Seminars from until

Quantum Signal Processing (QSP) is an algorithmic process by which one represents a function $f$ on the unit interval as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x ∈ [0,1]$ and some “phase factors” $\{ψ_k\}_{k ≥ 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the $SU(2)$-valued nonlinear Fourier transform, and the phase factors correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we extend QSP to represent any function $f$ satisfying a mild $\log$ integrability condition.

In this talk, we provide an introductory overview of fully nonlinear partial differential equations (PDEs), emphasizing the regularity of solutions and key analytical techniques. Fully nonlinear equations play a crucial role in diverse areas such as geometry, physics, and finance. They are distinguished from linear and semilinear equations by their intricate structure and the absence of the superposition principle. This nonlinearity introduces unique challenges in understanding the existence, uniqueness, and regularity of solutions. We explore foundational concepts, such as viscosity solutions and a priori estimates, which are central to the modern theory.

We study the initial value problem (IVP) associated with the semi-linear fractional Schrödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modeling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients.

This talk is based on a joint work with C. Kenig (Chicago), G. Ponce (Santa Barbara), and L. Vega (Bilbao).

The Hardy Uncertainty Principle states that if both a function f and its Fourier transform decay faster than the Gaussian function with a specific weight, then f=0. This result can be reformulated in terms of solutions to the free Schrödinger equation. In a series of work Escauriaza, Kenig, Ponce and Vega extended this result to the Schrödinger equation with potential and to NLS by the use of Carleman estimates. More precisely, it was proven that if u is a solution to the Schrödinger equation with potential, which at two times has Gaussian decay, and given the right conditions on the potential, then u=0.

The proof is based on Carleman estimates, which formally relies on calculus and convexity arguments. However, going from a formal level to a rigorous one is not straight forward, and if we do not justify the computations rigorously, we can prove wrong results.

We have adapted these techniques to the hyperbolic Schrödinger equation, which is physically relevant in for example Fluid mechanics. We will go through the main ideas of the proof with the Carleman estimates, and where the main changes are when going from the Laplace operator to the “hyperbolic Laplace” operator in the Schrödinger equation.

In this talk the derivation and dynamics of some 1D models for the propagation of internal waves are reviewed. From the starting point of the corresponding Euler equations and under certain physical hypotheses, Boussinesq-type systems are derived. Then a numerical analysis of the models, based on the approximation with spectral methods and efficient time integrators, is developed. This will be finally used to study, by computational means, some issues of their dynamics, mainly focused on the solitary wave solutions.

We consider a generalization of the classical perimeter, called nonlocal bi-axial discrete perimeter, where not only the external boundary of a polyomino $\mathcal{P}$ contributes to the perimeter, but all internal and external components of $\mathcal{P}$.

Formally, the nonlocal perimeter $Per_{\lambda}(\mathcal{P})$ of the polyomino $\mathcal{P}$ with parameter $\lambda>1$ is defined as:

$$ Per_{\lambda}(\mathcal{P}):=\sum_{x \in \mathbb{Z}^2 \cap \mathcal{P}, \, y \in \mathbb{Z}^2 \cap \mathcal{P}^c} \frac{1}{d^{\lambda}(x,y)} $$

where $d^{\lambda}(x,y)$ is the fractional bi-axial function defined by the relation:

$$ \frac{1}{d^{\lambda}(x,y)} := \frac{1}{|x_2-y_2|^\lambda}\textbf{1}_{\{ x_1=y_1, \, x_2 \neq y_2\}} + \frac{1}{|x_1-y_1|^{\lambda}} \textbf{1}_{\{ x_2=y_2, \, x_1 \neq y_1\}} $$

with $x=(x_1,x_2)$, $y=(y_1,y_2)$ and $\mathcal{P}^c=\mathbb{R}^2 \setminus \mathcal{P}$.

We tackle the nonlocal discrete isoperimetric problem analyzing and characterizing the minimizers within the class of polyominoes with a fixed area $n$.

The solution of this isoperimetric problem provides a foundation for rigorously investigating the metastable behavior of the long-range bi-axial Ising model.

A proof is the successful outcome of a proof search run, and such outcome is conveniently represented by a lambda-term. More generally, runs of goal-directed proof search are possibly infinite and are conveniently represented by terms of the coinductive lambda-calculus. For some logics, there is an equivalent, finitary representation, making use of formal fixed points. On such syntax one can base a new approach to the study of proof search. We will review the case study of intuitionistic implicational logic, that is, the theory of simple types. We will consider decision problems (existence of inhabitants, several concepts of finiteness), coherence questions about the uniqueness of inhabitants, and situations where a type either is empty of has infinitely many inhabitants. This is joint work with Ralph Matthes and Luís Pinto.

In this talk we explore the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincaré reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.