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Research of R. Vitolo:

Geometric Methods in Mathematical Physics

### Integrable Systems

Hamiltonian formalism for PDEs and Integrable Systems [22a,25a,26a,13c,29a,33a,34a,37a,38a,40a], differential equations which are uniquely characterized by their symmetry group [16a,17a,9c,10c,11c,15c,30a], generalized symmetries and applications [32a,35a,36a]. I contribute to the contents of the website http://gdeq.org/, which is a community of researchers who are interested in geometry and differential equations. I developed software for the research of integrability operators (Hamiltonian, symplectic and recursion operators) [1s,2s]. Together with A.C. Norman I wrote a technical manual on REDUCE internals and programming [3s].

### Classical and quantum mechanics

Spherically symmetric solutions in Galilei relativity [2a], existence and classification of quantizations [4a,7a,3c], covariant symmetries in mechanics [2c,8a,15a,28a], classical and quantum mechanics of the rigid body [18a,19a,5c], covariant quantization and geometric quantization [6c,31a], geometry of polarizations [39a]. I recently published a paper of historical interest on an early contribution of Levi-Civita to the correspondence between symmetries and conserved quantities [27a].

### Geometry of calculus of variations

I have studied variational sequences, on which I published a chapter of
the *Handbook of Global Analysis* [2b]. These are exact sequences
of spaces of differential forms on jets where one of the morphisms of the
sequence is the Euler--Lagrange operator. In particular I devoted myself to
variational sequences of finite order
[3a,5a,6a,9a,10a,11a,12a,13a,14a,15a,20a,1c,4c,7c,12c,Tesi] and to
variational multivectors, which are duals to forms in the variational
sequence [8c,22a,13c,29a].

### Classical field theory

Covariant Lagrangians for Einstein-Yang-Mills equations [1a,1b], inverse problem for Yang--Mills equations [21a].

Last updated: 23/04/2023