Mathemateician Edgard Pimentel, originally from São Paulo, has found a new home in Rio de Janeiro, although he still holds a special place in his heart for the São Paulo soccer club team. He pursued his interest in differential equations by earning a master’s degree in applied mathematics from the University of São Paulo. He further expanded his knowledge by obtaining a PhD in Mathematics from the Instituto Superior Técnico of the University of Lisbon, Portugal. Following his doctorate, Edgard completed a post-doctoral fellowship at the same institution, as well as at the Federal University of Ceará and the National Institute of Pure and Applied Mathematics (IMPA). He currently holds a position as a researcher at the Mathematics Center of the University of Coimbra and is a Professor at PUC-Rio. Edgard believes that some of his best ideas initially turned out to be completely incorrect. However, it was through revising these ideas that he discovered the most fruitful outcomes of his studies. His experimental nature extends beyond academia. Edgard is also an amateur chef who meticulously researches and experiments with recipes for hummus and pita bread.
The analysis of partial differential equations (PDEs) is a highly adaptable field within mathematics. It is appealing from an abstract perspective due to the depth and beauty of its fundamental questions, and it has practical applications in various disciplines such as physics, biology, and economics. For instance, PDEs are used in efforts to calculate the “age of the earth,” in studies on aerodynamics, and in models of opinion formation. Regularity theory for PDEs, one of the most refined and intricate research areas related to these equations, briefly examines the structure of a given equation. From its intrinsic properties, it uncovers universal characteristics of the solutions. For example, the ellipticity of an equation ensures that the derivative of its solutions exists and is (slightly more than) continuous. Our research is particularly interested in the implications of the failure or absence of these intrinsic properties. In other words, we aim to understand what can be said about the solutions of an equation when very basic and universally accepted assumptions are removed. We explore questions such as how the graphs of the solutions behave, whether they admit cusps or have corners, and if the ‘acceleration’ of these graphs can arbitrarily increase.