# Dr Matteo Casati

Research Associate in Mathematics

Matteo is a Research Associate in Mathematics, funded by the EPSRC Grant EP/P012698/1 “Exact Solutions for discrete and continuous nonlinear systems”, working under the supervision of Professor Jing Ping Wang.
He has previously been a Marie Curie fellow of Italian Institute of Higher Mathematics (INdAM), based at Loughborough University and at SISSA (Trieste, Italy).
He obtained his PhD in Mathematical Physics from SISSA in 2015, with the thesis Multidimensional Poisson vertex algebras and Poisson cohomology of Hamiltonian structures of hydrodynamic type. 

## Research interests

Integrable systems; in particular Poisson cohomology and theory of deformations for Hamiltonian PDEs.Geometry of PDEs

## Publications

### Article

• Casati, M. and Wang, J. (2019). A Darboux-Getzler theorem for scalar difference Hamiltonian operators. Communications in Mathematical Physics [Online]. Available at: https://dx.doi.org/10.1007/s00220-019-03497-2.
In this paper we extend the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators, to the difference case. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in the algebra of difference functions, endowing the space of local functionals with the structure of a Lie algebra. Its Poisson-Lichnerowicz cohomology carries information about the center, the symmetries and the admissible deformations of such an algebra. The analogue notion for the differential case has been widely investigated: the first and most important result is the triviality of all but the lowest cohomology for first order Hamiltonian differential operators, due to Getzler. We study the Poisson-Lichnerowicz cohomology for the operator K0 = S−S^{−1} , which is the normal form for (−1, 1) order scalar difference Hamiltonian operators; we obtain the same result as Getzler did, namely all the cohomology groups except from the 0th e 1st ones vanish. We then apply our main result to the classification of lower order scalar Hamiltonian operators recently obtained by De Sole, Kac, Valeri and Wakimoto.
• Casati, M. (2018). Higher order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type. Theoretical and Mathematical Physics [Online] 196:1129-1149. Available at: https://doi.org/10.1134/S0040577918080032.
The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with twodimensional, two-components Poisson brackets of hydrodynamic type at the third differential degree. This allows us to obtain their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of not equivalent infinitesimal deformation, nor in the third, corresponding to the obstructions to extend such deformations.
• Carlet, G., Casati, M. and Shadrin, S. (2018). Normal forms of dispersive scalar Poisson brackets with two independent variables. Letters in Mathematical Physics [Online]. Available at: https://doi.org/10.1007/s11005-018-1076-x.
We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well-known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.
• Casati, M. and Valeri, D. (2017). MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $$\mathcal {W}$$-algebras. Bollettino dell’Unione Matematica Italiana [Online] 11:503-531. Available at: https://dx.doi.org/10.1007/s40574-017-0146-9.
We give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute λ-brackets in Poisson vertex algebras, which play an important role in the theory of infinite-dimensional Hamiltonian systems. As an application, we give an introduction to the Mathematica package WAlg aimed to compute the λ-brackets among the generators of classical affine W-algebras. The use of these packages is shown by providing some explicit examples.
• Carlet, G., Casati, M. and Shadrin, S. (2017). Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets. Journal of Geometry and Physics [Online] 114:404-419. Available at: https://dx.doi.org/10.1016/j.geomphys.2016.12.008.
We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically nonvanishing in D > 1. Hence, in contrast with the D = 1 case, the deformation theory in the multivariable case is non-trivial.
• Casati, M. (2016). Dispersive deformations of the Hamiltonian structure of Euler’s equations. Theoretical and Mathematical Physics [Online] 188:1296-1304. Available at: https://doi.org/10.1134/S0040577916090026.
Euler’s equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler’s equation: we show that they are trivial up to the second order.
• Casati, M. (2015). On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type. Communications in Mathematical Physics [Online] 335:851-894. Available at: https://doi.org/10.1007/s00220-014-2219-2.
The theory of Poisson vertex algebras (PVAs) (Barakat et al. in Jpn J Math 4(2):141–252, 2009) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair of a differential algebra and a bilinear operation called the
Last updated