Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

Abstract We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them

Weighted inequalities in Fluid mechanics and general relativity: Carleman estimates and cusped travelling waves

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-07-2019Esta tesis tiene embargado el acceso al texto completo hasta el 16-01-202

Low Eigenvalues of Laplace and Schrödinger Operators

This workshop brought together researchers interested in eigenvalue problems for Laplace and Schr¨dinger operators. The main topics o of discussions and investigations covered Dirichlet and Neumann eigenvalue problems, inequalities for the spectral gap, isoperimertic problems and sharp Lieb–Thirring type inequalities. The focus included not only the analytic and geometric sides of the problems, but also related probabilistic and computational aspects

Some nonlocal operators and effects due to nonlocality

In this PhD thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and to some other types of fractional derivatives (the Caputo and the Marchaud derivatives). We make an extensive introduction to the fractional Laplacian, we present some related contemporary research results and we add some original material. Indeed, we study the potential theory of this operator, introduce a new proof of Schauder estimates using the potential theory approach, we study a fractional elliptic problem in $\mathbb{R}^n$ with convex nonlinearities and critical growth and we present a stickiness property of nonlocal minimal surfaces for small values of the fractional parameter. Also, we point out that the (nonlocal) character of the fractional Laplacian gives rise to some surprising nonlocal effects. We prove that other fractional operators have a similar behavior: in particular, Caputo-stationary functions are dense in the space of smooth functions, moreover, we introduce an extension operator for Marchaud-stationary functions.Comment: 255 pages, 35 figures. PhD thesis Univ Milan (2017

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq 2$ and general even exponents.Comment: 64 pages, 4 figures, 3 tables; v2: referee's suggestions incorporate

Quantum Field Theories, Isomonodromic Deformations and Matrix Models

Recent years have seen a proliferation of exact results in quantum field theories, owing mostly to supersymmetric localisation. Coupled with decades of study of dualities, this ensured the development of many novel nontrivial correspondences linking seemingly disparate parts of the mathematical landscape. Among these, the link between supersymmetric gauge theories with 8 supercharges and Painlev{\'e} equations, interpreted as the exact RG flow of their codimension 2 defects and passing through a correspondence with two-dimensional conformal field theory, was highly surprising. Similarly surprising was the realisation that three-dimensional matrix models coming from M-theory compute these solutions, and provide a non-perturbative completion of the topological string. Extending these two results is the focus of my work. After giving a review of the basics, hopefully useful to researchers in the field also for uses besides understanding the thesis, two parts based on published and unpublished results follow. The first is focused on giving Painlev{\'e}-type equations for general groups and linear quivers, and the second on matrix models