Research Associate in Mathematics, Dr Matteo Casati, from the School of Mathematics, Statistics and Actuarial Science (SMSAS) at the University of Kent, hosted a seminar at the Local and Nonlocal Geometry of PDEs and Integrability Conference.

Matteo hosted a seminar titled, ‘A Darboux-Getzler theorem for scalar difference Hamiltonian operators’ at the Conference to honour Joseph Krasil’shchik‘s birthday and his many contributions to the fields of geometry and algebra of differential equations, differential geometry, integrable systems and mathematical physics, held in Trieste, Italy.

**Abstract:**

The classification of Hamiltonian operators in the formal calculus of variations relies on their corresponding Poisson-Lichnerowicz cohomology. We consider the case of scalar difference Hamiltonian operators, such as the ones which constitute the biHamiltonian pair for the Volterra chain, and prove that 1) the normal form of a order 1 scalar difference operator P is constant; 2) Hp(P) = 0 ∀p > 1, so that in particular there are not nontrivial infinitesimal deformations and any infinitesimal deformation can be extended to an Hamiltonian operator; 3) as a consequence, any higher order compatible Hamiltonian operator can be brought to the constant, order 1 form by a (Miura-like) change of coordinates.

##### Click here for more information about the Conference.

## Dr Matteo Casati

### Research Associate in Mathematics

Matteo joined the School in May 2018 from Loughborough University where he was carrying out postdoctoral research as a Marie Curie Fellow. His previous positions and PhD studies were undertaken in Italy. He currently collaborates with Professor Jing Ping Wang. Matteo’s research interests include geometry and mathematical physics, specifically integrable systems, Frobenius manifolds, and nonlinear waves.