The theory of Riemann surfaces has many applications in mathematics and physics as surface theory, integrable systems and random matrix theory. Modern computational techniques allow the use of numerical methods for problems which were so far difficult to access both analytically and numerically. It is the aim of this research workshop to bring together experts from Riemann surface theory and the theory of moduli spaces with experts from symbolic and scientific computing. The exchange of knowledge should inform the analytical side on the possibilities of modern computing and the numerical side on current problems in Riemann surface theory which are computationally accessible. See also the Virtual Math Lab about Dynamical Systems.

Speakers

Andrei Bogatyrëv

INM RAS (Moscow), Russia

Harry Braden

University of Edinburgh, United Kingdom

Peter Buser

EPF Lausanne, Switzerland

Darren Crowdy

Imperial College (London), United Kingdom

Boris Dubrovin

Sissa Trieste, Italy

Viktor Enol'skii

Institute of Magnetism (Kiev), Ukraine and Concordia University (Montreal), Canada

Jörg Frauendiener

Universität Tübingen, Germany

Tamara Grava

Sissa Trieste, Italy

Martin Hassner

Hitachi (San Jose), USA

Alexei Kokotov

Concordia University (Montreal), Canada and MPI MIS (Leipzig), Germany

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.

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.

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.

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.

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.

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.

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.

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 $C_g$, such that for any surface of genus $g$, the complementary region to the Birman-Series set allows an isometrically embedded disk with radius $C_g$. The behavior of $C_g$ 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.

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.

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.

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.

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.

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.

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.

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 \verb!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 \verb!algcurves! package. The talk will conclude with Maple demonstrations of these procedures as time allows.

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.

Participants

Alexander Bobenko

Berlin

Andrei Bogatyrëv

Moscow

Ljudmilla Bordag

Halmstad

Harry Braden

Edinburgh

Peter Buser

Lausanne

Darren Crowdy

London

Boris Dubrovin

Trieste

Viktor Enol'skii

Kiev

Jörg Frauendiener

Tübingen

Tamara Grava

Trieste

Martin Hassner

San Jose

Christian Klein

Leipzig

Alexei Kokotov

Montreal

Dmitri Korotkin

Montreal

Jonathan Marshall

London

Christian Mercat

Montpellier

Matt Patterson

Seattle

Markus Schmies

Berlin

Mika Seppälä

Tallahassee

Vasilisa Shramchenko

Oxford

Frank Steiner

Ulm

Robert Todd

Tallahassee

Scientific Organizers

Alexander Bobenko

Technische Universität Berlin

Christian Klein

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

Administrative Contact

Regine Lübke

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig
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