Seminars from until

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

Using a fluctuation dissipation method, we construct an invariant measure for the surface quasi-geostrophic equation (SQG) and 3D Euler equation. Since the support of the measure contains entire solutions, we obtain a manifold containing solutions that do not blow-up. This complements results in which a blow-up solutions for SQG and grow up solutions for Euler are constructed. The method of the proof relies on an addition of a stochastic forcing and a small dissipation to the equation. For such stochastic equation, one can construct an invariant measure and by passing the strength of the forcing and the dissipation to zero, we obtain the desired invariant measure. We also discuss the size of the support of the measure, which relies on the number of conservation laws for the particular equation.

This is a joint project with Mouhamadou Sy.

The skew RSK dynamics is a discrete 2 dimensional deterministic dynamics introduced by Imamura, Sasamoto and myself. It is an integrable system in the sense that it possesses infinitely many conservation laws and a rich set of symmetries. The analysis of its scattering relations produces a nontrivial correspondence which generalizes the celebrated Robinson-Schensted-Knuth bijection. Such correspondence provides a bijective proof of summation identities between special symmetric polynomials, which describe the law of certain interacting particle systems or directed polymer models and allow their asymptotic analysis.

The replicator equation, originally used in evolutionary game theory, has been widely applied in evolutionary biology and ecology to model the complex dynamics of multi-type interacting species, such as multi-species ecological ensembles or multi-strain microbial pathogens. In this seminar we explore aspects of the competitive dynamics encoded in the replicator equation, paying special attention to invader-driven systems, in which all species are proactive. We focus on relating properties of the fitness matrix to the quality of species dynamics and to the diversity of these systems at equilibrium, and we find a mechanism for the selection of the final set of surviving species.

In the last few years the Gamow problem, namely

for , has attracted a lot of attention from mathematicians. Nowadays it is well understood that for small there exist a minimizer and it is a ball, while for very large there is no minimizer.

Although it is very easy to formulate, there are still several open problems about it (mostly concerning nonexistence of minimizers for large in a generalized -dimensional setting).

A variation of this model, which could be called ``spectral Gamow problem'' consists in using the first eigenvalue of the Dirichlet Laplacian instead of the Perimeter, namely to consider

and in this talk we will provide some new results on this case.

Moreover, we will consider a different problem but with a similar structure, which can be seen as the minimization of a Hartree functional settled in a box, namely

for .

The study of this functional arises when describing the ground state of a superconducting charge qubit.

We show that there is a threshold such that for all existence of minimizers occurs and minimizers are nearly spherical.

We will also give some ideas (although nonconclusive) on how to treat the nonexistence issue for this functional.

The techniques and tools needed in the proofs are very broad. We employ spectral quantitative inequalities, the regularity of free boundaries, spectral surgery arguments and shape variations.

This is a joint project with Cyrill Muratov (Pisa) and Berardo Ruffini (Bologna).

Distributed machine learning addresses the problem of training a model when the dataset is scattered across spatially distributed agents. The goal is to design algorithms that allow each agent to arrive at the model trained on the whole dataset, but without agents ever disclosing their local data.

This tutorial covers the two main settings in DML, namely, Federated Learning, in which agents communicate with a common server, and Decentralized Learning, in which agents communicate only with a few neighbor agents. For each setting, we illustrate synchronous and asynchronous algorithms.

We start by discussing convex models. Although distributed algorithms can be derived from many perspectives, we show that convex models allow to generate many interesting synchronous algorithms based on the framework of contractive operators. Furthermore, by stochastically activating such operators by blocks, we obtain directly their asynchronous versions. In both kind of algorithms agents interact with their local loss functions via the convex proximity operator.

We then discuss nonconvex models. Here, agents interact with their local loss functions via the gradient. We discuss the standard mini-batch stochastic gradient (SG) and an improved version, the loopless stochastic variance-reduced gradient (L-SVRG).

We end the tutorial by briefly mentioning our recent research on the vertical federated learning setting where the dataset is scattered, not by examples, but by features.

Distributed machine learning addresses the problem of training a model when the dataset is scattered across spatially distributed agents. The goal is to design algorithms that allow each agent to arrive at the model trained on the whole dataset, but without agents ever disclosing their local data.

This tutorial covers the two main settings in DML, namely, Federated Learning, in which agents communicate with a common server, and Decentralized Learning, in which agents communicate only with a few neighbor agents. For each setting, we illustrate synchronous and asynchronous algorithms.

We start by discussing convex models. Although distributed algorithms can be derived from many perspectives, we show that convex models allow to generate many interesting synchronous algorithms based on the framework of contractive operators. Furthermore, by stochastically activating such operators by blocks, we obtain directly their asynchronous versions. In both kind of algorithms agents interact with their local loss functions via the convex proximity operator.

We then discuss nonconvex models. Here, agents interact with their local loss functions via the gradient. We discuss the standard mini-batch stochastic gradient (SG) and an improved version, the loopless stochastic variance-reduced gradient (L-SVRG).

We end the tutorial by briefly mentioning our recent research on the vertical federated learning setting where the dataset is scattered, not by examples, but by features.

The phenomenon of dispersion in a physical system occurs whenever the elementary building blocks of the system, whether they are particles or waves, overall move away from each other, because each evolves according to a distinct momentum. This physical process limits the superposition of particles or waves, and leads to remarkable mathematical properties of the densities or amplitudes, including local and global decay, Strichartz estimates, and smoothing.

In kinetic theory, the effects of dispersion in the whole space were notably well captured by the estimates developed by Castella and Perthame in 1996, which, for instance, are particularly useful in the analysis of the Boltzmann equation to construct global solutions. However, these estimates are based on the transfer of integrability of particle densities in mixed Lebesgue spaces, which fails to apply to general settings of kinetic dynamics.

Therefore, we are now interested in characterizing the kinetic dispersive effects in the whole space in cases where only natural principles of conservation of mass, momentum and energy, and decay of entropy seem to hold. Such general settings correspond to degenerate endpoint cases of the Castella–Perthame estimates where no dispersion is effectively measured. However, by introducing a suitable kinetic uncertainty principle, we will see how it is possible to extract some amount of entropic dispersion and, in essence, measure how particles tend to move away from each other, at least when they are not restricted by a spatial boundary.

A simple application of entropic dispersion will then show us how kinetic dynamics in the whole space inevitably leads, in infinite time, to an asymptotic thermodynamic equilibrium state with no particle interaction and no available heat to sustain thermodynamic processes, thereby providing a provocative interpretation of the heat death of the universe.

We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.

We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed.Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate "Brownian-Motion-Like" trajectories as well as diffusive motion (displacement proportional to the square-root of time).This model satisfies two conditions that grant, for the temporal motion of the heavy particle:

(a) An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;

(b) A best quadratic fit with an "average" non-positive curvature in a proper time interval.

We note that Planck's constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates;to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient.Finally, we discuss whether cat states are present in the thermodynamic ensembles.

(Joint for with W.D. Wick)

File available at https://hal.science/hal-04838011

We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.

This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and common uses.

Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility. FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallis entropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering advantages in model interpretability and performance.

This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and
common uses.

Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility.FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallisentropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering advantages in model interpretability and performance.

We define a nonlinear Fourier transform which maps sequences of contractive matrices to -valued functions on the circle . We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on -valued functions whose diagonal blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.

Bibliography:

• L. Lin, Lecture Notes on Quantum Algorithms for Scientific Computation, https://arxiv.org/pdf/2201.08309
• D. Oliveira e Silva, Inequalities in nonlinear Fourier analysis. CIM Bulletin 38-39 (2017), 31-35
• T. Tao and C. Thiele, Nonlinear Fourier Analysis, https://arxiv.org/abs/1201.5129

We define a nonlinear Fourier transform which maps sequences of contractive matrices to -valued functions on the circle . We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on -valued functions whose diagonal blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.