Consider the Restricted Planar Elliptic Three Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. For eccentricity of Jupiter $e_0$ small enough, we show that there exists a family of probability measures supported at the $3 : 1$ mean motion resonance such that the push forward under the associated Hamiltonian flow has the following property. At the timescale $te_0^{-2}$, the distribution of the Jacobi constant of the Asteroid weakly converges to an Ito diffusion process on the line as $e_0 o 0$. This resonance corresponds to the biggest of the Kirkwood gap on the Asteroid belt in the Solar System. This is a joint work with V. Kaloshin, P. Martin and P. Roldan.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Maps to projective space are given by basepoint-free linear series, thus these are key to understanding the extrinsic geometry of algebraic curves. How does a linear series degenerate when the underlying curve degenerates and becomes nodal? Eisenbud and Harris gave a satisfactory answer to this question when the nodal curve is of compact type. I will report on a joint work in progress with Luca Battistella and Jonathan Wise, in which we review this question from a moduli-theoretic and logarithmic perspective. The logarithmic prospective helps understanding the rich polyhedral and combinatorial structures underlying degenerations of linear series; these are linked with the theory of matroids and Bruhat-Tits buildings.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

  Structure properties of free Lie algebras are a fundamental tool in group theory and its many applications. However, it is not always easy in practice to recognize that a Lie algebra is free. The talk will survey various results that allow to conclude to freeness, and various concrete examples.
  Based on joint work with L. Foissy.

In this talk I shall discuss some recent results about the construction of small and large amplitude quasi-periodic waves in Euler equations and other hydro-dynamical models in dimension greater or equal than two. I shall discuss quasi-peridic solutions and vanishing viscosity limit for forced Euler and Navier-Stokes equations and the problem of constructing quasi-periodic traveling waves bifurcating from Couette flow (and connections with inviscid damping). I also discuss some results concerning the construction of large amplitude quasi-periodic waves in rotating fluids. The techniques are of several kinds: Nash-Moser iterations, micro-local analysis, analysis of resonances in higher dimension, normal form constructions and spectral theory.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Let us consider a complex abelian scheme endowed with a section. On some suitable open subsets of the base it is possible to define the period map, i.e. a holomorphic map which marks a basis of the period lattice for each fiber. Since the abelian exponential map of the associated Lie algebra bundle is locally invertible, one can define a notion of abelian logarithm attached to the section. In general, the period map and the abelian logarithm cannot be globally defined on the base, in fact after analytic continuation they turn out to be multivalued functions: the obstruction to the global existence of such functions is measured by some monodromy groups. In the case when the abelian scheme has no fixed part and has maximal variation in moduli, we show that the relative monodromy group of ramified sections is non-trivial and, under some additional hypotheses, it is of full rank. As a consequence we deduce a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods. (Joint work with Paolo Dolce, Westlake University.)

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

Biexactness and the (AO)-property can be considered as analytic counterparts of hyperbolicity for discrete groups. Motivated by the problem of determining whether they are equivalent, I will discuss an approach to the study of regularity properties of boundary actions/representations based on measurable dynamics. This approach will be used to study SL(3,Z) and to answer a question posed by C. Anantharaman-Delaroche.

Some bibliography:
https://arxiv.org/abs/2111.13885
https://arxiv.org/abs/2305.16277
https://arxiv.org/abs/2410.01447

abstract

A basic problem in symplectic topology is the classification of Lagrangian submanifolds up to Hamiltonian isotopy. There is growing evidence that this is impossible to solve, but one can hope to have a coarser classification by proving that finitely many Lagrangians generate the Fukaya category. I will illustrate some concrete examples where we know how to do this, some in which we do not, and a new technique called decoupling that could partially bridge the gap.

In this talk I will present the recent paper by Fewster, Janssen, Loveridge, Waldron and myself: "Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory." In this work we show how mathematically rigorous notion of quantum reference frames allows to generalize the results of Chandrasekaran, Longo, Penington and Witten on observables in de Sitter space. The main idea is to study the joint algebra associated to the system together with the reference frame in the presence of symmetries. If both the system and the reference frame are covariant under some symmetry group, the construction of the invariant joint algebra very naturally involves crossed products.

We are concerned with a generalization to the singular case of a result of C.C. Chen e C.S. Lin [Comm. An. Geom. 1998] for Liouville-type equations with rough potentials. The singular problem is actually more delicate and results in a nontrivial variation of the regular case. Part of the arguments of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. The rest of the proof actually requires a different approach, due to the loss of translation invariance of the problem.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006