LMS Workshop on INTEGRABLE SYSTEMS,

 

Department of Mathematics

University of Glasgow, 5th June 2006

 

 

Programme:

 

      June 5th 2006

 

 

 

Time

Speaker/Event

Title

9:45-10:45

M. Blaszak

Reciprocal transformations for classical Stackel systems and related dispersionless systems

10:45-11:15

Coffee

11:15-12:15

A. Kokotov

Determinants of Laplacians and moduli spaces of Riemann surfaces and Abelian differentials.

12:15-1:30

Lunch

1:30-2:30

Y.Ohta

T.B.A

2:30-3:30

S. Silvestrov

Quasi-Lie algebras and twisted derivations

3:30-4:00

Tea

4:00-5:00

S. Simirnov

Discrete analogs of Veselov-Shabat dressing chain

Evening

Dinner: Bar Milano, Byers Road, 6:30pm

 

 

Location: Room 214, Department of Mathematics.

 

Titles and abstracts:

 

 

Maciej Blaszak

 

Reciprocal transformations for classical Stackel systems and related  dispersionless systems

Abstract:
Systematic construction of classical Stackel systems (these with all  constants of motion quadratic in momenta) in coordinate free way is presented. With each class of such  systems one can relates weakly nonlinear semi-hamiltonian dispersionless systems. It is shown that different classes of Stackel systems as well as connected dispersionless systems are related by an appropriate reciprocal transformations.

 

Alexey Kokotov


Determinants of Laplacians and moduli spaces of Riemann surfaces and Abelian differentials.
 

Abstract:
The  Ray-Singer formula for the determinant of the Laplacian on elliptic surfaces is generalized to Laplacians in flat metrics with conical singularities on compact Riemann surfaces of  genus greater than 1.  The result is given in terms of higher genus generalization of Dedekind's eta-function.  One of potential applications of the determinant of Laplacian is, according to the general approach of Osgood-Philips-Sarnak, to use it as a Morse function on the space of metrics on Riemann surfaces of a given genus and fixed area. We develop this idea for the case of  genus two. By a combination of analytical and numerical methods we find four stationary points of the determinant of Laplacian on the moduli space; three of them correspond to the surfaces with large groups of automorphisms.  This allows to calculate the orbifold Euler characteristics of the full moduli space of genus two curves, reproducing the number  -1/120 according to well-known Harer-Zagier formula. For symmetric strata of  Riemann surfaces with D2, D3 and Z2 symmetry groups the  orbifold Euler characteristic are given by      -1/2, -1/2 and -1/4  respectively.

 

S.Smirnov: 

 

Discrete analogs of Veselov-Shabat dressing chain

 

S. Silvestrov

 

Quasi-Lie algebras and twisted derivations
 
Abstract:
 In this talk the class of quasi-Lie algebras and some of its subclasses will be presented. The
definitions will be explained, and examples coming from twisted derivations, difference type
operators and other examples and constructions leading to quasi-Lie algebras will be described.


Updated 2nd June 2006