@mastersthesis {2015,
title = {Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type},
year = {2015},
note = {161 pages},
school = {SISSA},
abstract = {The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical
Poisson brackets, defined on an infinite dimensional space of maps Σ {\textrightarrow} M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson{\textendash}Lichnerowicz cohomology. From this computations it
follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.},
keywords = {Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems},
author = {Matteo Casati}
}