A topological proof of the ShapiroShapiro Conjecture Jake Levinson - PowerPoint PPT Presentation

A topological proof of the Shapiro–Shapiro Conjecture Jake Levinson (U. Washington) joint with Kevin Purbhoo (U. Waterloo) NU / UIC / UofC Online Seminar August 6, 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) .

A higher-dimensional simple flex (in P 3 ) ◮ C meets its tangent line to order 2 (generic behavior) ◮ C meets its tangent plane to order 3 + 1 = 4 (flex!)

Real and complex flexes Example (Critical points of rational functions)

Real and complex flexes Example (Critical points of rational functions) Let φ : P 1 → P 1 be given by √ t 3 + i 3 t : t 2 + � � i φ ( t ) = √ 3 √ � t 3 + i 3 t � = √ : 1 t 2 + i / 3 √ = t 3 + i 3 t √ (as rational function). t 2 + i / 3

Real and complex flexes Example (Critical points of rational functions) Let φ : P 1 → P 1 be given by √ t 3 + i 3 t : t 2 + � � i φ ( t ) = √ 3 √ � t 3 + i 3 t � = √ : 1 t 2 + i / 3 √ = t 3 + i 3 t √ (as rational function). t 2 + i / 3 The Wronskian computes the critical points where φ ′ ( t ) = 0: � f 0 � f 1 = f 0 f ′ 1 − f ′ 0 f 1 = 1 − t 4 . Wr ( φ ) = det f ′ f ′ 0 1 4 critical points at t = ± 1 , ± i .

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 curves (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! Goal for today: 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.

(Wronskians and) Schubert calculus The Grassmannian is the space of planes: Gr ( k , C n ) = { vector subspaces S ⊂ C n : dim( S ) = k } . For us : subspaces � f 0 , . . . , f k � of the space of polynomials C [ t ] ≤ n .

(Wronskians and) Schubert calculus The Grassmannian is the space of planes: Gr ( k , C n ) = { vector subspaces S ⊂ C n : dim( S ) = k } . For us : subspaces � f 0 , . . . , f k � of the space of polynomials C [ t ] ≤ n . Simplest (codimension 1) Schubert variety : X ( F n − k ) = { S ∈ Gr ( k , n ) : S ∩ F n − k � = 0 } .

(Wronskians and) Schubert calculus The Grassmannian is the space of planes: Gr ( k , C n ) = { vector subspaces S ⊂ C n : dim( S ) = k } . For us : subspaces � f 0 , . . . , f k � of the space of polynomials C [ t ] ≤ n . Simplest (codimension 1) Schubert variety : X ( F n − k ) = { S ∈ Gr ( k , n ) : S ∩ F n − k � = 0 } . General Schubert varieties: consider S ∩ F , for a complete flag : F : C n = F n ⊃ F n − 1 ⊃ · · · ⊃ F 1 .

(Wronskians and) Schubert calculus The Grassmannian is the space of planes: Gr ( k , C n ) = { vector subspaces S ⊂ C n : dim( S ) = k } . For us : subspaces � f 0 , . . . , f k � of the space of polynomials C [ t ] ≤ n . Simplest (codimension 1) Schubert variety : X ( F n − k ) = { S ∈ Gr ( k , n ) : S ∩ F n − k � = 0 } . General Schubert varieties: consider S ∩ F , for a complete flag : F : C n = F n ⊃ F n − 1 ⊃ · · · ⊃ F 1 . For us : “divisibility flags” F ( z ) for z ∈ C : F ( z ) : { f divisible by ( t − z ) } ⊃ { f divisible by ( t − z ) 2 } ⊃ · · · .

Wronskians and Schubert calculus The Wronskian of f 0 , . . . , f k ∈ C [ t ] ≤ n is  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k Up to scalar, depends only on span C ( f 0 , . . . , f k ).

Wronskians and Schubert calculus The Wronskian of f 0 , . . . , f k ∈ C [ t ] ≤ n is  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k Up to scalar, depends only on span C ( f 0 , . . . , f 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 ) � .

Wronskians and Schubert calculus The Wronskian of f 0 , . . . , f k ∈ C [ t ] ≤ n is  f 0 ( t ) · · · f k ( t )  . . ... . . Wr ( f 0 , . . . , f k ) = det  . .    f ( k ) f ( k ) ( t ) · · · ( t ) 0 k Up to scalar, depends only on span C ( f 0 , . . . , f 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 ) � . ◮ Fiber of the Wronski map ⇔ set of φ with specified flexes. ◮ Wr ( φ ) = ( t − t 1 ) · · · ( t − t N ) ⇐ ⇒ t 1 , . . . , t N are flexes of the curve φ ( t ).

Fibers of the Wronski map Wronski problem (reformulated) Understand fibers of the Wronski map.

Fibers of the Wronski map Wronski problem (reformulated) Understand fibers of the Wronski map. Consider a fiber Z = Wr − 1 (( t − t 1 ) · · · ( t − t N )). This is an intersection of Schubert varieties: ◮ Wr ( φ ) has a root at t i ⇐ ⇒ φ ∈ X ( F ( t i )) . ◮ So, Z = X ( F ( t 1 )) ∩ · · · ∩ X ( F ( t N )) .

Fibers of the Wronski map Wronski problem (reformulated) Understand fibers of the Wronski map. Consider a fiber Z = Wr − 1 (( t − t 1 ) · · · ( t − t N )). This is an intersection of Schubert varieties: ◮ Wr ( φ ) has a root at t i ⇐ ⇒ φ ∈ X ( F ( t i )) . ◮ So, Z = X ( F ( t 1 )) ∩ · · · ∩ X ( F ( t N )) . Schubert calculus: Such an intersection is counted (with multiplicity) by standard Young tableaux : � � 1 2 4 1 3 5 SYT ( ) = 3 5 6 , 2 4 6 , · · ·

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

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then the fiber is reduced and every point in it is real. “If every flex of φ is real, then φ is real.” And, the intersection of Schubert varieties is transverse!

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then the fiber is reduced and every point in it is real. “If every flex of φ is real, then φ is real.” And, the intersection of Schubert varieties is transverse! Many consequences : ◮ Fiber cardinality is exactly # SYT ( ) ◮ Each φ is canonically identified by a tableau [Purbhoo ’09].

Shapiro–Shapiro / M–T–V, geometric statement Theorem (Mukhin–Tarasov–Varchenko ’05, ’09) If Wr ( φ ) has all real roots, then the fiber is reduced and every point in it is real. “If every flex of φ is real, then φ is real.” And, the intersection of Schubert varieties is transverse! Many consequences : ◮ Fiber cardinality is exactly # SYT ( ) ◮ Each φ is canonically identified by a tableau [Purbhoo ’09]. ◮ Certain covering spaces of M 0 , n ( R ) exist [Speyer ’14] ◮ and more (Purbhoo, Halacheva–Rybnikov–Kamnitzer–Weeks, L–Gillespie, White, . . . )