Integrability, algebra and geometry


13 December 2019, School of Mathematics and Statistics
University of Glasgow

About the workshop:

A one day workshop at the University of Glasgow aimed to bring together people working in integrable systems from various perspectives, in particular with an emphasis on the interplay with algebra and geometry.

Speakers:

Schedule

All talks will be in Room 116.

  • 13:30 -- Daniele Valeri: Newbie approach to classical R-matrices
  • 14:30 -- Anne Moreau: Arc spaces and vertex algebras
  • 15:30 -- Coffee
  • 16:00 -- Andy Hone: Continued fractions and Hankel determinants from hyperelliptic curves
  • 17:00 -- Jenya Ferapontov: Quadratic line complexes in mathematical physics
  • 18:00 -- Reception

After the reception there will be a conference differ.

Titles and abstracts

  • Jenya Ferapontov: Quadratic line complexes in mathematical physics
    Quadratic line complex is a hypersurface in the Grassmannian of lines defined by a single quadratic equation in the Plücker coordinates. Quadratic complexes have been extensively investigated in the classical works by Plücker, Kummer, Klein and many other prominent geometers of the 19-20th centuries. Large part of this theory has nowadays become textbook material. In this talk I will describe two problem of mathematical physics where quadratic complexes naturally occur:
    Problem 1. Classification of second-order linearly degenerate PDEs.
    Problem 2. Classification of third-order Hamiltonian operators of differential-geometric type.
    In both cases, the link to quadratic complexes allows to obtain partial classification results.

  • Andy Hone: Continued fractions and Hankel determinants from hyperelliptic curves
    Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the simplest case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in a particular form by Chang, Hu and Xin. We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus two satisfy a Somos-8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system.

  • Anne Moreau: Arc spaces and vertex algebras
    To any vertex algebra, one can attach in a canonical way a certain Poisson variety, called the associated variety. The geometrical properties of the associated variety and of its arc space reflect some properties of the vertex algebra. In this talk I will illustrate this phenomenon on some examples, and will explain the notion of chiral symplectic leaves which is an important tool, introduced by Tomoyuki Arakawa and myself, in this context.

  • Daniele Valeri: Newbie approach to classical R-matrices
    In this talk I will review the definition of a classical R-matrix and a construction of certain Poisson structures for finite dimensional associative algebras given by Oevel and Ragnisco. Using Poisson vertex algebras I will show how to construct an affine analogue of this construction. The latter is intimately related to classical W-algebras and integrable hierarchies of Lax type equations.

Funding:

The event is supported by a London Mathematical Society Scheme 9 "Celebrating New Appointments" grant awarded to the organiser, the Glasgow Mathematical Journal Learning and Research Support Fund, and the School of Mathematics and Statistics of the University of Glasgow.

Registration:

Attendance is free, but in order to help with the organisation it is appreciated if you could register filling this form. Limited funds are available, if you want to apply please indicate it in the form. In case you have any questions, please send an email to organisername.organisersurname@glasgow.ac.uk.

Organiser:

Daniele Valeri

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