Prossimi seminari del Dipartimento di Matematica

In this talk, I will give a brief introduction to Carleman estimates and show their application to some problems in PDEs, such as, for instance, the uniqueness of solutions and the local solvability.

Mean-field theory approximates many-body quantum dynamics by an effective single-particle evolution governed by nonlinear equations. As an introduction, I will present a quantum algorithm for solving nonlinear differential equations that relies on this mean-field approximation. Extending mean-field theory to non-Hermitian Hamiltonians could allow us to address more general classes of equations. The main part of the talk will focus on this non-Hermitian generalization, illustrating the associated challenges and potential approaches through concrete examples, and outlining the current progress of our ongoing work.

One of the most basic and important questions in PDE is that of regularity. It is also a unifying problem in the field, since it affects all kinds of PDEs. A classical example is Hilbert’s XIXth problem (1900), which roughly speaking asked to determine whether all solutions to uniformly elliptic variational PDEs are smooth. Starting from De Giorgi’s groundbreaking approach to this problem (1957), the first part of this talk will review the core ideas of elliptic regularity theory, emphasizing the main differences between the linear and nonlinear settings. We will then turn to the more recent theory of elliptic PDEs with p, q-growth — that is, elliptic equations whose ellipticity and growth are governed by different powers of the gradient. In this setting, a central feature is that regularity does not always hold: as shown by counterexamples due to Marcellini (1987) and Giaquinta (1987), certain variational integrals admit unbounded minimizers as soon as p and q are to far apart. Ensuring regularity for all solutions therefore requires an appropriate balance between p and q. Finally, we will discuss some current developments in which the growth of the stress field is prescribed by distinct Young functions, leading to an Orlicz-type framework that captures a broad range of nonstandard behaviors and provides a natural setting for genuinely non-homogeneous problems. - Seminario all'interno del ciclo di seminari ASK -

In this work, we initiate the study of Hamiltonian learning for positive temperature bosonic Gaussian states, the quantum generalization of the widely studied problem of learning Gaussian graphical models. We obtain efficient protocols, both in sample and computational complexity, for the task of inferring the parameters of their underlying quadratic Hamiltonian under the assumption of bounded temperature, squeezing, displacement and maximal degree of the interaction graph. Our protocol only requires heterodyne measurements, which are often experimentally feasible, and has a sample complexity that scales logarithmically with the number of modes. Furthermore, we show that it is possible to learn the underlying interaction graph in a similar setting and sample complexity. In addition, we use our techniques to obtain the first results on learning Gaussian states in trace distance with a quadratic scaling in precision and polynomial in the number of modes, albeit imposing certain restrictions on the Gaussian states. Our main technical innovations are several continuity bounds for the covariance and Hamiltonian matrix of a Gaussian state, which are of independent interest, combined with what we call the local inversion technique. In essence, the local inversion technique allows us to reliably infer the Hamiltonian of a Gaussian state by only estimating in parallel submatrices of the covariance matrix whose size scales with the desired precision, but not the number of modes. This way, we bypass the need to obtain precise global estimates of the covariance matrix, controlling the sample complexity. Based on https://arxiv.org/abs/2411.03163 with Cambyse Rouzé and Daniel Stilck França

This talk focuses on a family of Hardy-Sobolev doubly critical p-Laplace systems defined on the whole Euclidean space. A key tool in our analysis is the moving plane method, which will serve as our starting point. We will outline the origins of this method, explain its main features, and discuss the current state of the art in the context of more general problems, culminating in the system under consideration. Particular attention will be paid to how the nonlinear nature of the operator, the presence of Hardy’s potential, and the coupled structure of the system make the application of the method highly nontrivial. - Seminario all'interno del ciclo di seminari ASK-

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