ICMS

The higher-genus sigma function and applications

Oct 11, 2010 - Oct 15, 2010

ICMS, 15 South College Street

Organisers

Name	Institution
Athorne, Chris	University of Glasgow
Braden, Harry	University of Edinburgh
Buchstaber, Victor
Eilbeck, Chris	Heriot-Watt University
Enolski, Victor	National Academy of Sciences of Ukraine
Gibbons, John	Imperial College
Previato, Emma	Boston University

Organisers:
Victor Buchstaber [Chair] (Steklov Institute, Moscow)
Emma Previato[Vice-Chair] (Boston)

Local Organisers:
Chris Eilbeck [Vice-Chair] (Heriot-Watt University)
Chris Athorne (University of Glasgow)
Harry Braden (University of Edinburgh)
Victor Enolski (Institute of Magnetism, Kiev)
John Gibbons (Imperial College)

 

Computer algebra has given great impetus to the progress of many areas of mathematics in recent years.  The traditional boundaries of knowledge in number theory, group theory, algebraic geometry and differential equation theory are moving forward very rapidly, progress n part due to the design of new algorithms and to increasing computer power: for this to continue, two cultures have to come together -- the theoretically-trained and the "experimentalists".

We propose to bring together experts from different areas where computer algebra is contributing to major progress, specifically: the areas of algebraic curves and their moduli spaces; transcendental (special) functions; and their applications in applied mathematics and mathematical physics.  We believe the chosen participants can establish common ground, and give impetus to a variety of projects.  While some have collaborated before, there will be new directions to explore where the full power of two different settings has not yet been combined.  In addition, new areas have sprung out of classical mathematics, and it is again the theory of algebraic curves that has so far been found most effective in such contemporary issues as coding theory and cryptography; those too will be represented at this workshop.

It seems appropriate to centre the meeting on the "higher-genus sigma$-function".  In classical complex analysis, Weierstrass' $sigma$-function plays a central role because it captures both the algebraic (addition rule) and the transcendental properties of elliptic curves, with consequent applications.  This theme updates and develops the successful ICMS meeting in 1998, inspired by the work of H.F. Baker on the higher-genus sigma-function.

 The workshop report is available here.

 

 

Arrangements

Chris Eilbeck's website
Chris has published some talks and photos from the workshop on his website.

Attendance
Attendance is by invitation only. Invitations are sent out continuously from the end of May 2010.

Venue
The workshop will take place at 15 South College Street, ICMS head quarters. 

Accommodation
Delegates who wish to have accommodation booked for them are asked to tick the appropriate box on the registration form. Accommodation is block booked for single occupancy although delegates who plan to bring along partners and families are advised to contact Helene Frossling to discuss this. Accommodation details will be sent to delegates during the week commencing 27 September.

Talks
Speakers are advised that the lecture theatre has the following equipment: data projector (preferably used with the computer located in the lecture theatre but laptops may also be used; to use a Mac with the data projector you may need to bring an adaptor), visualiser (x1) and blackboards (x2). If you require further equipment please inform Helene Frossling in advance.

Abstracts
Please bear in mind that not all participants are experts in all the theory and applications of the different facets of sigma function theory covered by the conference.  Those working in number theory, algebraic geometry, and mathematical physics often use different approaches. Please try to make at least the first part of your talks not too technical in order that we initiate a fruitful dialogue across the topic boundaries.

With this principle as a guide, please look again at your abstract and consider whether it can be extended to be more informative to others from different areas.  In particular please add references to a few recent or classical papers to enable other delegates to dig deeper into the topic of your talk.  You are encouraged to send updated abstracts to Helene Frossling at ICMS as soon as possible. 

Poster session
All delegates (speakers included) are welcome to present posters during the informal wine reception and poster session on Monday 11 October.  Posters should preferably not be bigger than A0 (portrait) size.

Insurance
All delegates must ensure their travel insurance cover is adequate. ICMS cannot be held responsible for costs incurred in case of flight delays or similar. 

Sunday 10 October
For those who arrive to Edinburgh ahead of the workshop start, there will be a chance to meet up on Sunday night. The pub Cafe Royal has been in business since 1826, and on the present premises since 1863.  This Victorian bar serves excellent beers, wines, spirits and coffees, and has both a great bar food menu as well as a restaurant section. Address: 17 West Register St. This is a small lane behind Princes St and might not be marked on all tourist maps. It starts at the east end of Princes St, opposite the Balmoral Hotel where you should head down the right side of Burger King and follow the gentle slope up. You will be passing another beautiful old pub, the Abbotsford, on your left. Time to be confirmed.

Conference Co-ordinator
This workshop will be co-ordinated by Helene Frossling: helene[dot]frossling[at]icms[dot]org[dot]uk

 

 

Programme

Programme updated 4 October 2010.

Changes may be made to this programme without prior notice. Speakers are encouraged to check this programme now and then for changes.

Monday 11 October

08.30 - 09.15	Registration and coffee
09.15 - 09.30	Victor Enolskii (National Academy of Sciences of Ukraine) Introduction
09.30 - 10.30	Atsushi Nakayashiki (Kyushu University) On a deformation of Baker's addition theorem
10.30 - 11.00	Coffee (served in the Chapterhouse)
11.00 - 12.00	Christophe Ritzenhalter (IML) Algorithmic number theory and the allied theory of theta functions
12.00 - 13.00	Yoshihiro Onishi (University of Yamanashi) Frobenius-Stickelberger-type formulae for general curves
13.00 - 14.30	Lunch (served in the Chapterhouse)
14.30 - 15.30	David Grant (University of Colorado at Boulder) A level 1 genus 2 Jacobi's derivative formula and applications to the analytic theory of genus 2 curves
15.30 - 16.00	Coffee (served in the Chapterhouse)
16.00 - 17.00	Christian Klein (Institut de Mathématiques de Bourgogne) Algebraic curves and Riemann surfaces in Matlab
17.00 - 18.30	Wine reception and poster session (The Atrium)

Tuesday 12 October

09.30 - 10.30	Victor Enolski (National Academy of Sciences of Ukraine) On the existence of non-abelian monopoles: the algebro-geometric approach
10.30 - 11.00	Coffee (served in the Chapterhouse)
11.00 - 12.00	Harry Braden (University of Edinburgh) Symmetry, curves and monopoles
12.00 - 13.00	Jutta Kunz (University of Oldenburg) Monopoles and monopole-antimonopole systems
13.00 - 14.30	Lunch (served in the Chapterhouse)
14.30 - 15.30	Yury Fedorov (Politechnic University of Catalonia) Matrix Jacobi-Mumform systems, discrete Neumann systems on Stiefel varieties, and the addition law on Prym varieties
15.30 - 16.00	Coffee (served in the Chapterhouse)
16.00 - 17.00	Armando Treibich (Universite d'Artois) Systems of polynomial equations and hyperelliptic tangential covers

Wednesday 13 October

09.30 - 10.30	Claus Lämmerzahl (University Bremen) Analytic solutions of the geodesic equation in axially symmetric space-times and observables
10.30 - 11.00	Coffee (served in the Chapterhouse)
11.00 - 12.00	Betti Hartmann (Jacobs University Bremen) Geodesic motion in space-times containing cosmic stringsb
12.00 - 13.00	Valeriya Kagramanova (University of Oldenburg) Hyperelliptic curve of arbitrary genus in geodesic equations of higher dimensional space-times
13.00 - 14.30	Lunch (served in the Chapterhouse)
14.30 - 15.30	Emma Previato (Boston University) Subvarieties of moduli spaces of curves
15.30 - 16.00	Coffee (served in the Chapterhouse)
16.00 - 17.00	Dmitry Korotkin (Concordia University) Isomonodromic tau-function as higher genus analog of Dedekind eta-function

Thursday 14 October

09.30 - 10.30	Matthew England (University of Glasgow) Bases and addition formulae associated with higher genus Abelian functions
10.30 - 11.00	Coffee (served in the Chapterhouse)
11.00 - 12.00	Chris Athorne (University of Glasgow) A conjecture for covariant quadratic identities in P{ijk} for hyperelliptic curves of arbitrary genus
12.00 - 13.00	Chris Eilbeck (Heriot-Watt University) Some computational challenges in Abelian function theory at genus 3 and 4
13.00 - 14.30	Lunch (served in the Chapterhouse)
14.30 - 15.30	John Gibbons (Imperial College London) Explicit calculation of some reductions of the Benney equations
15.30 - 16.00	Coffee (served in the Chapterhouse)
16.00 - 17.00	Caroline Kalla (Université de Bourgogne) Quasi-periodic solutions of the Davey-Stewartson equation and the vector NLS equation

Friday 15 October

09.30 - 10.30	Andrew Hone (University of Kent) Sigma function solutions of bilinear recurrence relations
10.30 - 11.00	Coffee (served in the Chapterhouse)
11.00 - 12.00	Elena Bunkova (Russian Academy of Sciences) Sigma functions of genus 2 and their applications
12.00 - 13.00	Sergey Shpectorov (University of Birmingham) Parallel computation of braid orbits
13.00 - 14.30	Lunch (served in the Chapterhouse)
14.30 - 15.30	Andrey Mironov (Sobolev Institute of Mathematics) Commuting ordinary differential operators of rank 2
15.30 - 16.00	Coffee (served in the Chapterhouse)

Presentations:

Presentation Details
Athorne, Chris
A conjecture for covariant quadratic identities in P{ijk} for hyperelliptic curves of arbitrary genus
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The quadratic identities for the covariantly defined P{ijk}-functions in genus one, two and three have been shown to take the shapes of multiply bordered determinants involving sets of arbitrary parameters, Studying the representation theory implicit in this structure suggests a conjecture for the whole family of hyperelliptic curves.
Braden, Harry
Symmetry, Curves and Monopoles
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The modern approach to integrability proceeds via a Riemann surface, the spectral curve. In many applications this curve is specified by transcendental constraints in terms of periods. In some cases, including the construction of magnetic monopoles, physical symmetries are inherited by the spectral curve and there may be consequent simplifications of the transcendental constraints. We shall look at this interplay of ideas.
Bunkova, Elena
Elliptic formal group laws and differential equations
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The elliptic formal group law corresponds to the geometric group structure on an elliptic curve in Tate coordinates. In the focus of our interest will be the differential equations that appear as we study the structure of the formal group law. These are the equations on the exponential f of the formal group law, and the equations on the function Phi, the logarithmic derivative of which is f. In a special case, the equation on Phi will be the Lame equation. We will obtain the generalized Lame equation and study its properties. As an application, we will introduce the general Krichever genus, which is given by a generalized Baker-Akhiezer function. The talk is based on the work by Victor M. Buchstaber and Elena Yu. Bunkova "Elliptic formal group laws, integral Hirzebruch genera and Krichever genera" arXiv:1010.0944v1
Eilbeck, Chris
Some computational challenges in Abelian function theory at genus 3 and 4
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I will discuss two recent strands in work to understand the calculation and the structure of PDEs satisfied by Abelian functions associated with curves of genus 3 and 4. These Abelian functions are defined as logarithmic derivatives of the corresponding sigma function. The first strand is the connection between Fay's algebro-geometric tau-function and the sigma function, which provides two new alternative ways to calculate the family of PDEs associated with each curve. Hopefully this approach will lead to new insights, but computationally the technique is currently very intensive and much more work is required before it can be offered as a competitive algorithm in comparison with previous techniques. This work is documented in the recent paper by Eilbeck, Enolski and Gibbons, arXiv:1006.5219. The second strand in the talk is an attempt to understand the Kummer varieties associated with these curves, using constructive computer algebra calculations. In genus 2 this is classical work, well known since the books of Hudson and Baker, building on work by the German school. In higher genus, in the hyperelliptic case, much progress was made some time ago in this area by Reid and other algebraic geometers, although the connection with the sets of PDEs studied by mathematical physicists is not always clear. In the trigonal case relatively little is known, although there are some surprises to report. England's talk at this meeting and the paper arXiv:1008.0289 would be useful background to my talk.
England, Matthew
Bases and addition formulae associated with higher genus Abelian functions
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We define Abelian functions as those which are periodic with respect to the period matrices associated with classes of algebraic curves. We can realise them as generalisations of the Weierstrass P-function, defined in turn as logarithmic derivatives of the sigma-function. This approach, following the original ideas of Klein, was pioneered in [1]. In recent years the theory of these functions has been developed to study those associated with higher genus curves. These functions arise naturally as solutions to some of the important equations of mathematical physics, and so a greater understanding of their properties is desirable. The differential equations, series expansions, addition formulae and applications of the functions are the topics of several of the presentations at this meeting. Progress has followed both from new theory, and from the growth in computer power and an advancement of efficient symbolic computation techniques, as discussed in [2]. In this presentation we will start with an introduction to the functions and their key properties before presenting some recent results. These will focus on the construction of new bases for the vector spaces of the functions, which in turn can be used to derive the addition formulae they satisfy. Specifically, we will present new bases and addition formulae for the functions associated to trigonal curves of genus 3 and 4. This follows from the ideas in [3], with many of these new results recently made available in [4]. [1] V.M. Buchstaber, V.Z. Enolskii, and D.V. Leykin. Kleinian functions, hyperelliptic Jacobians and applications. Reviews in Math. and Math. Physics, 10:1–125, 1997. [2] M. England and J. C. Eilbeck. Abelian functions associated with a cyclic tetragonal curve of genus six. Journal of Physics A: Mathematical and Theoretical, 42:095210 (27pp), 2009. [3] J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Onishi, and E. Previato. Abelian functions for trigonal curves of genus three. International Mathematics Research Notices, page Art.ID: rnm140 (38 pages), 2007. [4] J.C. Eilbeck, M. England, Y. Onishi. Abelian functions associated with genus three algebraic curves. Preprint, arXiv:1008.0289v1
Enolski, Victor
On the existence of non-abelian monopoles: the algebro-geometric approach
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We develop the Atiyah-Drinfeld-Manin-Hitchin-Nahm construction to study $SU(2)$ non-abelian charge 3 monopoles within the algebro-geometric method. The method starts with finding an algebraic curve, the monopole spectral curve, subject to Hitchin's constraints. We take as the monopole curve the genus four curve that admits a $C_3$ symmetry, $eta3+alphaetazeta2+betazeta6+gammazeta3-beta=0$, with real parameters $alpha$, $beta$ and $gamma$. In the case $alpha=0$ we prove that the only suitable values of $gamma/beta$ are $pm 5sqrt{2}$ ($beta$ is given below) which corresponds to the tetrahedrally symmetric solution. We then extend this result by continuity to non-zero values of the parameter $alpha$ and find finally a {em new} one-parameter family of monopole curves with $C_3$ symmetry.
Fedorov, Yury
Matrix Jacobi-Mumform systems, discrete Neumann systems on Stiefel varieties, and the addition law on Prym varieties
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Abstract available as PDF: http://www.icms.org.uk/downloads/sigma/Fedorov.pdf
Gibbons, John
Explicit calculation of some reductions of the Benney equations
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The Benney equations are a set of dispersionless pde's in a countably infinite set of dynamical variables. They admit reductions, in which only finitely many, N, of these variables are independent. These reductions correspond to conformal mappings of a domain, typically the half-plane, into a similar domain with N slits. These slits run from the boundary, along fixed non-intersecting Jordan curves, to variable end points. An important class of these correspond to Schwartz-Christoffel maps; these have straight slits. If the angles between the slits and the boundary are rational multiples of pi, then the Schwartz-Christoffel mapping may be expressed as an Abelian function associated with an algebraic curve. We discuss several examples, where this has been explicitly calculated in terms of quotients of derivatives of the associated sigma function.
Grant, David
A level 1 genus 2 Jacobi's derivative formula and applications to the analytic theory of genus 2 curves
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We use a recent generalization of Jacobi's derivation formula for theta functions in two variables to come up with a "modular" set of parameters on principly polarized complex abelian surfaces whose polarization is irreducible, and use these to derive a model for complex curves of genus 2 as subvarieties of their jacobians. In this way we obtain a model whose coefficients are modular functions of a lower level than previously obtained, and hence is better suited for arithmetic applications.
Hartmann, Betti
Geodesic motion in space-times containing cosmic strings
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Cosmic strings have gained renewed interest over the past years due to their possible connection to the fundamental entities of string theory. While it was believed for a long time that these latter objects are not observable on cosmic scales, recent years have seen the introduction of inflationary models resulting from string theory in which cosmic strings form generically at the end of inflation. It has thus vital to understand how such objects can be detected in the universe. In this talk, I will discuss one of the possible ways to detect them, namely through the way that test particles move in cosmic string space-times. I will first present solutions to the geodesic equation in the space-time of black holes pierced by infinitely thin cosmic strings in which case the solutions can be given explicitely in terms of elliptic functions. I will then discuss our results for the space-time of a finite width cosmic string. In the latter case, the solutions of the geodesic equation have to be computed numerically.
Hone, Andrew
Sigma function solutions of bilinear recurrence relations
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Birational maps defined by bilinear recurrence relations appear in many contexts, including number theory, statistical mechanics, and most recently in Fomin and Zelevinsky's theory of cluster algebras. Sequences generated by bilinear recurrences are known as Somos sequences, after Michael Somos, who noticed the surprising fact that certain rational recurrences generate sequences of integers. After reviewing Somos sequences and the way that they arise from acyclic quivers, it is explained how Somos-6 and Somos-7 sequences arise as a reduction of Miwa's equation (discrete BKP). Analytic formulae in terms of genus two sigma functions are presented, before finishing with some open problems.
Kagramanova, Valeriya
Hyperelliptic curve of arbitrary genus in geodesic equations of higher dimensional space-times
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We present analytical solution of geodesic equations with underlying polynomial being a hyperelliptic curve of arbitrary genus. The problem is reduced to a theta-divisor and the solution is expressed in terms of derivatives of Kleinian sigma function. With the developed method we solve geodesic equations in higher dimensional spherically symmetric space-times, e.g. Schwarzschild, Schwarzschild-de Sitter, Reissner-Nordstroem and Reissner-Nordstroem-de Sitter, and in axially symmetric space-times, e.g. Myers-Perry.
Kalla, Caroline
Quasi-periodic solutions of the Davey-Stewartson equation and the vector NLS equation
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It is well known that almost periodic solutions of some integrable systems can be given in terms of multi-dimensional theta functions associated to compact Riemann surfaces. We present such solutions for the Davey-Stewartson equation and the vector NLS equation using a method based on degenerated versions of Fay’s famous trisecant identity, relating the theta function and its partial derivatives. A degenerated version of the Fay identity leads to quasi-periodic solutions of the DS equation and the vector NLS equation. The Riemann surface associated to the theta function has to satisfy reality properties. Degenerating the Riemann surface into a genus zero surface, we obtain solutions like solitons, breathers, dromions, lumps, etc...
Klein, Christian
Algebraic curves and Riemann surfaces in Matlab
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We present a fully numerical approach to Riemann surfaces defined via plane algebraic curves. The code in Matlab computes for a given algebraic equation in two variables the branch points and singularities, the holomorphic differentials and a base of the homology. The monodromy group for the surface is determined via analytic continuation of the roots of the algebraic equation on a set of contours forming the generators of the fundamental group. The periods of the holomorphic differentials are computed with Gauss-Legendre integration along these contours. The Abel map is obtained in a similar way. The performance of the code is illustrated for many examples. As an application we study quasi-periodic solutions to certain integrable partial differential equations.
Korotkin, Dmitry
Isomonodromic tau-function as higher genus analog of Dedekind eta-function
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In this talk we discuss various properties of the isomonodromic tau-funtion corresponding to a special class of Riemann-Hilbert problems; these Riemann-Hilbert problems arise in the theory of Frobenius manifold structures on Hurwitz spaces. On each stratum of Hurwitz space the tau-function becomes a section of the Hodge line bundle, non-singular everywhere except the boundary (in genus one it essentially coincides with the Dedekind eta-function). The tau-function can be computed explicitly in terms of theta-functions in any genus. We discuss various applications of this object - from holomorphic factorization of determinant of Laplacian on Riemann surfaces to random matrices. The talk is based on results obtained jointly with A.Kokotov and P.Zograf.
Kunz, Jutta
Monopoles and monopole-antimonopole systems
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The Georgi-Glashow model has a plethora of classical solutions. These include monopoles, multimonopoles and monopole-antimonopole systems. Most of these solutions are known only numerically. I discuss their properties as well as the effect of electric charge and gravity.
Laemmerzahl, Claus
Analytic solutions of the geodesic equation in axially symmetric space-times and observables
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The complete set of analytic solutions of the geodeisc equation in the class of Plebanski-Demianski space-times are presented in terms of the Kleinian sigma function. The corresponding observables like Perihelion shift, Lense-Thirring effect, deflection of light, and others, are given in terms of complete hyperelliptic integrals.
Mironov, Andrey
Commuting ordinary differential operators of rank 2
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We suggest a method of constructing of commuting ordinary differential operators of rank 2 corresponding to spectral curves of genus 2,3 and 4. In the case of genus 2 and 4 the operators have polynomial coefficients. We also find spectral data corresponding to self-adjoint operators.
Nakayashiki, Atsushi
On a deformation of Baker's addition theorem
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A deformation of Baker's addition theorem for Klein's sigma function of genus two is proposed. An extension of three term identity of the elliptic sigma function to the case of genus two is also discussed.
Onishi, Yoshihiro
Frobenius-Stickelberger-type formulae for general curves
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In this talk, Frobenius-Stickelberger-type determinatal formula is developed for the general curves of type of Buchstaber, Enolskii, and Leykin.
Previato, Emma
Subvarieties of moduli spaces of curves
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We review special subvarieties of the moduli space of compact Riemann surfaces, with emphasis on applications to differential equations and computer algebra. For example: curves with automorphisms can be characterized by theta-function equations in genus two, partially in genus three (work with T. Shaska and S. Wijesiri); the sigma function has a special vanishing pattern for cyclic curves (work with S. Matsutani); the Klein curve has an `implementable' Arithmetic-geometric Mean (work with E.S.A. Farrington).
Ritzenhalter, Christophe
Algorithmic number theory and the allied theory of theta functions
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This talk will be a survey about the different uses of theta functions in algorithmic number theory, motivated by its applications to cryptography and coding theory. We will focus on explicit relations between the curve and its Jacobian for curves of small genus. The existence of fast algorithms for computing theta constants and the period matrix will be addressed as well.Finally, we will present open questions with the hope they have been considered in other areas. No knowledge in number theory is required to follow the talk.
Shpectorov, Sergey
Parallel computation of braid orbits
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Hurwitz loci of curves of genus g admitting a group G are classified by the orbits of the surface braid group on the set of generating tuples in G. We will discuss the implementation, based on SCSCP, of the main orbit construction routine from the package BRAID for the parallel computation of braid orbits on multiprocessor systems.
Treibich, Armando
systems of polynomial equations and hyperelliptic tangential covers
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Download abstract from http://www.icms.org.uk/downloads/sigma/Treibich.pdf

Participants

Name	Institution
Athorne, Chris	University of Glasgow
Braden, Harry	University of Edinburgh
Bunkova, Elena	Russian academy of sciences
Eilbeck, Chris	Heriot-Watt University
England, Matthew	University of Glasgow
Enolski, Victor	National Academy of Sciences of Ukraine
Fedorov, Yury	Politecnic university of Catalonia
Fordy, Allan	University of Leeds
Gibbons, John	Imperial College
Grant, David	University of Colorado at Boulder
Hartmann, Betti	Jacobs University Bremen
Hone, Andrew	University of Kent
Kagramanova, Valeriya	Oldenburg University
Kalla, Caroline	Université de Bourgogne
Klein, Christian	Institut de Mathématiques de Bourgogne
Korotkin, Dmitry	Concordia University
Kunz, Jutta	University of Oldenburg
Laemmerzahl, Claus	University Bremen
Mironov, Andrey	Sobolev Institute of Mathematics
Nakayashiki, Atsushi	Kyushu University
Nijhoff, Frank	University of Leeds
Norman, Sophie	University of Glasgow
Onishi, Yoshihiro	University of Yamanashi
Previato, Emma	Boston University
Reid, Miles	University of Warwick
Ritzenhalter, Christophe	IML
Shpectorov, Sergey	University of Birmingham
Strachan, Ian	University of Glasgow
Swierczewski, Chris	University of Washington
Szabo, Richard	Heriot-Watt University
Treibich, Armando	Universite d'Artois