graph Ex: { laminar families } 22[?] stable sets in the graph of the crossing relation on 2[?] 11

  12. Orthoscheme Complex of Semilattices ? 2?: abstract simplicial complex Obs. ? ? = ? ? 0,1?= ? ?? 2?= ?(?) Lem: A flag complex Boolean semilattice principal ideal is a Boolean lattice & [LF] ? ?,? ?,? ? ? ? ? Thm [CCHO2014]: : median semilattice ? : CAT(0) principal ideal is a distributive lattice & [LF] 12

  13. semimodular lattice This work Not CAT(0) (Hayashi 2019) modular semilattice geometric lattice supersolvable lattice median semilattice modular lattice ?? CAT(0) ?? CAT(0) noncrossing partition lattice BM conjecture partition lattice distributive lattice Boolean semilattice tree space Boolean lattice 13

  14. Main Result Def.[Bandelt-Van de Vel-Verheul 1993] modular semilattice principal ideal is a modular lattice & [LF] ? ?,? ?,? ? ? ? ? Thm. [H2019] Conj [CCHO2014] : modular semilattice ? : CAT(0). 14

  15. Proof Outline Recall: CAT(0) property Unique Geodesic Property For ? , We prove UGP in constructive way Key Fact: Owen-Provan algorithm (Owen11, Owen-Provan 11) for geodesics in tree space ?? can prove UGP (without knowing CAT(0)) H.19 Miller-Owen-Provan 15 flag complex median semilattice Boolean semilattice [H.19] ?,? ?( ), median subsemilattice of s.t. (?,?)-geodesic belongs to ?( ) ?( ). 15

  16. Concluding Remarks Challenge to BM-conjecture difficulty: violation of modularity Algorithmic aspect for geodesics: tree space/flag complex max. weight stable set in bipartite graph median semilattice max. weight ideal in poset modular semilattice Generalization of weighted MVSP H. Hirai: A Nonpositive Curvature Property of Modular Semilattices, arXiv1905.01449, 2019 Papers & slides: http://www.misojiro.t.u-tokyo.ac.jp/~hirai/ Thank you for your attention 16

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