Publications
Preprints
My preprints can also be found on Google Scholar and on arXiv. They have been all submitted for publication.

Normal forms of dispersive scalar Poisson brackets with two independent variables, with M. Casati and S. Shadrin (2017).
 Abstract: 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.

Central invariants revisited, with R. Kramer and S. Shadrin (2016).
 Abstract: We give a new proof of the statement of DubrovinLiuZhang that the Miuraequivalence classes of the deformations of semisimple biHamiltonian structures of hydrodynamic type are parametrized by the socalled central invariants.
Journal papers
The titles link to the published version of the pdf file.
 Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed, with H. Posthuma and S. Shadrin (2017), accepted for publication in Journal of Differential Geometry.
 Abstract: We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.
 Rational reductions of the 2DToda hierarchy and mirror symmetry, with A. Brini, S. Romano and P. Rossi. J. Eur. Math. Soc. 19 (2016) 835880.
 Abstract: We introduce and study a twoparameter family of symmetry reductions of the twodimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the AblowitzLadik hierarchy and E. Frenkel's qdeformed GelfandDickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the TakasakiTakebe classical limit gives rise to a family of nonconformal Frobenius manifolds with flat identity. We use this to generalize the relation of the AblowitzLadik hierarchy to GromovWitten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dualtype Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric CalabiYau threefolds obtained from minimal resolutions of the local orbifold line.
 Poisson cohomology of scalar multidimensional DubrovinNovikov brackets, with M. Casati and S. Shadrin. J. Geom. Phys. 114 (2017) 404419.
 Abstract: We compute the Poisson cohomology of a scalar Poisson bracket of DubrovinNovikov 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 nontrivial.
 Remark: A Mathematica package to explicitly compute the dimension of Poisson cohomology groups is available.
 The biHamiltonian cohomology of a scalar Poisson pencil, with H. Posthuma and S. Shadrin. Bull. London Math. Soc. 48 (2016) 617627.
 Abstract: We compute the biHamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p\sub{d}(\hat{F}, d\sub{1},d\sub{2})$ is isomorphic to $\R$ for (p,d)=(0,0), to $C^\infty (\R)$ for (p,d)=(1,1), (2,1), (2,3), (3,3), and vanishes otherwise.
 Bihamiltonian cohomology of KdV brackets, with H. Posthuma and S. Shadrin. Comm. Math. Phys. 341 (2016) 805819.
 Abstract: Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.
 Principal hierarchies of infinitedimensional Frobenius manifolds: The extended 2D Toda lattice, with L. Ph. Mertens. Adv. Math. 278 (2015) 137181.
 Abstract: We define a dispersionless tausymmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\infty$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinitedimensional Frobenius manifold $M\sub{0}$ introduced by Carlet, Dubrovin and Mertens in Math. Ann. 349 (2011) 75115 and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of $M\sub{0}$.
 Towards Lax formulation of integrable hierarchies of topological type, with J. van de Leur, H. Posthuma and S. Shadrin. Comm. Math. Phys. 326 (2014) 815849.
 Abstract: To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudodifferential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [BurPosSha1, BurPosSha2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.
 Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of $\mathbb{C}P^1$ orbifolds, with J. van de Leur. J. Phys. A: Math. Theor. 46 (2013) 405205405220.
 Abstract: We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this confirms a conjecture that relates the total descendent potential of the orbifold $C\sub{k,m}$ with a tau function of the extended bigraded Toda hierarchy.
 Integrable hierarchies and the mirror model of local $\mathbb{C}P^1$, with A. Brini and P. Rossi. Physica D 241 (2012) 21562167.
 Abstract: We study structural aspects of the AblowitzLadik (AL) hierarchy in the light of its realization as a twocomponent reduction of the twodimensional Toda hierarchy, and establish new results on its connection to the GromovWitten theory of local $\mathbb{C}P^1$. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with nonconstant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local biHamiltonian structure of the AblowitzLadik system in the zerodispersion limit. Secondarily, we make precise the relation between this canonical Frobenius structure and the one that underlies the GromovWitten theory of the resolved conifold in the equivariantly CalabiYau case; a key role is played by Dubrovin's notion of "almost duality" of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local $\mathbb{C}P^1$ in terms of a dual logarithmic LandauGinzburg model.
 Infinitedimensional Frobenius manifolds for 2+1 integrable systems, with B. Dubrovin and L. Ph. Mertens. Math. Ann. 349 (2011) 75 115.
 Abstract: We introduce a structure of an infinitedimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0/\infty$ respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.
 On the Lax representation of the 2component KP and 2D Toda hierarchies, with M. Mañas. J. Phys. A: Math. Theor. 43 (2010) 434011.
 Abstract: We identify a set of free dependent variables of the 2component KP and 2D Toda hierarchies by explicitly solving the constraints appearing in their Lax formulation.
 Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures, with P. Lorenzoni and A. Raimondo. J. Phys. A: Math. Theor. 43 (2010) 045201.
 Abstract: We study the finitedimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.
 The Extended Bigraded Toda Hierarchy. J. Phys. A: Math. Gen. 39 (2006) 94119435.
 Abstract: We generalize the Toda lattice hierarchy by considering $N+M$ dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [CDZ04]. Using Rmatrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups $\tilde{W}^{(N)}(A\sub{N+M1})$ of the $A$ series, defined in [DZ98].
 The Hamiltonian structures of the twodimensional Toda lattice and Rmatrices. Lett. Math. Phys. 71 (2005) 209226.
 Abstract: We construct the triHamiltonian structure of the twodimensional Toda hierarchy using the Rmatrix theory.
 The Extended Toda Hierarchy, with B. Dubrovin and Y. Zhang. Moscow Math. J. 4 (2004) 313332.
 Abstract: Using construction of logarithm of a difference operator, we present the Lax pair formalism for certain extension of the continuous version of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the $\mathbb{C}P^1$ topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy.
 Extended Toda Lattice. Theor. Math. Phys. 137 (2003) 13901395.
 Abstract: We introduce nonlocal flows that commute with those of the classical Toda hierarchy. We define a logarithm of the difference Lax operator and use it to obtain a Lax representation of the new flows.
PhD and Master thesis
 Extended Toda hierarchy and its Hamiltonian structure, PhD thesis, supervisor B. Dubrovin, International School for Advanced Studies (2003).
 M5brana: confronto tra l'approccio PST e quello doppiamente supersimmetrico, Master thesis, supervisor M. Tonin, University of Padova (1998).