Monodromy and Real Wronskians Jake Levinson (Simon Fraser - PowerPoint PPT Presentation

Monodromy and Real Wronskians Jake Levinson (Simon Fraser University) joint with Kevin Purbhoo (U. Waterloo) ICERM September 2, 2020

Parametric curves and Wronskians ◮ Parametric curve φ : P 1 → P k : t �→ φ ( t ) = [ f 0 ( t ) : · · · : f k ( t ) ] , where f i ( t ) ∈ C [ t ] ≤ n . ◮ The Wronskian of f 0 , . . . , f k is given by  f 0 ( t ) · · · f k ( t )  f ′ f ′ 0 ( t ) · · · k ( t )   Wr ( f 0 , . . . , f k ) = det . . ...   . .  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k

Parametric curves and Wronskians ◮ Parametric curve φ : P 1 → P k : t �→ φ ( t ) = [ f 0 ( t ) : · · · : f k ( t ) ] , where f i ( t ) ∈ C [ t ] ≤ n . ◮ The Wronskian of f 0 , . . . , f k is given by  f 0 ( t ) · · · f k ( t )  f ′ f ′ 0 ( t ) · · · k ( t )   Wr ( f 0 , . . . , f k ) = det . . ...   . .  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k ◮ Detects flexes : t such that φ, φ ′ , φ ′′ , . . . , φ ( k ) is linearly dependent (e.g. inflection point, cusp, ...) ◮ Simple flex : Rank deficiency at φ ( k ) , fixed at φ ( k +1) .

What is a simple flex in P 3 ? ◮ C meets its tangent line to order 2 (not special) ◮ C meets its tangent plane to order 3 + 1 = 4 (flex!)

The Wronski problem Wronski problem Describe the curves φ with a given Wronskian.

The Wronski problem Wronski problem Describe the curves φ with a given Wronskian. Basic combinatorial question: how many? Theorem (Classical) There are only finitely-many parametric curves φ with flexes at prescribed t i ∈ P 1 (up to PGL k +1 ).

The Wronski problem Wronski problem Describe the curves φ with a given Wronskian. Basic combinatorial question: how many? Theorem (Classical) There are only finitely-many parametric curves φ with flexes at prescribed t i ∈ P 1 (up to PGL k +1 ). Deep connection to Schubert calculus: The number of such φ (counted with multiplicity) is the number of standard Young tableaux : � � 1 2 4 1 3 5 SYT ( ) = 3 5 6 , 2 4 6 , · · ·

Over R , things are remarkably nice! Shapiro–Shapiro Conjecture (’95) / M–T–V Theorem: Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then φ itself is defined over R (up to coordinate change on P k ). Very unusual real algebraic geometry problem with real solutions!

Over R , things are remarkably nice! Shapiro–Shapiro Conjecture (’95) / M–T–V Theorem: Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then φ itself is defined over R (up to coordinate change on P k ). Very unusual real algebraic geometry problem with real solutions! Theorem (L–Purbhoo ’19) Let Wr ( φ ) have n 1 distinct real roots, n 2 complex conjugate pairs. Over R , the number of such φ , counted with signs, is the (2 n 2 , 1 n 1 ) . symmetric group character χ Recovers M–T–V in the case n 2 = 0.

The Wronski map Gr → P N Consider the Grassmannian of subspaces � f 0 , . . . , f k � ⊆ C [ t ] ≤ n .

The Wronski map Gr → P N Consider the Grassmannian of subspaces � f 0 , . . . , f k � ⊆ C [ t ] ≤ n . Up to scalar, the Wronskian depends only on � f 0 , . . . , f k � :  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k

The Wronski map Gr → P N Consider the Grassmannian of subspaces � f 0 , . . . , f k � ⊆ C [ t ] ≤ n . Up to scalar, the Wronskian depends only on � f 0 , . . . , f k � :  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k Gives the Wronski map : Wr : Gr ( k +1 , C [ t ] ≤ n ) → P ( C [ t ] ≤ ( k +1)( n − k ) ) , � f 0 , . . . , f k � �→ � Wr ( f 0 , . . . , f k ) � . Note: Fiber of Wr = set of all φ with specified flexes.

The Wronski map Gr → P N Consider the Grassmannian of subspaces � f 0 , . . . , f k � ⊆ C [ t ] ≤ n . Up to scalar, the Wronskian depends only on � f 0 , . . . , f k � :  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k Gives the Wronski map : Wr : Gr ( k +1 , C [ t ] ≤ n ) → P ( C [ t ] ≤ ( k +1)( n − k ) ) , � f 0 , . . . , f k � �→ � Wr ( f 0 , . . . , f k ) � . Note: Fiber of Wr = set of all φ with specified flexes. Aside #1: Fibers are intersections of Schubert varieties.

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has distinct real roots, then every point of the fiber is real and reduced.

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions.

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions. Many consequences : ◮ The Wronski map is a covering map over the locus of distinct real roots: ◮ UC N ( RP 1 ) := { sets of n distinct points on RP 1 } ⊆ P ( R [ t ] ≤ n ) .

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions. Many consequences : ◮ The Wronski map is a covering map over the locus of distinct real roots: ◮ UC N ( RP 1 ) := { sets of n distinct points on RP 1 } ⊆ P ( R [ t ] ≤ n ) . ◮ Fiber cardinality is exactly # SYT ( ) ◮ Each φ is canonically identified by a tableau [Purbhoo ’09].

Configuration spaces of RP 1 and CP 1 ◮ UC N ( RP 1 ) is much simpler than UC N ( CP 1 ): ◮ Fundamental group π 1 ( UC N ( RP 1 )) ∼ = Z by rotation by 2 π N . ◮ Open subset UC N ( R ) is a simplex.

Labeling fibers and monodromy ◮ Purbhoo ’09: Over UC N ( R ), label sheets by tableaux. ◮ Label a “limit fiber” near “ { 0, 0, . . . , 0 } ” / ∈ UC N ( R ). 1 3 4 ◮ Orders of vanishing of Pl¨ ucker coordinates � 2 5 6

Labeling fibers and monodromy ◮ Purbhoo ’09: Over UC N ( R ), label sheets by tableaux. ◮ Label a “limit fiber” near “ { 0, 0, . . . , 0 } ” / ∈ UC N ( R ). 1 3 4 ◮ Orders of vanishing of Pl¨ ucker coordinates � 2 5 6 ◮ Monodromy over ∞ acts by combinatorial bijections! RP 1 ∞ 0 “tableau promotion”

Labeling fibers and monodromy ◮ Purbhoo ’09: Over UC N ( R ), label sheets by tableaux. ◮ Label a “limit fiber” near “ { 0, 0, . . . , 0 } ” / ∈ UC N ( R ). 1 3 4 ◮ Orders of vanishing of Pl¨ ucker coordinates � 2 5 6 ◮ Monodromy over ∞ acts by combinatorial bijections! RP 1 ∞ ∞ 0 RP 1 0 “tableau promotion” “tableau evacuation” Aside #2: ◮ Parallel story over M 0 , N ( R ) (Kamnitzer, Speyer, Rybnikov) ◮ Topology, genus of curves in Gr ( k +1 , n +1) (L, Gillespie–L) ◮ Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo)

A challenge and a new approach Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then φ is defined over R (up to change of coordinates). Challenge for geometers : ◮ M–T–V proof uses the Bethe ansatz ◮ Subsequent geometry work used M–T–V as black box. ◮ Many open generalizations of interest!

A challenge and a new approach Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then φ is defined over R (up to change of coordinates). Challenge for geometers : ◮ M–T–V proof uses the Bethe ansatz ◮ Subsequent geometry work used M–T–V as black box. ◮ Many open generalizations of interest! Now : conjugate roots in C and a topological approach. ( − ) 1 2 5 ( + ) 1 3 5 3 4 6 , 2 4 6 , · · · Oriented Young tableaux.

Generalization: complex conjugate roots for Wr ( φ ) Definition (Cutting up R [ t ] ≤ N ) For a partition µ = (2 n 2 , 1 n 1 ), let P ( µ ) be � polynomials with n 1 distinct real roots, � P ( µ ) = n 2 complex conjugate pairs ⊆ R [ t ] ≤ N . Base case: µ = (1 N ), all real roots. Look at fibers of the restricted Wronski map: Wr µ : Wr − 1 ( P ( µ )) → P ( µ ) . (Note: no roots at ∞ .)

Topological and algebraic degrees How many real points in the fiber of Wr µ ? ◮ Upper bound from (algebraic) degree = # SYT ( ).

Topological and algebraic degrees How many real points in the fiber of Wr µ ? ◮ Upper bound from (algebraic) degree = # SYT ( ). ◮ Lower bound from topological degree : + + − Wr − 1 ( P ( µ )) Algebraic degree: 3 Topological degree: 1 + + P ( µ )

Topological and algebraic degrees How many real points in the fiber of Wr µ ? ◮ Upper bound from (algebraic) degree = # SYT ( ). ◮ Lower bound from topological degree : + + − Wr − 1 ( P ( µ )) Algebraic degree: 3 Topological degree: 1 + + P ( µ ) ◮ We use a new “character” orientation on the Schubert cell.