# Research Workshop: Riemann Surfaces - Analytical and Numerical Methods

## Abstracts for the talks

### Geometry and combinatorics of PS-3 integral equations

**Andrei Bogatyrëv** *(Russia)*

Thursday, June 01, 2006

More than a hundred years ago H.Poincare and V.A.Steklov
considered a problem for the Laplace equation with spectral
parameter in the boundary conditions. Today similar problems for
two adjacent domains with the spectral parameter in the
conditions on the common boundary of the domains arises in a
variety of situations: in justification and optimization of
domain decomposition method, simple 2D models of oil
extraction. Singular 1D integral Poincare-Steklov equation
naturally emerges after reducing this 2D problem to the common
boundary of the domains.

This latter equation has one functional parameter R(x) - the
smooth change of variable on the interval of integration. When
R(x) is a rational function, the considered equation is
equivalent to some Riemann monodromy problem [1], so powerful
geometrical techniques may be applied for the investigation.
Explicit representations for the eigenvalues/eigenfunctions
may be given: in terms of Jacobi elliptic functions when deg
R=2 [2] or in terms of Kleinian membranes when deg R=3 [3].

In the latter case the Riemann monodromy problem may be reformulated
in terms of branched projective structures on genus 2 Riemann
surface. The problem has essentially combinatorial nature and
may be solved with the help of the technique resembling
Grothendieck's dessins.

[1] Bogatyrev A.B. Poincare-Steklov integral equations
and Riemann monodromy problem //Functional Analysis and
Applications, 34:2 (2000), pp. 9-22.

[2] Bogatyrev A.B. A geometric method for solving a series of
integral PS equations// Math. Notes, 63:3 (1998), pp. 302-310.

[3] Bogatyrev A.B. PS-3 integral equations and projective
structures on Riemann Surfaces //Math. Sbornik, 192:4
(2001), pp. 3--36.

### Monopoles and an identity of Ramanujan

**Harry Braden** *(United Kingdom)*

Thursday, June 01, 2006

Magnetic monopoles, or the topological soliton solutions of
Yang-Mills-Higgs gauge theories in three space dimensions, have
been objects of fascination for over a quarter of a century. BPS
monopoles in particular have been the focus of much research. Many
striking results are now known, yet, disappointingly, explicit
solutions are rather few. We bring techniques from the study of
finite dimensional integrable systems to bear upon the
construction. The transcendental constraints of Hitchin may be
replaced by (also transcendental) constraints on the period
matrix. For a class of curves we show how to these may be reduced
to a number theoretic problem. A recently proven result of
Ramanujan related to the hypergeometric function enables us to solve these and construct the
corresponding monopoles.

### Gaps left by simple closed geodesics on surfaces

**Peter Buser** *(Switzerland)*

Thursday, June 01, 2006

This talk concerns joint work with Hugo Parlier.
For a Riemann surface endowed with a hyperbolic metric, J. Birman and C.
Series have shown that the set of all points lying on any simple closed
geodesic is nowhere dense on the surface. (This set is sometimes
referred to as the Birman-Series set).
The talk will discuss the existence of positive constants , such
that for any surface of genus *g*, the complementary region to the
Birman-Series set allows an isometrically embedded disk with radius
. The behavior of as function of *g*, as well as some bounds
will be discussed.

The talk will also discuss a new algorithm for the enumeration of
the simple closed goedesics.

### Conformal mapping of multiply connected domains: theory and applications

**Darren Crowdy** *(United Kingdom)*

Wednesday, May 31, 2006

The theory of conformal mappings plays an important
role in many applications and, recently, has been shown
to arise in integrable systems theory. This talk will
focus on the mathematical construction of conformal
mappings to multiply connected domains. For example, one of the
few general constructive techniques of conformal
mapping theory is based on the Schwarz-Christoffel mapping formula.
In recent years, powerful new software has been
developed to construct such mappings, based on this
classical formula, in the simply connected case. In this talk,
a new general formula for the Schwarz-Christoffel mapping
to multiply connected polygonal domains will be derived.
The idea of the construction is to perform the analysis
on a compact Riemann surface known as the Schottky double
of a conformally equivalent circular domain
and to make use of an associated prime function.
Applications and numerical issues will be discussed.

### Towards the Algebro-Geometric Integration of the Bogomolny Equations

**Viktor Enol'skii** *(Ukraine, and Canada)*

Thursday, June 01, 2006

Finding explicit solutions of the Bogomolny equations for (the BPS limit)
of Yang-Mills-Higgs systems represents an important
and challenging problem in mathematical physics because of
its links to gauge theory and the standard model of elementary
particles. Although the complete integrability of these equations was
proven a quarter of a century ago and the relation of this integrability
to algebraic curves was also elucidated, far less is known about the
algebro-geometric (theta-function) solutions. In our investigation we
develop the approach by Ercolani-Sinha (1989) based on the famous
Atyah-Drinfeld-Hitchin-Manin-Nahm construction.

In the talk we first explain the whole construction in the case when Bogomolny equations are solvable in terms of elliptic functions where all answers are explicit. Then we discus the generalization to higher genera when Hitchin constraints appear. This leads to complications which make the problem both mathematically interesting and difficult.

Taking into account the subject of the meeting we will then concentrate on the computer algebra problems relevant to solving Hitchin's constraints and the calculation of objects of physical interest - gauge and Higgs field. These include the implementing Weierstrass-Poincaré reduction of Riemann period matrix to standard form, and calculating theta-functions whose period matrices yield poor convergence. Both problem are good candidates for incorporation within Maple packages.

More details on the work and results will be done in the next lecture of
H.W.Braden.

### Spectral methods and Riemann surfaces

**Jörg Frauendiener** *(Germany)*

Wednesday, May 31, 2006

In this talk I will describe the basics about spectral methods. I
will then go on to discuss an implementation of theta-functions on
hyperelliptic Riemann surfaces using some of these ideas. Some
applications will be mentioned.

### Numerical simulation of the small dispersion limit of the KdV equation, Whitham equations and Painleve' equations

**Tamara Grava** *(Italy)*

Friday, June 02, 2006

In this work we compare numerically the solution of the
small dispersion limit of the KdV equation with the first order
asympototics formulas given in terms of the Whitham equations.
The first order asymptotics fails to be a good asymptotic in some regions
of the (x,t) plane. In these regions a better asympotitic is provided by
special solutions of the Painleve' equations.

### Elliptic Filters and Genera

**Martin Hassner** *(USA)*

Friday, June 02, 2006

Zolotarev polynomials have been used in electrical engineering as
realizable electrical filters with prescribed frequency behaviour.
More recently such polynomials have appeared as elliptic genera of
manifolds. In this talk I describe an application of their
underlying mathematical structure to practical problems in digital
data coding and signal processing.

### Flat conical metrics on Riemann surfaces and determinants of Laplacians

**Alexei Kokotov** *(Canada, and Germany)*

Friday, June 02, 2006

Regularized determinants of Laplacians in flat singular metrics
on compact Riemann surfaces of arbitrary genus are discussed.
Analytic surgery theorem and variational formulas for these
determinants are proved. This leads to an explicit calculation
of the determinants, the resulting expression generalizes the
well-known Ray-Singer formula for the determinant of Laplacian
in flat smooth metric on an elliptic surface.

### Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

**Dmitri Korotkin** *(Canada)*

Friday, June 02, 2006

We study extremal properties
of the determinant of the Laplacian
in the Bergman metric on the moduli space of compact genus two Riemann
surfaces.
By a combination of analytical and numerical methods we identify four
non-degenerate
critical points of this function and compute the signature of the
Hessian at these points.
The curve with the maximal number of automorphisms (the Burnside curve)
turns out to be the
point of the absolute maximum. Our results agree with the mass formula
for virtual Euler characteristics
of the moduli space. A similar analysis is performed for
Bolza's strata of symmetric Riemann surfaces of genus two.
This is a joint work with Christian Klein and Alexey Kokotov.

### Function theory on the Schottky double

**Jonathan Marshall** *(United Kingdom)*

Wednesday, May 31, 2006

In this talk we develop some constructive function
theory aimed at solving a number of problems, arising in
physics and applied mathematics, associated with multiply connected
domains. The analysis is performed on a compact Riemann surface
known as the Schottky double of the domain. The practical
importance of a function known as the Schottky-Klein
prime function will be emphasized. This is joint work with
D. Crowdy.

### Discrete Riemann surfaces

**Christian Mercat** *(France)*

Thursday, June 01, 2006

Surfaces found in computers are usually discrete surfaces like
3D-meshes. We will present a way to consider them as Riemann surfaces
with holomorphic functions and forms living on its vertices and edges,
period matrices that can be numerically computed and explicit basis of
local holomorphic functions like polynomials and exponentials.

### Computing with Maple on Riemann surfaces arising from algebraic curve

**Matt Patterson** *(USA)*

Friday, June 02, 2006

Our group is interested in computations on Riemann surfaces that arise
from irreducible plane algebraic curves, and this talk will highlight some
of the work that we have done. I will begin with a brief outline of how a
Riemann surface is obtained from an algebraic curve, along the way
introducing some methods that have already been implemented in the Maple
package `algcurves`

. For example, procedures to compute bases of both
the homology and cohomology, as well as the Riemann matrix of a Riemann
surface originating from an algebraic curve. Subsequently I will discuss
the Abel map, the vector of Riemann constants and algorithms to compute
both; implementations of these algorithms will be included in future
versions of the `algcurves`

package. The talk will conclude with
Maple demonstrations of these procedures as time allows.

### Computational Methods for Schottky Uniformization and the Java Oorange Environment

**Markus Schmies** *(Germany)*

Wednesday, May 31, 2006

The numerical treatment of Riemann surfaces of higher genus is a
challenging problem. We developed methods based on Schottky uniformization
which we used in many applications involving examples up to genus 12. The
talk gives a short introduction to Schottky uniformization and those
closely related Poincaree Theta series which express differentials and
integrals. We will briefly discuss numerical issues of the evaluation of
these series and will present the results of our evaluation algorithms.
Finally, we give a software demonstration and develop from scratch a non-trivial
application utilizing our methods in the Java Oorange Environment.

### Riemann-Hilbert problems on Hurwitz-Frobenius manifolds

**Vasilisa Shramchenko** *(United Kingdom)*

Wednesday, May 31, 2006

There are two (dual) Riemann-Hilbert problems naturally associated to
every Frobenius manifold. In this talk I will present solutions to the
Riemann-Hilbert problems corresponding to Frobenius structures on
Hurwitz spaces (moduli spaces of functions over Riemann surfaces). The
solutions are given in terms of meromorphic bidifferentials defined on the
underlying surface. In the case of Hurwitz spaces corresponding to
hyperelliptic coverings the Stokes matrix and monodromy matrices are
computed.

### Quantum chaos and the Selberg trace formula on Riemann surfaces

**Frank Steiner** *(Germany)*

Friday, June 02, 2006

The classical dynamics of a point particle sliding freely
on a Riemann surface (RS) of constant negative curvature
(hyperbolic flow) is strongly chaotic (ergodic, mixing and
Bernoullian). The corresponding quantum dynamics is given
by the eigenvalue problem of the Laplace-Beltrami operator
on the given RS. The Selberg trace formula is a deep
relation in spectral geometry which expresses the quantal
energy spectrum by the length spectrum of the classical
periodic orbits (closed geodesics) on the given RS. For
compact RSs of genus two (arithmetic and non-arithmetic ones)
and the non-compact modular surface, we present analytical
and numerical results on the length spectrum, the eigenvalues
and eigenfunctions, spectral statistics and a comparison
with random matrix theory. Finally, we discuss a conjecture
on the value distribution of the Selberg zeta function on
the critical line.

## Date and Location

**May 31 - June 02, 2006**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Alexander Bobenko**

Technische Universität Berlin

Institut für Mathematik

Berlin

**Christian Klein**

Max Planck Institute for Mathematics in the Sciences

Leipzig

## Administrative Contact

**Regine Lübke**

Max Planck Institute for Mathematics in the Sciences

Leipzig

Contact by Email