# Geometry and Mathematical Physics Seminar 2012/2013

**Extraordinary meeting
**

Thursday 18 July 2013. *The Frank Adams Room (Room 1.212), the Alan Turing Building. 3pm *
##
Factorization of Darboux Transformations

Ekaterina Shemyakova (State University of New York at New Paltz)

`shemyake@newpaltz.edu`

The aim of the talk is to give an introduction to the theory of
Darboux-Laplace transformations
(first part) as well as to present a proof (second part) of a long term
conjecture hinted by Darboux at the end of 19th
century.

Darboux transformations are the major tool in modern theory of
integrable systems (soliton theory).
Invented by Darboux in the context of classical differential geometry, these
transformations have been rediscovered in 1970s.
Many different versions have been since introduced. The general theory
has not been settled yet.

In the second part of the talk we prove that for a hyperbolic
partial differential operator or order $2$ on the plane every Darboux
transformation of arbitrary order $d$ is a product
of elementary Darboux transformations of order $1$.

The analogous statement for $1$-dimensional Schrödinger operator was
proved earlier in four steps
(Shabat, Veselov and Bagrov, Samsonov). In this case the factorization
is not
unique, and different factorizations imply discrete symmetries related
to the Yang-Baxter maps (Adler and Veselov).