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Totally positive spaces: topology and applications Pavel Galashin April 26, 2019 Joint work with Steven Karp, Thomas Lam, and Pavlo Pylyavskyy arXiv:1707.02010 , arXiv:1807.03282 , arXiv:1904.00527 w 0 w 0 s 2 s 1 s 1 s 2 s 2 s 1 s 1 s 2

  1. Flag variety Let G := GL n ( R ) and B := { upper triangular n × n matrices } . Definition Flag variety: { V 0 ⊂ V 1 ⊂ · · · ⊂ V n = R n | dim V i = i for all 0 ≤ i ≤ n } . G / B = gB ↔ ( V 0 , V 1 , . . . , V n ) , where V i := span of first i columns of g . Definition (Lusztig (1994)) Let G � 0 = { totally nonnegative matrices in G } and ( G / B ) � 0 := { gB | g ∈ G � 0 } = { gB | g ∈ U − � 0 } . Pavel Galashin Totally positive spaces 04/26/2019 8 / 24

  2. Flag variety Let G := GL n ( R ) and B := { upper triangular n × n matrices } . Definition Flag variety: { V 0 ⊂ V 1 ⊂ · · · ⊂ V n = R n | dim V i = i for all 0 ≤ i ≤ n } . G / B = gB ↔ ( V 0 , V 1 , . . . , V n ) , where V i := span of first i columns of g . Definition (Lusztig (1994)) Let G � 0 = { totally nonnegative matrices in G } and ( G / B ) � 0 := { gB | g ∈ G � 0 } = { gB | g ∈ U − � 0 } . Example All n ! coordinate flags { wB | w ∈ S n } belong to ( G / B ) � 0 . Pavel Galashin Totally positive spaces 04/26/2019 8 / 24

  3. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  4. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Write v ′ ≤ v ≤ w ≤ w ′ . ( v , w ) � ( v ′ , w ′ ) ⇐ ⇒ Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  5. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Write v ′ ≤ v ≤ w ≤ w ′ . ( v , w ) � ( v ′ , w ′ ) ⇐ ⇒ Theorem (Rietsch (1999, 2006)) ( Q , � ) is the “face poset” of ( G / B ) � 0 . Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  6. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Write v ′ ≤ v ≤ w ≤ w ′ . ( v , w ) � ( v ′ , w ′ ) ⇐ ⇒ Theorem (Rietsch (1999, 2006)) ( Q , � ) is the “face poset” of ( G / B ) � 0 . Theorem (Williams (2007)) The poset ( Q , � ) is thin and shellable. Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  7. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Write v ′ ≤ v ≤ w ≤ w ′ . ( v , w ) � ( v ′ , w ′ ) ⇐ ⇒ Theorem (Rietsch (1999, 2006)) ( Q , � ) is the “face poset” of ( G / B ) � 0 . Theorem (Williams (2007)) The poset ( Q , � ) is thin and shellable. Thus there exists some regular CW complex with face poset ( Q , � ) . Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  8. Face poset of ( G / B ) � 0 Definition Let Q := { ( v , w ) ∈ S n × S n | v ≤ w } . Write v ′ ≤ v ≤ w ≤ w ′ . ( v , w ) � ( v ′ , w ′ ) ⇐ ⇒ Theorem (Rietsch (1999, 2006)) ( Q , � ) is the “face poset” of ( G / B ) � 0 . Theorem (Williams (2007)) The poset ( Q , � ) is thin and shellable. Thus there exists some regular CW complex with face poset ( Q , � ) . Conjecture (Williams (2007)) ( G / B ) � 0 is a regular CW complex. Pavel Galashin Totally positive spaces 04/26/2019 9 / 24

  9. → ( G / B ) � 0 U � 0 ֒ ( Q , � ) (id , w 0 ) (id , s 1 s 2 ) (id , s 2 s 1 ) ( s 1 , w 0 ) ( s 2 , w 0 ) (id , s 1 ) (id , s 2 ) ( s 1 , s 1 s 2 ) ( s 1 , s 2 s 1 ) ( s 2 , s 1 s 2 ) ( s 2 , s 2 s 1 ) ( s 1 s 2 , w 0 ) ( s 2 s 1 , w 0 ) (id , id) ( s 1 , s 1 ) ( s 2 , s 2 ) ( s 1 s 2 , s 1 s 2 ) ( s 2 s 1 , s 2 s 1 ) ( w 0 , w 0 ) Pavel Galashin Totally positive spaces 04/26/2019 10 / 24

  10. → ( G / B ) � 0 U � 0 ֒ ( Q , � ) ( G / B ) � 0 w 0 (id , w 0 ) s 2 s 1 s 1 s 2 (id , s 1 s 2 ) (id , s 2 s 1 ) ( s 1 , w 0 ) ( s 2 , w 0 ) (id , s 1 ) (id , s 2 ) ( s 1 , s 1 s 2 ) ( s 1 , s 2 s 1 ) ( s 2 , s 1 s 2 ) ( s 2 , s 2 s 1 ) ( s 1 s 2 , w 0 ) ( s 2 s 1 , w 0 ) s 1 s 2 (id , id) ( s 1 , s 1 ) ( s 2 , s 2 ) ( s 1 s 2 , s 1 s 2 ) ( s 2 s 1 , s 2 s 1 ) ( w 0 , w 0 ) id Pavel Galashin Totally positive spaces 04/26/2019 10 / 24

  11. → ( G / B ) � 0 U � 0 ֒ ( Q , � ) ( G / B ) � 0 w 0 (id , w 0 ) s 2 s 1 s 1 s 2 (id , s 1 s 2 ) (id , s 2 s 1 ) ( s 1 , w 0 ) ( s 2 , w 0 ) (id , s 1 ) (id , s 2 ) ( s 1 , s 1 s 2 ) ( s 1 , s 2 s 1 ) ( s 2 , s 1 s 2 ) ( s 2 , s 2 s 1 ) ( s 1 s 2 , w 0 ) ( s 2 s 1 , w 0 ) s 1 s 2 (id , id) ( s 1 , s 1 ) ( s 2 , s 2 ) ( s 1 s 2 , s 1 s 2 ) ( s 2 s 1 , s 2 s 1 ) ( w 0 , w 0 ) id w 0 s 2 s 1 s 1 s 2 ( S n ≤ ) U � 0 s 1 s 2 id id Pavel Galashin Totally positive spaces 04/26/2019 10 / 24

  12. → ( G / B ) � 0 U � 0 ֒ ( Q , � ) ( G / B ) � 0 w 0 (id , w 0 ) (id , w 0 ) s 2 s 1 s 2 s 1 s 1 s 2 s 1 s 2 (id , s 1 s 2 ) (id , s 1 s 2 ) (id , s 2 s 1 ) (id , s 2 s 1 ) ( s 1 , w 0 ) ( s 2 , w 0 ) (id , s 1 ) (id , s 1 ) (id , s 2 ) (id , s 2 ) ( s 1 , s 1 s 2 ) ( s 1 , s 2 s 1 ) ( s 2 , s 1 s 2 ) ( s 2 , s 2 s 1 ) ( s 1 s 2 , w 0 ) ( s 2 s 1 , w 0 ) s 1 s 1 s 2 s 2 (id , id) (id , id) ( s 1 , s 1 ) ( s 2 , s 2 ) ( s 1 s 2 , s 1 s 2 ) ( s 2 s 1 , s 2 s 1 ) ( w 0 , w 0 ) id w 0 s 2 s 1 s 1 s 2 ( S n ≤ ) U � 0 s 1 s 2 id id Pavel Galashin Totally positive spaces 04/26/2019 10 / 24

  13. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  14. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  15. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Lusztig (1994) : ( G / P ) � 0 := π (( G / B ) � 0 ) . Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  16. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Lusztig (1994) : ( G / P ) � 0 := π (( G / B ) � 0 ) . Example � GL k ( R ) � ∗ Maximal parabolic subgroup: P = . 0 GL n − k ( R ) Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  17. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Lusztig (1994) : ( G / P ) � 0 := π (( G / B ) � 0 ) . Example � GL k ( R ) � ∗ Maximal parabolic subgroup: P = . 0 GL n − k ( R ) In this case G / P = Gr ( k , n ), and the projection is π : G / B → Gr ( k , n ) , ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ V k . Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  18. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Lusztig (1994) : ( G / P ) � 0 := π (( G / B ) � 0 ) . Example � GL k ( R ) � ∗ Maximal parabolic subgroup: P = . 0 GL n − k ( R ) In this case G / P = Gr ( k , n ), and the projection is π : G / B → Gr ( k , n ) , ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ V k . Postnikov (2006): Gr � 0 ( k , n ) := { V k ∈ Gr ( k , n ) | ∆ I ( V k ) � 0 for all I ⊂ [ n ] of size k } . Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  19. Partial flag variety Let P ⊃ B be a parabolic subgroup of G . We get a projection flag partial flag π : G / B → G / P ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ ( V 0 , V j 1 , . . . , V j m , V n ) . Lusztig (1994) : ( G / P ) � 0 := π (( G / B ) � 0 ) . Example � GL k ( R ) � ∗ Maximal parabolic subgroup: P = . 0 GL n − k ( R ) In this case G / P = Gr ( k , n ), and the projection is π : G / B → Gr ( k , n ) , ( V 0 , V 1 , . . . , V n − 1 , V n ) �→ V k . Postnikov (2006): Gr � 0 ( k , n ) := { V k ∈ Gr ( k , n ) | ∆ I ( V k ) � 0 for all I ⊂ [ n ] of size k } . Surprising fact: When G / P = Gr ( k , n ), we have ( G / P ) � 0 = Gr � 0 ( k , n ). Pavel Galashin Totally positive spaces 04/26/2019 11 / 24

  20. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  21. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  22. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  23. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  24. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  25. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  26. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  27. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Theorem (G.–Karp–Lam) Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  28. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Theorem (G.–Karp–Lam) 2017: Gr � 0 ( k , n ) is homeomorphic to a closed ball. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  29. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Theorem (G.–Karp–Lam) 2017: Gr � 0 ( k , n ) is homeomorphic to a closed ball. 2018: ( G / P ) � 0 is homeomorphic to a closed ball. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  30. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Theorem (G.–Karp–Lam) 2017: Gr � 0 ( k , n ) is homeomorphic to a closed ball. 2018: ( G / P ) � 0 is homeomorphic to a closed ball. 2019: Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  31. Regularity theorem Conjecture (Postnikov (2006), Williams (2007)) Gr � 0 ( k , n ) is a regular CW complex homeomorphic to a ball. ( G / P ) � 0 is a regular CW complex homeomorphic to a ball. Lusztig (1998) : ( G / P ) � 0 is contractible. Williams (2007) : The face poset is thin and shellable. Postnikov–Speyer–Williams (2009) : Gr � 0 ( k , n ) is a CW complex. Rietsch–Williams (2008) : ( G / P ) � 0 is a CW complex. Rietsch–Williams (2010) : The closure of each cell is contractible. Theorem (G.–Karp–Lam) 2017: Gr � 0 ( k , n ) is homeomorphic to a closed ball. 2018: ( G / P ) � 0 is homeomorphic to a closed ball. 2019: Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. Corollary of proof (Hersh (2014)) : Lk � 0 id ⊂ U � 0 is a regular CW complex. Pavel Galashin Totally positive spaces 04/26/2019 12 / 24

  32. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  33. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  34. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Affine flag variety Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  35. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Subtraction-free MR Affine flag variety Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  36. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Subtraction-free MR Link induction Affine flag variety Generalized Poincar´ e Conjecture Smooth vs Topological Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  37. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Subtraction-free MR Link induction Affine flag variety Generalized Poincar´ e Conjecture Smooth vs Topological Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  38. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  39. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = g ∈ Q Π > 0 Recall: ( G / P ) � 0 = � g . w 0 s 2 s 1 s 1 s 2 s 1 s 2 id Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  40. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = g . For g ∈ Q , define Star � 0 h � g Π > 0 g ∈ Q Π > 0 Recall: ( G / P ) � 0 = � := � h . g w 0 s 2 s 1 s 1 s 2 Π > 0 g s 1 s 2 id Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  41. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = g . For g ∈ Q , define Star � 0 h � g Π > 0 g ∈ Q Π > 0 Recall: ( G / P ) � 0 = � := � h . g w 0 s 2 s 1 s 1 s 2 Π > 0 g s 1 s 2 id Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  42. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = g . For g ∈ Q , define Star � 0 h � g Π > 0 g ∈ Q Π > 0 Recall: ( G / P ) � 0 = � := � h . g ∼ ν g : Star � 0 → Π > 0 × Cone ( Lk � 0 FS atlas: For each g ∈ Q , a map ¯ − g ). g g w 0 s 2 id s 2 s 1 s 1 s 2 w 0 s 1 s 2 ¯ ν g − → Π > 0 g s 1 s 2 s 2 s 1 Π > 0 s 1 g id Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  43. Proof idea Theorem (G.–Karp–Lam (2019)) Gr � 0 ( k , n ) and ( G / P ) � 0 are regular CW complexes. ⇒ Fomin–Shapiro atlas = ⇒ Regular CW complex Bruhat atlas = g . For g ∈ Q , define Star � 0 h � g Π > 0 g ∈ Q Π > 0 Recall: ( G / P ) � 0 = � := � h . g ∼ ν g : Star � 0 → Π > 0 × Cone ( Lk � 0 FS atlas: For each g ∈ Q , a map ¯ − g ). g g w 0 s 2 id s 2 s 1 s 1 s 2 w 0 s 1 s 2 Lk � 0 ¯ ν g − → g Π > 0 g s 1 s 2 s 2 s 1 Π > 0 s 1 g id Pavel Galashin Totally positive spaces 04/26/2019 13 / 24

  44. Part 2. Applications

  45. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. b 3 b 2 b 4 b 1 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  46. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  47. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  48. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  49. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  50. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  51. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  52. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 • More spins aligned = ⇒ higher probability J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  53. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 J e 2 J e 9 J e 3 • More spins aligned = ⇒ higher probability J e 8 J e 1 b 4 J e 5 b 1 J e 7 J e 4 J e 6 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  54. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 • More spins aligned = ⇒ higher probability • Mathematical model for ferromagnetism b 4 b 1 b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  55. Ising model Definition Planar Ising network: planar weighted graph embedded in a disk. Ising model: probability measure on spin configurations. b 3 b 2 • More spins aligned = ⇒ higher probability • Mathematical model for ferromagnetism b 4 b 1 • Phase transitions, critical temperatures, . . . b 5 b 6 Pavel Galashin Totally positive spaces 04/26/2019 15 / 24

  56. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  57. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  58. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 . Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  59. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 . Griffiths (1967) : Correlations are always nonnegative. Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  60. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 . Griffiths (1967) : Correlations are always nonnegative. Kelly–Sherman (1968) : How to describe correlation matrices by inequalities? Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  61. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 . Griffiths (1967) : Correlations are always nonnegative. Kelly–Sherman (1968) : How to describe correlation matrices by inequalities? Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  62. Ising model: boundary correlations Let b 1 , . . . , b n be the boundary vertices. Definition Correlation: m ij := Prob(Spin b i = Spin b j ) − Prob(Spin b i � = Spin b j ). Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 . Griffiths (1967) : Correlations are always nonnegative. Kelly–Sherman (1968) : How to describe correlation matrices by inequalities? Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. Pavel Galashin Totally positive spaces 04/26/2019 16 / 24

  63. Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. Pavel Galashin Totally positive spaces 04/26/2019 17 / 24

  64. Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. We define a simple doubling map φ : X n ֒ → Gr ( n , 2 n ): Pavel Galashin Totally positive spaces 04/26/2019 17 / 24

  65. Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. We define a simple doubling map φ : X n ֒ → Gr ( n , 2 n ):  1 m 12 m 13 m 14   1 1 m 12 − m 12 − m 13 m 13 m 14 − m 14  m 12 1 m 23 m 24 − m 12 m 12 1 1 m 23 − m 23 − m 24 m 24      �→     m 13 m 23 1 m 34 m 13 − m 13 − m 23 m 23 1 1 m 34 − m 34    m 14 m 24 m 34 1 − m 14 m 14 m 24 − m 24 − m 34 m 34 1 1 Pavel Galashin Totally positive spaces 04/26/2019 17 / 24

  66. Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. We define a simple doubling map φ : X n ֒ → Gr ( n , 2 n ):  1 m 12 m 13 m 14   1 1 m 12 − m 12 − m 13 m 13 m 14 − m 14  m 12 1 m 23 m 24 − m 12 m 12 1 1 m 23 − m 23 − m 24 m 24      �→     m 13 m 23 1 m 34 m 13 − m 13 − m 23 m 23 1 1 m 34 − m 34    m 14 m 24 m 34 1 − m 14 m 14 m 24 − m 24 − m 34 m 34 1 1 Pavel Galashin Totally positive spaces 04/26/2019 17 / 24

  67. Definition (G.–Pylyavskyy (2018)) X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. We define a simple doubling map φ : X n ֒ → Gr ( n , 2 n ):  1 m 12 m 13 m 14   1 1 m 12 − m 12 − m 13 m 13 m 14 − m 14  m 12 1 m 23 m 24 − m 12 m 12 1 1 m 23 − m 23 − m 24 m 24      �→     m 13 m 23 1 m 34 m 13 − m 13 − m 23 m 23 1 1 m 34 − m 34    m 14 m 24 m 34 1 − m 14 m 14 m 24 − m 24 − m 34 m 34 1 1 Pavel Galashin Totally positive spaces 04/26/2019 17 / 24

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