At the beginning of Enrico Fermi's scientific activity there is a study of the ergodic hypothesis in which he proved ergodicity for a generic dynamical system, generalising a theorem by Poincaré. Its proof was incomplete but, guided by physical intuition, he was anyway convinced of the result. In 1953-1954 he (together with J. Pasta, S. Ulam and M. Tsingou) started a series of numerical experiments to check that essentially any non-linearity would lead to a system satisfying the ergodic hypothesis.

The outcome of the experiments was against the conjectured conclusion and a lot of effort has been spent in the last sixty years to understand the phenomenon, often paving the way to new fields of Mathematics and Physics (e.g. infinite dimensional integrable systems or prethermalisation).

In this seminar I will introduce the problem, some basic notions of ergodic theory and I will try to show what the state of the art of the problem is trying to explain the most recent results.

An initial-boundary value problem for a model of stimulated Raman Scattering was considered in [Moscovchenko Kotlyarov 2010]. The authors showed that in the long-time range t → +∞ the x > 0, t > 0 quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results in [Moscovchenko Kotlyarov 2010].

The class of (mildly) nonassociative algebras known as Jordan algebras were firstly introduced during the '30s of last century in order to formalize the algebraic structure underlying a finite dimensional quantum space. In this talk I shall give some standard theorems in the theory of Jordan algebras together with their representation theory, then I will present some recent results in the theory of differential calculus and of connections on Jordan modules.

Since their introduction by Kontsevich in 1995, stable maps and their moduli have revealed to be increasingly important both in mathematics and in physics, because of their applications in various fields (enumerative geometry, symplectic topology, string theory, etc). This talk will be devoted to the study of the cohomology of moduli spaces of genus 0 stable maps into a smooth projective variety X. The case where X is a complex projective space was studied by Getzler and Pandharipande, using the combinatorial properties of these spaces. We will show that their method can be adapted to consider the more general case where X is a complex Grassmannian. This is joint work in progress with Fabio Perroni. In the spirit of GMPS seminars, we will focus more on ideas and motivations rather than technicalities.

I will outline the intersection theory in the Deligne-Mumford compactification of the moduli space of Riemann surfaces, with examples, and I will explain how the remarkable connection with the KdV equation (Kontsevich-Witten Theorem) provides a way to compute all the intersection numbers. Then I will present the connection with isomonodromic tau functions, giving as an application a derivation of explicit non recursive formulae for the intersection numbers first derived by M. Bertola, B. Dubrovin, and D. Yang. Finally, I will present ongoing generalizations.

From linear algebra we know that a generic real symmetric matrix A=(a_ij)_i,j=1ⁿ has n distinct real eigenvalues. A geometric reason for this is that the set of symmetric matrices with repeated eigenvalues has codimension two in the space of all symmetric matrices. Let's denote by f_A(x)=x^tAx=Σ_i,j=1ⁿ a_ijx_ix_j the quadratic form associated to the symmetric matrix A =(a_ij). Then critical points and critical values of f_A : S^n-1 → K(R) (the restriction of f_A to the unit sphere) are exactly unit eigenvectors and eigenvalues of A. In particular, the number of critical points of f_A | _S^n-1 equals 2n for any generic A. If, now, f=Σ_i1,...,id=1ⁿ a_{i1... id} x_{i1... id} is a homogeneous form of degree at least d ≤ 3 the number C(f) of critical point of f|_S^n-1 is not generically constant anymore. However, there is an upper bound on this number:C(f) ≤ 2 (d-1)^n-1 (d-2)^-1 where f is a generic degree d ≥ 1 form in n ≥ 2 variables. I will explain how to construct for any d and n generic forms attaining this bound. Moreover, I will show that the bound is attained by harmonic forms, known as spherical harmonics. The presentation will be as elementary as possible.

One main goal to study Calabi-Yau manifolds is to determine their enumerative invariants, Gromov-Witten, Gopakumar-Vafa, Donaldson-Thomas and so on. In physics, these invariants are encoded in the partition function of the topological string theory on such manifolds. For the so called local Calabi-Yau, the enumerative invariants can be further generalized to refined BPS invariants. The traditional technique to compute such invariants includes the refined topological vertex in A model and refined holomorphic anomaly equations in B model. Recently a new method called blowup equations was proposed, which was generalized from the Gottsche-Nakajima-Yoshioka K-theoretic blowup equations for supersymmetric gauge theories. These blowup equations also play an important role on the quantization of mirror curve of local Calabi-Yau and result in the equivalence between Nekrasov-Shatashivili quantization and the Grassi-Hatsuda-Marino conjecture. As I will explain from the very beginning, no prerequisites on Calabi-Yau are needed.

I will discuss how moduli spaces of Gieseker-semistable sheaves on certain projective varieties can be realized as quiver moduli spaces by using t-structures and exceptional collections in the derived category. This construction can be used to prove easily some of the geometric properties of the moduli space of sheaves, and to do some explicit computations. I will first give an introduction to all these notions to make the talk as self-contained as possible.

During the talk we will briefly look at what the main definitions are in (the mathematics of) Quantum Mechanics and then we will dig deeper in the Gross-Pitaevskii effective theory for Bose-Einstein Condensates. In particular we will also focus on the GP variational problem and how this relates to Superfluidity in BEC.

We review the categorical approach to the BPS sector of a 4d N=2 QFT, clarifying many tricky issues and presenting a few novel results. To a given N=2 QFT one associates several triangulated categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. The aim of this talk is to give an introduction to the above topics, focussing more on examples, ideas and applications rather than technicalities: no particular prerequisites are needed.