1. 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

2. 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.

3. 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 : ˆIle de R´e, France Figure : Model of San Diego Bay

4. 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.

5. 4
The Riemann Theta Function
θ(z, Ω) =
n∈Zg
e
2πi
1
2n·Ωn+n·z

6. 4
The Riemann Theta Function
θ(z, Ω) =
n∈Zg
e
2πi
1
2n·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

7. 5
Abelian Functions
Periodic, meromorphic functions
f : Cg
→ C
with 2g independent periods.

8. 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.

9. 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!

10. 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

11. 7
Demo
Riemann theta functions.

12. 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.

13. 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 .

14. 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 (*)

15. 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 (*)

16. 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.

17. 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?

18. 12
Riemann Surfaces
(Compact) Riemann Surfaces X:
Connected, 1-dimensional complex manifold.

19. 12
Riemann Surfaces
(Compact) Riemann Surfaces X:
Every neighborhood of P ∈ X looks like U ⊂ C.

20. 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

21. 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.

22. 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.

23. 14
Geometry of Riemann Surfaces
Riemann surface X

24. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ

25. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ = 0

26. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ

27. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ = 0

28. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ

29. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ

30. 14
Geometry of Riemann Surfaces
H1(X, Z) = closed, oriented, homologous cycles on X
γ = γ1 + γ2
γ1 γ2

31. 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 }

32. 14
Geometry of Riemann Surfaces
Aside: what is γ homologous to?
γ

33. 15
Demo
Basis of cycles.

34. 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.

35. 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.

36. 17
Holomorphic Differentials
Holomorphic 1-forms:
Γ(X, Ω1
X ).

37. 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∗
?

38. 18
Demo
Basis of 1-forms.

39. 19
Period Matrices
Define A, B ∈ Cg×g
:
Aij =
aj
ωi Bij =
bj
ωi
“Period matrix” τ = [A | B] ∈ Cg×2g
.

40. 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].

41. 20
Period Matrices and Riemann matrices
Amazing Fact
Ω is a Riemann matrix.

42. 20
Period Matrices and Riemann matrices
Amazing Fact
Ω is a Riemann matrix.
dimC{period matrices} = 3g − 3
dimC hg = g(g + 1)/2

43. 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?

44. 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.

45. 21
Demo
Period / Riemann matrices.

46. 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 (*)

47. 23
Return to KP
Actually constructing solutions
u(x, y, t) = 2∂2
x log θ(Ux + Vy + Wt + z0, Ω) + c.
Ingredients:

48. 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,

49. 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
ni Pi , Pi ∈ X.
Goal: Develop a fast algorithm for producing and evaluating these
solutions.

50. 24
Generalization to Other PDEs
Analytically determine finite genus solution formula:
u =
u(x, t) (1D case),
u(x, y, t) (2D case).

51. 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”.

52. 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
other integrable PDEs.

53. 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.

54. 26
LMRs for Helton–Vinnikov Curves
Combinatorial Approach (Plaumann, Sturmfels, Vinzant)
O 2(d−2
2 ) compute time

55. 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.

56. 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

57. 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 .

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