Computational Applications of Riemann Surfaces and Abelian Functions - - PowerPoint PPT Presentation

Computational Applications of Riemann Surfaces and Abelian Functions

General Examination March 14, 2014 Chris Swierczewski cswiercz@uw.edu

Department of Applied Mathematics University of Washington Seattle, Washington

1

Acknowledgments

◮ Committee:

◮ Bernard Deconinck (advisor), ◮ Randy Leveque, ◮ Bob O’Malley, ◮ William Stein, ◮ Rekha Thomas (GSR).

◮ Research Group:

◮ Olga Trichthenko, ◮ Natalie Sheils, ◮ Ben Segal.

◮ Bernd Sturmfels (UC Berkeley), ◮ Jonathan Hauenstein (NCSU), ◮ Daniel Shapero (UW), ◮ Grady Williams (UW), ◮ Megan Karalus.

2

The Kadomtsev–Petviashvili Equation

u(x, y, t) = surface height of a 2D periodic shallow water wave.

3 4uyy = ∂

∂x

• ut − 1

4 (6uux + uxxx)

• Figure : ˆ

Ile de R´ e, France Figure : Model of San Diego Bay

3

Theta Function Solutions

Family of solutions: ∀g ∈ Z+ u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c, ◮ c ∈ C, ◮ U, V , W , z0 ∈ Cg, ◮ Ω ∈ Cg×g. ◮ “Riemann theta function” θ : Cg × Cg×g → C

Finite genus solutions:

◮ dense in space of periodic solutions to KP.

4

The Riemann Theta Function

θ(z, Ω) =

• n∈Zg

e2πi

1 2 n·Ωn+n·z

4

The Riemann Theta Function

θ(z, Ω) =

• n∈Zg

e2πi

1 2 n·Ωn+n·z

• Convergence

◮ Requires Im(Ω) > 0. ◮ Also need only consider ΩT = Ω. ◮ Space of Riemann matrices:

hg =

• Ω ∈ Cg×g | ΩT = Ω and Im(Ω) > 0
• (Siegel upper half space.)

θ : Cg × hg → C

5

Abelian Functions

Periodic, meromorphic functions f : Cg → C with 2g independent periods.

5

Abelian Functions

Periodic, meromorphic functions f : Cg → C with 2g independent periods.

◮ Example g = 1:

℘(z), sn(z), cn(z), tn(z).

◮ Example g:

u(x, y, t) ∀g > 0.

◮ Can be written in terms of θ functions.

5

Abelian Functions

Periodic, meromorphic functions f : Cg → C with 2g independent periods.

◮ Example g = 1:

℘(z), sn(z), cn(z), tn(z).

◮ Example g:

u(x, y, t) ∀g > 0.

◮ Can be written in terms of θ functions.

These things can be computed!

6

abelfunctions

A Python library for computing with Abelian functions, Riemann surfaces, and complex algebraic curves. https://github.com/cswiercz/abelfunctions https://www.cswiercz.info/abelfunctions

7

Demo

Riemann theta functions.

8

Connection to Algebraic Geometry

u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c

U, V , W , z0, c, Ω not arbitrary.

8

Connection to Algebraic Geometry

u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c

U, V , W , z0, c, Ω not arbitrary. Derived from a complex plane algebraic curve: given f (λ, µ) = αn(λ)µn + αn−1(λ)µn−1 + · · · + α0(λ) the curve C is the set C =

• (λ, µ) ∈ C2 : f (λ, µ) = 0
• .

9

Goal of This Talk

Algebraic Curves and Riemann Surfaces Introduction Geometry: Basis of Cycles Algebra: Holomorphic 1-forms Period Matrices Goals and Applications Periodic Solutions to Integrable PDEs Linear Matrix Representations The Constructive Schottky Problem (*)

10

Goal of This Talk

Algebraic Curves and Riemann Surfaces Introduction Geometry: Basis of Cycles Algebra: Holomorphic 1-forms Period Matrices Goals and Applications Periodic Solutions to Integrable PDEs Linear Matrix Representations The Constructive Schottky Problem (*)

11

Algebraic Curves

C =

• (x, y) ∈ C2 : f (x, y) = 0
• ⊂ C2 .

C as a y-covering of Cx:

◮ x independent, varies over Cx. ◮ y as dependent variable.

11

Algebraic Curves

C =

• (x, y) ∈ C2 : f (x, y) = 0
• ⊂ C2 .

C as a y-covering of Cx:

◮ x independent, varies over Cx. ◮ y as dependent variable. ◮ What are all possible y-roots to f (x, y) = 0?

x → y(x) = (y1(x), . . . , yd(x)) Q: Is there some surface other than Cx where y(x) is single-valued?

12

Riemann Surfaces

(Compact) Riemann Surfaces X:

◮ Connected, 1-dimensional complex manifold.

12

Riemann Surfaces

(Compact) Riemann Surfaces X:

◮ Every neighborhood of P ∈ X looks like U ⊂ C.

12

Riemann Surfaces

(Compact) Riemann Surfaces X:

◮ Every neighborhood of P ∈ X looks like U ⊂ C. ◮ Homeomorphic to a doughnut with g holes.

◮ g = genus

12

Riemann Surfaces

(Compact) Riemann Surfaces X:

◮ Every neighborhood of P ∈ X looks like U ⊂ C. ◮ Homeomorphic to a doughnut with g holes.

◮ g = genus

◮ The genus of a curve = the genus of x-surface on which y(x)

is single-valued.

◮ Branch cuts, etc. ◮ Caveats: singular points and points at infinity.

13

Algebraic Curves and Riemann Surfaces

C : f (x, y) = 0 ↓ “desingularize” and “compactify” ↓ Riemann surface X

◮ Desingularize:

◮ C is singular at (α, β) ∈ C if

∇f (α, β) = 0

◮ Puiseux series parameterize curves at singularities.

◮ Compactify: add points at infinity.

14

Geometry of Riemann Surfaces

Riemann surface X

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Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ = 0

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ = 0

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ

14

Geometry of Riemann Surfaces

H1(X, Z) = closed, oriented, homologous cycles on X γ = γ1 + γ2 γ1 γ2

14

Geometry of Riemann Surfaces

a1 a2 b1 b2 ai ◦ aj = 0 bi ◦ bj = 0 ai ◦ bj = δij H1(X, Z) = span{a1, . . . , ag, b1, . . . , bg}

14

Geometry of Riemann Surfaces

Aside: what is γ homologous to? γ

15

Demo

Basis of cycles.

16

Integration on X

Integration: natural use for paths. 1-forms: ω ∈ Ω1

X,

where, it is locally written ω

• Uα⊂X = hα
• x, y(x)
• dx,

hα meromorphic.

16

Integration on X

Integration: natural use for paths. 1-forms: ω ∈ Ω1

X,

where, it is locally written ω

• Uα⊂X = hα
• x, y(x)
• dx,

hα meromorphic. Given a path γ ∈ H1(X, Z) we can compute

• γ

ω.

17

Holomorphic Differentials

Holomorphic 1-forms: Γ(X, Ω1

X).

17

Holomorphic Differentials

Holomorphic 1-forms: Γ(X, Ω1

X).

Finite dimensional vector space: dimC Γ(X, Ω1

X) = g

Γ(X, Ω1

X) = span {ω1, . . . , ωg}

Aside: why are there no holomorphic differentials on all of X = C∗?

18

Demo

Basis of 1-forms.

19

Period Matrices

Define A, B ∈ Cg×g: Aij =

• aj

ωi Bij =

• bj

ωi “Period matrix” τ = [A | B] ∈ Cg×2g.

19

Period Matrices

Define A, B ∈ Cg×g: Aij =

• aj

ωi Bij =

• bj

ωi “Period matrix” τ = [A | B] ∈ Cg×2g. Possible to choose ωi’s such that

• aj

ωi = δij. (“normalized 1-forms”) Normalized period matrix τ = [I | Ω].

20

Period Matrices and Riemann matrices

Amazing Fact

Ω is a Riemann matrix.

20

Period Matrices and Riemann matrices

Amazing Fact

Ω is a Riemann matrix.

◮ dimC{period matrices} = 3g − 3 ◮ dimC hg = g(g + 1)/2

20

Period Matrices and Riemann matrices

Amazing Fact

Ω is a Riemann matrix.

◮ dimC{period matrices} = 3g − 3 ◮ dimC hg = g(g + 1)/2

Schottky Problem (1880s)

Given a Riemann matrix can we tell if it’s a period matrix?

20

Period Matrices and Riemann matrices

Amazing Fact

Ω is a Riemann matrix.

◮ dimC{period matrices} = 3g − 3 ◮ dimC hg = g(g + 1)/2

Schottky Problem (1880s)

Given a Riemann matrix can we tell if it’s a period matrix?

Novikov Conjecture (1965) / Shiota Theorem (1986)

A Riemann matrix Ω is a period matrix if and only if ∃U, V , W , z0 ∈ Cg, c ∈ C such that u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c

satisfies the KP equation.

21

Demo

Period / Riemann matrices.

22

Goal of This Talk

Algebraic Curves and Riemann Surfaces Introduction Geometry: Basis of Cycles Algebra: Holomorphic 1-forms Period Matrices Goals and Applications Periodic Solutions to Integrable PDEs Linear Matrix Representations The Constructive Schottky Problem (*)

23

Return to KP

Actually constructing solutions u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c.

Ingredients:

23

Return to KP

Actually constructing solutions u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c.

Ingredients:

• 1. Curve C : f (λ, µ) = 0,

23

Return to KP

Actually constructing solutions u(x, y, t) = 2∂2

x log θ(Ux + Vy + Wt + z0, Ω) + c.

Ingredients:

• 1. Curve C : f (λ, µ) = 0,
• 2. Divisor D on X: a finite, formal sum of places

D =

• i

niPi, Pi ∈ X. Goal: Develop a fast algorithm for producing and evaluating these solutions.

24

Generalization to Other PDEs

Analytically determine finite genus solution formula: u = u(x, t) (1D case), u(x, y, t) (2D case).

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Generalization to Other PDEs

Analytically determine finite genus solution formula: u = u(x, t) (1D case), u(x, y, t) (2D case). All necessary parameters are computable using abelfunctions.

◮ i.e. KP is “generic enough”.

24

Generalization to Other PDEs

Analytically determine finite genus solution formula: u = u(x, t) (1D case), u(x, y, t) (2D case). All necessary parameters are computable using abelfunctions.

◮ i.e. KP is “generic enough”.

Goal: Develop a framework for computing finite genus solutions to

• ther integrable PDEs.

25

Linear Matrix Representations

∀f ∈ C[x, y]: f (x, y) = det(A + Bx + Cy), A, B, C symmetric. Applications:

◮ Control theory, ◮ solving polynomial inequalities,

◮ Can use positive (semi) definite programming if A, B, C ≥ 0.

◮ study of two-dimensional spectrahedra: regions in R2 bounded

by Helton–Vinnikov curves.

26

LMRs for Helton–Vinnikov Curves

Combinatorial Approach (Plaumann, Sturmfels, Vinzant) O

• 2(d−2

2 )

compute time

26

LMRs for Helton–Vinnikov Curves

Combinatorial Approach (Plaumann, Sturmfels, Vinzant) O

• 2(d−2

2 )

compute time Helton–Vinnikov Theta Function Approach O

• g2

≈ O

• d4

compute time Uses Riemann theta, Abel map, and Schottky–Klein prime form. Goal: Develop high-performance algorithms for computing LMRs.

27

Constructive Schottky Problem

Recall

All period matrices are Riemann matrices, but not vice versa.

◮ dimC{period matrices} = 3g − 3 ◮ dimC hg = g(g + 1)/2

27

Constructive Schottky Problem

Recall

All period matrices are Riemann matrices, but not vice versa.

◮ dimC{period matrices} = 3g − 3 ◮ dimC hg = g(g + 1)/2

The Constructive Schottky Problem

Given a Riemann matrix Ω can we produce a curve C : f (x, y) = 0 with Ω as its period matrix? Goal (Long Term): Compute such an f .

Thank you!

Code: https://github.com/cswiercz/abelfunctions Documentation: https://www.cswiercz.info/abelfunctions General Exam: https://github.com/cswiercz/general-exam