Communications in Mathematics |

Editors: Ana Cristina Moreira Freitas, Carlos Florentino, Diogo Oliveira e Silva & Ivan Kaygorodov

These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agregaç\~ao em Matemática e Applicações (University of Beira Interior, Covilh\~a, Portugal, 13-14/03/2023).

This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra $\mathbb{A}_{1}$.

These are detailed notes for a lecture on "Sharp restriction theory" which I presented as part of my "Agregação em Matemática" in Instituto Superior Técnico, Lisboa, Portugal (9-10 February, 2023).

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.

We show that a measurable function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, with $d\geq 3$, satisfies the functional relation \begin{equation*} g(\omega)+g(\omega_*)=g(\omega')+g(\omega_*'), \end{equation*} for all admissible $\omega,\omega_*,\omega',\omega_*'\in\mathbb{S}^{d-1}$ in the sense that \begin{equation*} \omega+\omega_*=\omega'+\omega_*', \end{equation*} if and only if it can be written as \begin{equation*} g(\omega)=A+B\cdot\omega, \end{equation*} for some constants $A\in \mathbb{R}$ and $B\in\mathbb{R}^d$. Such functions form a family of quantized collision invariants which play a fundamental role in the study of hydrodynamic regimes of the Boltzmann--Fermi--Dirac equation near Fermionic condensates, i.e., at low temperatures. In particular, they characterize the elastic collisional dynamics of Fermions near a statistical equilibrium where quantum effects are predominant.

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to satisfy upper bounds given by products of two functions, one in each variable. The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the $\mathscr{H}_\alpha$ (or Struve) transform in the case $\alpha>\frac{1}{2}$, or the characterization of power weights for which the Laplace transform is bounded in the limiting cases $p=1$ or $q=\infty$.

These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and their underling foliations, singularity theory and dynamical systems. Compared to the original text, a section containing some background material on holomorphic foliations was added. Also some new results obtained in the past three years that are in line with the one presented in the habilitation were included.

The Portuguese School of Extremes and Applications is nowadays well recognised by the international scientific community, and in my opinion, the organisation of a NATO Advanced Study Institute on Statistical Extremes and Applications, which took place at Vimeiro in the summer of 1983, was a landmark for the international recognition of the group. The dynamic of publication has been very high and the topics under investigation in the area of Extremes have been quite diverse. In this article, attention will be paid essentially to some of the scientific achievements of the author in this field.

These are lecture notes from a course given at the summer school "Heat kernels and spectral geometry: from manifolds to graphs" in Bregenz, Austria, 2022. They are designed to be accessible to doctoral level students, and include background chapters on Laplacians on domains and quantum graphs before moving on to specialised topics involving the dependence and optimisation of operator eigenvalues on a metric graph in function of the graph geometry, drawn in part from the recent literature.