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 : P1 → Pk:

t → φ(t) = [ f0(t) : · · · : fk(t) ], where fi(t) ∈ C[t]≤n.

◮ The Wronskian of f0, . . . , fk is given by

Wr(f0, . . . , fk) = det      f0(t) · · · fk(t) f ′

0(t)

· · · f ′

k(t)

. . . ... . . . f (k) (t) · · · f (k)

k

(t)     

Parametric curves and Wronskians

◮ Parametric curve φ : P1 → Pk:

t → φ(t) = [ f0(t) : · · · : fk(t) ], where fi(t) ∈ C[t]≤n.

◮ The Wronskian of f0, . . . , fk is given by

Wr(f0, . . . , fk) = det      f0(t) · · · fk(t) f ′

0(t)

· · · f ′

k(t)

. . . ... . . . f (k) (t) · · · f (k)

k

(t)     

◮ 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 P3)

◮ 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 : P1 → P1 be given by φ(t) =

• t3 + i

√ 3t : t2 +

i √ 3

• =

t3 + i √ 3t t2 + i/ √ 3 : 1

• = t3 + i

√ 3t t2 + i/ √ 3 (as rational function).

Real and complex flexes

Example (Critical points of rational functions)

Let φ : P1 → P1 be given by φ(t) =

• t3 + i

√ 3t : t2 +

i √ 3

• =

t3 + i √ 3t t2 + i/ √ 3 : 1

• = t3 + i

√ 3t t2 + i/ √ 3 (as rational function). The Wronskian computes the critical points where φ′(t) = 0: Wr(φ) = det f0 f1 f ′ f ′

1

• = f0f ′

1 − f ′ 0f1 = 1 − t4.

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 ti ∈ P1 (up to PGLk+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 ti ∈ P1 (up to PGLk+1). Deep connection to Schubert calculus: The number of such curves (counted with multiplicity) is the number of standard Young tableaux: SYT( ) =

• 1 2 4

3 5 6 , 1 3 5 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 Pk). 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 Pk). Very unusual real algebraic geometry problem with real solutions! Goal for today:

Theorem (L–Purbhoo ’19)

Let Wr(φ) have n1 distinct real roots, n2 complex conjugate pairs. Over R, the number of such φ, counted with signs, is the symmetric group character χ (2n2, 1n1). Recovers M–T–V in the case n2 = 0.

(Wronskians and) Schubert calculus

The Grassmannian is the space of planes: Gr(k, Cn) = {vector subspaces S ⊂ Cn : dim(S) = k}. For us: subspaces f0, . . . , fk of the space of polynomials C[t]≤n.

(Wronskians and) Schubert calculus

The Grassmannian is the space of planes: Gr(k, Cn) = {vector subspaces S ⊂ Cn : dim(S) = k}. For us: subspaces f0, . . . , fk of the space of polynomials C[t]≤n. Simplest (codimension 1) Schubert variety: X (Fn−k) = {S ∈ Gr(k, n) : S ∩ Fn−k = 0}.

(Wronskians and) Schubert calculus

The Grassmannian is the space of planes: Gr(k, Cn) = {vector subspaces S ⊂ Cn : dim(S) = k}. For us: subspaces f0, . . . , fk of the space of polynomials C[t]≤n. Simplest (codimension 1) Schubert variety: X (Fn−k) = {S ∈ Gr(k, n) : S ∩ Fn−k = 0}. General Schubert varieties: consider S ∩ F, for a complete flag: F : Cn = Fn ⊃ Fn−1 ⊃ · · · ⊃ F1.

(Wronskians and) Schubert calculus

The Grassmannian is the space of planes: Gr(k, Cn) = {vector subspaces S ⊂ Cn : dim(S) = k}. For us: subspaces f0, . . . , fk of the space of polynomials C[t]≤n. Simplest (codimension 1) Schubert variety: X (Fn−k) = {S ∈ Gr(k, n) : S ∩ Fn−k = 0}. General Schubert varieties: consider S ∩ F, for a complete flag: F : Cn = Fn ⊃ Fn−1 ⊃ · · · ⊃ F1. 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 f0, . . . , fk ∈ C[t]≤n is Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)    Up to scalar, depends only on spanC(f0, . . . , fk).

Wronskians and Schubert calculus

The Wronskian of f0, . . . , fk ∈ C[t]≤n is Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)    Up to scalar, depends only on spanC(f0, . . . , fk). Gives the Wronski map: Wr : Gr(k + 1, C[t]≤n) → P(C[t]≤(k+1)(n−k)), f0, . . . , fk → Wr(f0, . . . , fk).

Wronskians and Schubert calculus

The Wronskian of f0, . . . , fk ∈ C[t]≤n is Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)    Up to scalar, depends only on spanC(f0, . . . , fk). Gives the Wronski map: Wr : Gr(k + 1, C[t]≤n) → P(C[t]≤(k+1)(n−k)), f0, . . . , fk → Wr(f0, . . . , fk).

◮ Fiber of the Wronski map ⇔ set of φ with specified flexes. ◮ Wr(φ) = (t − t1) · · · (t − tN)

⇐ ⇒ t1, . . . , tN 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 − t1) · · · (t − tN)). This is an intersection of Schubert varieties:

◮ Wr(φ) has a root at ti ⇐

⇒ φ ∈ X (F(ti)).

◮ So, Z = X (F(t1)) ∩ · · · ∩ X (F(tN)).

Fibers of the Wronski map

Wronski problem (reformulated)

Understand fibers of the Wronski map. Consider a fiber Z = Wr−1((t − t1) · · · (t − tN)). This is an intersection of Schubert varieties:

◮ Wr(φ) has a root at ti ⇐

⇒ φ ∈ X (F(ti)).

◮ So, Z = X (F(t1)) ∩ · · · ∩ X (F(tN)).

Schubert calculus: Such an intersection is counted (with multiplicity) by standard Young tableaux: SYT( ) =

• 1 2 4

3 5 6 , 1 3 5 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 M0,n(R) exist [Speyer ’14] ◮ and more (Purbhoo, Halacheva–Rybnikov–Kamnitzer–Weeks,

L–Gillespie, White, . . . )

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 M0,n(R) exist [Speyer ’14] ◮ and more (Purbhoo, Halacheva–Rybnikov–Kamnitzer–Weeks,

L–Gillespie, White, . . . )

How to find combinatorics in geometry

Key idea

Degenerate the problem until it breaks into pieces.

How to find combinatorics in geometry

Key idea

Degenerate the problem until it breaks into pieces. Take the roots of Wr to be (t1, . . . , tN) = (z, z2, . . . , zN). Take limz→0.

1 R z → 0 z1 z2 z3 z4

How to find combinatorics in geometry

Key idea

Degenerate the problem until it breaks into pieces. Take the roots of Wr to be (t1, . . . , tN) = (z, z2, . . . , zN). Take limz→0.

1 R z → 0 z1 z2 z3 z4

What will happen to the fiber at z = 0?

Critical points of rational functions, redux

Degree-3 maps φ : P1 → P1, φ(t) = a3t3 + a2t2 + a1t + a0 b3t3 + b2t2 + b1t + b0 :

◮ Gr(2, 4) = Gr(2, C[t]≤3) ∋

a3 a2 a1 a0 b3 b2 b1 b0

Critical points of rational functions, redux

Degree-3 maps φ : P1 → P1, φ(t) = t2 + a1t + a0 t3 + b1t + b0 :

◮ Gr(2, 4) = Gr(2, C[t]≤3) ∋

1 a1 a0 1 b1 b0

Critical points of rational functions, redux

Degree-3 maps φ : P1 → P1, φ(t) = t2 + a1t + a0 t3 + b1t + b0 :

◮ Gr(2, 4) = Gr(2, C[t]≤3) ∋

1 a1 a0 1 b1 b0

• ◮ Numerology:

◮ Wronskian has 4 roots z, z2, z3, z4, critical points of φ. ◮ Fiber = two points, counted by

• 1 2

3 4 , 1 3 2 4

• .

Critical points of rational functions, redux

Degree-3 maps φ : P1 → P1, φ(t) = t2 + a1t + a0 t3 + b1t + b0 :

◮ Gr(2, 4) = Gr(2, C[t]≤3) ∋

1 a1 a0 1 b1 b0

• ◮ Numerology:

◮ Wronskian has 4 roots z, z2, z3, z4, critical points of φ. ◮ Fiber = two points, counted by

• 1 2

3 4 , 1 3 2 4

• .

z → 0 Wr(φ) z1 z2 z3 z4 Z ⊂ Gr(2, 4) (two solutions for φ)

Critical points of rational functions, redux

Degree-3 maps φ : P1 → P1, φ(t) = t2 + a1t + a0 t3 + b1t + b0 :

◮ Gr(2, 4) = Gr(2, C[t]≤3) ∋

1 a1 a0 1 b1 b0

• ◮ Numerology:

◮ Wronskian has 4 roots z, z2, z3, z4, critical points of φ. ◮ Fiber = two points, counted by

• 1 2

3 4 , 1 3 2 4

• .

z → 0 Wr(φ) z1 z2 z3 z4 Z ⊂ Gr(2, 4) (two solutions for φ)

• 1

≈ z ≈ z3 1 ≈ z4 ≈ z6

• 1

≈ z ≈ z4 1 ≈ z3 ≈ z6

• 1 2

3 4 1 3 2 4

Tableau labels from Pl¨ ucker coordinates

Limiting matrix:

• 1

≈ z1 ≈ z3 1 ≈ z4 ≈ z6

• 1 2

3 4

Pl¨ ucker coordinates (minors) on Gr(2, 4):

∅

det12 = 1 det13 = O(z1) det14 = O(z3) det23 = O(z4) det24 = O(z6) det34 = O(z10)

Tableau labels from Pl¨ ucker coordinates

Limiting matrix:

• 1

≈ z1 ≈ z3 1 ≈ z4 ≈ z6

• 1 2

3 4

Pl¨ ucker coordinates (minors) on Gr(2, 4):

∅

1 1 2 1 3 1 2 3 1 2 3 4

det12 = z0 det13 = O(z1) det14 = O(z1+2) det23 = O(z1+3) det24 = O(z1+2+3) det34 = O(z1+2+3+4)

Combinatorics and geometry

Theorem (Purbhoo ’09, Speyer ’14)

This procedure gives a bijection lim

z→0 Z ←

→ SYT( ).

Combinatorics and geometry

Theorem (Purbhoo ’09, Speyer ’14)

This procedure gives a bijection lim

z→0 Z ←

→ SYT( ). By M–T–V, it extends to all fibers Z where Wr has all roots in R.

Combinatorics and geometry

Theorem (Purbhoo ’09, Speyer ’14)

This procedure gives a bijection lim

z→0 Z ←

→ SYT( ). By M–T–V, it extends to all fibers Z where Wr has all roots in R. Other deformations of Z act by combinatorial bijections!

RP1 ∞ tableau promotion RP1 ∞ tableau evacuation

Combinatorics and geometry

Theorem (Purbhoo ’09, Speyer ’14)

This procedure gives a bijection lim

z→0 Z ←

→ SYT( ). By M–T–V, it extends to all fibers Z where Wr has all roots in R. Other deformations of Z act by combinatorial bijections!

RP1 ∞ tableau promotion RP1 ∞ tableau evacuation

And more:

◮ Topology and genus when dim(Z) = 1 (L, Gillespie–L) ◮ Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo) ◮ Vector bundles on M0,n (Kamnitzer, Rybnikov)

A challenge and a new approach

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(f0, . . . , fk)(t) has all real roots, then φ is defined over R (up to change of coordinates). Challenge for geometers:

◮ M–T–V proof uses integrable systems, the Bethe ansatz

A challenge and a new approach

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(f0, . . . , fk)(t) has all real roots, then φ is defined over R (up to change of coordinates). Challenge for geometers:

◮ M–T–V proof uses integrable systems, 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(f0, . . . , fk)(t) has all real roots, then φ is defined over R (up to change of coordinates). Challenge for geometers:

◮ M–T–V proof uses integrable systems, 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 3 4 6 , (+) 1 3 5 2 4 6 , · · · Oriented Young tableaux.

Generalization: complex conjugate roots for Wr(φ)

Definition (Cutting up R[t]≤N)

For a partition µ = (2n2, 1n1), let P(µ) be P(µ) =

• polynomials with n1 distinct real roots,

n2 complex conjugate pairs

• ⊆ R[t]≤N.

Base case: µ = (1N), all real roots.

Generalization: complex conjugate roots for Wr(φ)

Definition (Cutting up R[t]≤N)

For a partition µ = (2n2, 1n1), let P(µ) be P(µ) =

• polynomials with n1 distinct real roots,

n2 complex conjugate pairs

• ⊆ R[t]≤N.

Base case: µ = (1N), all real roots.

◮ We study the restricted Wronski map

Wrµ : Wr−1(P(µ)) → P(µ).

Note: we’re always in the Schubert cell X ∅(∞)◦ (no roots at ∞).

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree... (=?):

Wr−1(P(µ)) P(µ)

Algebraic degree: 3

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree... (=?):

Wr−1(P(µ)) P(µ)

+ − + + +

Algebraic degree: 3

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree... (=?):

Wr−1(P(µ)) P(µ)

+ − + + +

Algebraic degree: 3 Topological degree: 1

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree... (=?):

Wr−1(P(µ)) P(µ)

+ − + + +

Algebraic degree: 3 Topological degree: 1

◮ We use a new “character” orientation on the Schubert cell.

The topological degree of Wrµ

Character table of S4. (χλ(µ)) λ, µ (4) (3, 1) (22) (2, 12) (14) 1 1 1 1 1 1

• 1
• 1

3

• 1

2 2

• 1
• 1

1 3

• 1

1 1

• 1

1

The topological degree of Wrµ

Character table of S4. (χλ(µ)) λ, µ (4) (3, 1) (22) (2, 12) (14) 1 1 1 1 1 1

• 1
• 1

3

• 1

2 2

• 1
• 1

1 3

• 1

1 1

• 1

1 For Sk(n−k), let be the k × (n − k) rectangle, µ = (2n2, 1n1).

The topological degree of Wrµ

Character table of S4. (χλ(µ)) λ, µ (4) (3, 1) (22) (2, 12) (14) 1 1 1 1 1 1

• 1
• 1

3

• 1

2 2

• 1
• 1

1 3

• 1

1 1

• 1

1 For Sk(n−k), let be the k × (n − k) rectangle, µ = (2n2, 1n1).

Theorem (L, Purbhoo ‘19)

Under the character orientation, the restricted Wronski map Wrµ has topological degree χ (µ).

Signed Young tableaux

Theorem (L, Purbhoo ‘19)

Under the character orientation, the restricted Wronski map Wrµ has topological degree χ (µ).

Signed Young tableaux

Theorem (L, Purbhoo ‘19)

Under the character orientation, the restricted Wronski map Wrµ has topological degree χ (µ). Murnaghan–Nakayama rule for χλ(µ), µ = (2n2, 1n1): χλ(µ) =

• T

(−1)# (T) : µ-domino tableaux (+) 1 2 4

3 5 6 , (−) 1 3 4 2 5 6 , · · ·

shape(T) = λ.

Signed Young tableaux

Theorem (L, Purbhoo ‘19)

Under the character orientation, the restricted Wronski map Wrµ has topological degree χ (µ). Murnaghan–Nakayama rule for χλ(µ), µ = (2n2, 1n1): χλ(µ) =

• T

(−1)# (T) : µ-domino tableaux (+) 1 2 4

3 5 6 , (−) 1 3 4 2 5 6 , · · ·

shape(T) = λ.

◮ Special case: µ = (1N), no dominos χ

(1N) = #SYT.

◮ Corollary: Shapiro–Shapiro Conjecture.

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Wr

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Z Z Wr

◮ ∆ lifts to Z

∪ Z , two kinds of “double flex”:

◮ Type : rank deficiency at φ(k−1) rather than φ(k). ◮ Type

: rank deficiency at φ(k) and again at φ(k+1).

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ + + + + + + + + − − − P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Z Z Wr

◮ ∆ lifts to Z

∪ Z , two kinds of “double flex”:

◮ Type : rank deficiency at φ(k−1) rather than φ(k). ◮ Type

: rank deficiency at φ(k) and again at φ(k+1).

◮ Character orientation: multiply by the equation of Z .

Labeling fibers by signed Young tableaux

Proof sketch (signs agree with M–N rule):

◮ Label limit fibers by tableaux. ◮ Track +/− signs along a network of paths:

↔ (−1).

Labeling fibers by signed Young tableaux

Proof sketch (signs agree with M–N rule):

◮ Label limit fibers by tableaux. ◮ Track +/− signs along a network of paths:

↔ (−1). Case 1:

1 2 ← → 1 2 / 1 2 ← → 1 2

+ + + + + + + + + − − − P(µ = (16)) P(µ′ = (2, 14)) P(µ) P(µ′) Z Z

1 2 4 3 5 6 1 3 4 2 5 6 1 2 4 3 5 6 1 3 4 2 5 6 (+1) (−1)

Labeling fibers by signed Young tableaux

Proof sketch (signs agree with M–N rule):

◮ Label limit fibers by tableaux. ◮ Track +/− signs along a network of paths.

Case 2:

3 2

← →

2 3

+ + + + + + + + − − P(µ = (16)) P(µ = (2, 14)) P(µ) P(µ′) Z Z

1 3 4 2 5 6 1 2 4 3 5 6

(empty)

(+1)

Open questions

◮ (Representation theory).

Do all SN character values χλ(µ) give topological degrees of real Schubert problems? (µ = (2a1b))

◮ (Complex geometry).

Explicit geometry of Wrµ over P(µ) for µ = (1N)?

◮ (Stable curves).

How does the geometry look over the moduli space M0,N?

◮ M0,N(R) is non-orientable!

Many interesting relationships to find between geometry and combinatorics.

Thank you!