Dubrovin and Novikov initiated the study of local homogeneous differential-geometric Poisson brackets of arbitrary degree $k$ in their seminal 1984 paper. Despite several results in low degree, very little is known about their structure for arbitrary $k$. After an introduction to the topic we report on our recent results on the structure of DN brackets of degree $k$. We show that certain...
We consider the monodromy-free opers corresponding to solutions of the Affine Gaudin Bethe Ansatz equations.
We define and study the spectral determinants (called Q functions) for these opers. We conjecture that the Q-functions obtained from the Affine Gaudin Bethe Ansatz coincide with the Q-functions of the Bazhanov-Lukyanov-Zamolodchikov opers with the monster potential, which are related...
I will present a new approach to the computation of the large
genus asymptotics of intersection numbers, in particular of
Witten—Kontsevich, Theta and r-spin intersection numbers. Our technique
is based on a resurgent analysis of the generating series of such
intersection numbers, and relies on the presence of a quantum curve and
the determinant formulae. With this approach, we are able...
I will review recent progress in the study of the spectrum of the quantum double ramification hierarchy associated with the trivial cohomological field theory, i.e., of the quantum KdV hierarchy in its first Poisson structure. A remarkable relation to quasimodular forms naturally suggests a conjecture for the expansion to all orders of the spectrum, and we verify the conjecture explicitly at...
The qDEs define a class of ordinary differential equations in the complex domain, whose study represents a challenging and active area in both contemporary geometry and mathematical physics. The qDEs define rich invariants attached to smooth projective varieties. These equations encapsulate information not only about the enumerative geometry of varieties but also, conjecturally, about their...
The aim of this talk is to explain the connections between the discrete Painlevé I-II equations and certain random and combinatoric models. In the first part of the talk, we will start from a classical result by Borodin, which allows us to calculate the probability distributions of the first parts of random partitions with Poissonized Plancherel measure via a recurrence relation using...
In statistical physics systems of interacting particles can be used as toy models to study non equilibrium phenomena. A way to do that is to place the system in contact with an external environment that generates a current through it, place the system out of equilibrium and so reversibility is lost. In these boundary driven models the first natural question one would like to answer regards the...
The dust equation, also known as homogeneous or pressureless Euler equation (Zel'dovich, 1970), is a toy-model for the fluid dynamic Euler equation in which pressure is neglected. This PDE is the multidimensional analogue of the Burgers-Hopf equation and is a good training ground for the analysis of the catastrophe structures in more than one spatial dimension. In the talk, the gradient...
We study the problem of propagation of an input electromagnetic pulse through a long two-level laser amplifier under trivial initial conditions. we consider an unstable model described by the Maxwell–Bloch equations without spectral broadening. We obtain rigorous asymptotic results at large times, in two regions: in a region near the light cone and in the tail region. The region near the light...
I will talk about our recent joint work with Xavier Blot, where we related the quantum intersection numbers on the moduli space of curves to the stationary relative Gromov-Witten invariants of the projective line with an insertion of a Hodge class.
As a corollary, this gives a new geometric interpretation of the standard intersection numbers of psi-classes on the moduli space of curves. We...
In this talk, I will explain how to construct quasimodular forms from Betti numbers of moduli spaces of one-dimensional coherent sheaves on the projective plane. This gives a proof of some predictions from theoretical physics about the refined topological string theory in the Nekrasov-Shatashvili limit. The proof of this result combines various tools of modern enumerative algebraic geometry,...
We define a certain extension of the Ablowitz-Ladik hierarchy, and show that it possesses a tau structure and Virasoro symmetries. We prove that this integrable hierarchy coincides with the topological deformation of the Principal Hierarchy of a generalized Frobenius manifold with non-flat unity.
I will present a conjecture about large deviations of the partition function of the log-Gamma polymer. We can rigorously prove our result, except for one step for which we only have heuristic evidence. I will show that the conjectured large deviation rate function matches with that of last passage percolation with exponential weights in the zero-temperature limit, and with the lower tail of...
We construct a new class of (gas) solutions of the mKdV equation as a limit of $2N$ solitons, eventually in the presence of an additional dispersive coefficient. We also analyze the long-time behaviour of the profile: the solution is asymptotically an elliptic solution at both $x \to \pm \infty$, with same parameters, but different shift $x^{\pm}_0$.
This is a joint work with Ken McLaughlin...
I will present a series of examples of soliton gasses, in which the interplay of Riemann-Hilbert analysis and randomness provides a detailed description of interesting phenomena. Time will be reserved to discuss future research challenges. Our research group includes Manuela Girotti (Emory), Aikaterini Gkogkou (Tulane), Tamara Grava (SISSA), Robert Jenkins (University of Central Florida),...
Despite needing a complete mathematical foundation, the theory of Generalized Hydrodynamics has been used to obtain precise approximations of the correlation functions for several integrable models. For example, H. Spohn used this theory to compute the correlation function of the Toda lattice.
We consider another integrable model, namely the Volterra lattice. We introduce the Generalized...
Hydrodynamics has motivated many of the advances that lie at the foundations of the mathematics of completely integrable models; the interplay between a fluid and its boundaries adds to the richness, both from a mathematical and a physical perspective, of phenomena that can arise in this area. This talk will present examples of how boundary effects lead to notable outcomes. These effects can...
Classical affine W-algebras W(g,O) are algebraic structures associated to a simple Lie algebra g and a nilpotent orbit O. In this talk we want to review their definition, the construction (for all but seven nilpotent orbits for exceptional Lie algebras) of an integrable hierarchy of PDEs known as generalized Drinfeld-Sokolov hierarchy and how to derive the tau-structure of the hierarchy.
We study some relations between trees decorated by the double ramification cycles times \lambda_g and trees decorated by the Omega class, via virtual localisation. We then employ virtual localisation to deduce some polynomiality properties of the Omega class. If time permits, some connections with the DR/DZ conjecture will be discussed.
We introduce some geometric structures related to the notion of bi-flat F-manifolds and we show their applications to the theory of Integrable Systems.
Based on joint works with Alessandro Arsie, Franco Magri and Sara Perletti.
We define a certain extension of the Ablowitz-Ladik hierarchy, and show that it possesses a tau structure and Virasoro symmetries. We prove that this integrable hierarchy coincides with the topological deformation of the Principal Hierarchy of a generalized Frobenius manifold with non-flat unity.
