# Sala Conferenze (Puteano, Centro De Giorgi)

## A randomized version of the Littlewood Conjecture - Part I

The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, but an even stronger statement with an extra factor of a logarithm also holds.

## A randomized version of the Littlewood Conjecture - Part II: discussion

The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, but an even stronger statement with an extra factor of a logarithm also holds.

## Lagrange Spectrum of Veech surfaces

The Lagrange spectrum is a classical object in Diophantine approximation on the real line that has been generalised to many

different settings. In particular, recently it has been generalised to a similar object for translation surfaces, which attracted quite some

attention in the field. We study the Lagrange spectrum in the contest of Veech translation surfaces. These are particular translation surfaces

with many symmetries, that can be thought as a dynamical equivalent of the torus in higher genera.

## When panic makes you blind: a chaotic route to systemic risk

We introduce a slow-fast dynamical system to study the role of the expectation feedbacks on systemic stability of financial markets. We study a simple stylization of a financial market, that is a set of financial institutions having capital requirements in the form of the Value at Risk constraint, following standard mark-to-market and risk management rules and investing in some risky assets.

## Linearization of multi-dimensional dynamical systems through tree-expansions

joint work with F.Fauvet and F.Menous

## Quasiperiodic sums and products

Quasiperiodic Sums and Products arise in many areas of mathematics including the study of Strange Non-Chaotic Attractors, Critical KAM Theory, Quantum Chaos, q-series, Partition Theory, and Diophantine Approximation.

The graphs of these functions form intriguing geometrically strange and self-similar structures. They are easy and rewarding to investigate numerically, and suggest many avenues for investigation. However they prove resistant to rigorous analysis.

## Integrability of bi-homogeneous potentials

We present a new notion of homogeneity for rational potentials in the plane, we called rotation homogeneity. These potentials can be seen as singular limits of rational potentials, and in particular, a necessary condition for integrability is that the dominant term and lower order term should be integrable. This new notion of homogeneity can combine with the classical one, producing a bihomogeneous case, which is a 2-integer parameter family.