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

Acknowledgments 1 ◮ 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.

The Kadomtsev–Petviashvili Equation 2 u ( x , y , t ) = surface height of a 2D periodic shallow water wave. 4 u yy = ∂ 3 u t − 1 � � 4 (6 uu x + u xxx ) ∂ x Figure : ˆ Ile de R´ e, France Figure : Model of San Diego Bay

Theta Function Solutions 3 Family of solutions: ∀ g ∈ Z + u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c , ◮ c ∈ C , ◮ U , V , W , z 0 ∈ C g , ◮ Ω ∈ C g × g . ◮ “Riemann theta function” θ : C g × C g × g → C Finite genus solutions : ◮ dense in space of periodic solutions to KP.

The Riemann Theta Function 4 � 1 � e 2 π i 2 n · Ω n + n · z � θ ( z , Ω) = n ∈ Z g

The Riemann Theta Function 4 � 1 � e 2 π i 2 n · Ω n + n · z � θ ( z , Ω) = n ∈ Z g Convergence ◮ Requires Im(Ω) > 0. ◮ Also need only consider Ω T = Ω. ◮ Space of Riemann matrices : Ω ∈ C g × g | Ω T = Ω and Im(Ω) > 0 � � h g = (Siegel upper half space.) θ : C g × h g → C

Abelian Functions 5 Periodic, meromorphic functions f : C g → C with 2 g independent periods.

Abelian Functions 5 Periodic, meromorphic functions f : C g → C with 2 g 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.

Abelian Functions 5 Periodic, meromorphic functions f : C g → C with 2 g 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!

abelfunctions 6 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.

Connection to Algebraic Geometry 8 u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c U , V , W , z 0 , c , Ω not arbitrary.

Connection to Algebraic Geometry 8 u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c U , V , W , z 0 , c , Ω not arbitrary. Derived from a complex plane algebraic curve : given f ( λ, µ ) = α n ( λ ) µ n + α n − 1 ( λ ) µ n − 1 + · · · + α 0 ( λ ) the curve C is the set ( λ, µ ) ∈ C 2 : f ( λ, µ ) = 0 � � C = .

Goal of This Talk 9 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 (*)

Goal of This Talk 10 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 (*)

Algebraic Curves 11 ( x , y ) ∈ C 2 : f ( x , y ) = 0 ⊂ C 2 . � � C = C as a y-covering of C x : ◮ x independent, varies over C x . ◮ y as dependent variable.

Algebraic Curves 11 ( x , y ) ∈ C 2 : f ( x , y ) = 0 ⊂ C 2 . � � C = C as a y-covering of C x : ◮ x independent, varies over C x . ◮ y as dependent variable. ◮ What are all possible y -roots to f ( x , y ) = 0? x �→ y ( x ) = ( y 1 ( x ) , . . . , y d ( x )) Q: Is there some surface other than C x where y ( x ) is single-valued?

Riemann Surfaces 12 (Compact) Riemann Surfaces X : ◮ Connected, 1-dimensional complex manifold.

Riemann Surfaces 12 (Compact) Riemann Surfaces X : ◮ Every neighborhood of P ∈ X looks like U ⊂ C .

Riemann Surfaces 12 (Compact) Riemann Surfaces X : ◮ Every neighborhood of P ∈ X looks like U ⊂ C . ◮ Homeomorphic to a doughnut with g holes. ◮ g = genus

Riemann Surfaces 12 (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.

Algebraic Curves and Riemann Surfaces 13 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.

Geometry of Riemann Surfaces 14 Riemann surface X

Geometry of Riemann Surfaces 14 γ H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ = 0 H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ � = 0 H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 γ 1 γ 2 γ = γ 1 + γ 2 H 1 ( X , Z ) = closed, oriented, homologous cycles on X

Geometry of Riemann Surfaces 14 a 1 a 2 a i ◦ a j = 0 b i ◦ b j = 0 b 1 b 2 a i ◦ b j = δ ij H 1 ( X , Z ) = span { a 1 , . . . , a g , b 1 , . . . , b g }

Geometry of Riemann Surfaces 14 γ Aside: what is γ homologous to?

15 Demo Basis of cycles.

Integration on X 16 Integration: natural use for paths. 1-forms : ω ∈ Ω 1 X , where, it is locally written � � � ω U α ⊂ X = h α x , y ( x ) dx , h α meromorphic . � �

Integration on X 16 Integration: natural use for paths. 1-forms : ω ∈ Ω 1 X , where, it is locally written � � � ω U α ⊂ X = h α x , y ( x ) dx , h α meromorphic . � � Given a path γ ∈ H 1 ( X , Z ) we can compute � ω. γ

Holomorphic Differentials 17 Holomorphic 1-forms: Γ( X , Ω 1 X ) .

Holomorphic Differentials 17 Holomorphic 1-forms: Γ( X , Ω 1 X ) . Finite dimensional vector space: dim C Γ( X , Ω 1 X ) = g Γ( X , Ω 1 X ) = span { ω 1 , . . . , ω g } Aside: why are there no holomorphic differentials on all of X = C ∗ ?

18 Demo Basis of 1-forms.

Period Matrices 19 Define A , B ∈ C g × g : � � A ij = ω i B ij = ω i a j b j “Period matrix” τ = [ A | B ] ∈ C g × 2 g .

Period Matrices 19 Define A , B ∈ C g × g : � � A ij = ω i B ij = ω i a j b j “Period matrix” τ = [ A | B ] ∈ C g × 2 g . Possible to choose ω i ’s such that � ω i = δ ij . (“normalized 1-forms”) a j Normalized period matrix τ = [ I | Ω] .

Period Matrices and Riemann matrices 20 Amazing Fact Ω is a Riemann matrix.

Period Matrices and Riemann matrices 20 Amazing Fact Ω is a Riemann matrix. ◮ dim C { period matrices } = 3 g − 3 ◮ dim C h g = g ( g + 1) / 2

Period Matrices and Riemann matrices 20 Amazing Fact Ω is a Riemann matrix. ◮ dim C { period matrices } = 3 g − 3 ◮ dim C h g = g ( g + 1) / 2 Schottky Problem (1880s) Given a Riemann matrix can we tell if it’s a period matrix?

Period Matrices and Riemann matrices 20 Amazing Fact Ω is a Riemann matrix. ◮ dim C { period matrices } = 3 g − 3 ◮ dim C h g = 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 , z 0 ∈ C g , c ∈ C such that u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c satisfies the KP equation.

21 Demo Period / Riemann matrices.

Goal of This Talk 22 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 (*)

Return to KP 23 Actually constructing solutions u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c . Ingredients:

Return to KP 23 Actually constructing solutions u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c . Ingredients: 1. Curve C : f ( λ, µ ) = 0,

Return to KP 23 Actually constructing solutions u ( x , y , t ) = 2 ∂ 2 x log θ ( Ux + Vy + Wt + z 0 , Ω) + c . Ingredients: 1. Curve C : f ( λ, µ ) = 0, 2. Divisor D on X : a finite, formal sum of places � D = n i P i , P i ∈ X . i Goal: Develop a fast algorithm for producing and evaluating these solutions.