Volume 32 (2024), Issue 3 (Special issue: Portuguese Mathematics)

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

1. Non-associative algebraic structures: classification and structure

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).

2. Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Samuel A. Lopes.
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 $\mathbb{A}_{1}$.

3. Topics in elliptic problems: from semilinear equations to shape optimization

Hugo Tavares.
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.