(DMAT) Received June 20, 2023. The natural world is full of large systems of interacting units. A human brain, for example, contains around 86 billion (86 · 10^{9}) neurons whose activities evolve in time as a function of the activities of the neurons they are connected to. An ecosystem can also be regarded as an […]
(School of Mathematics, Birmingham). Received on 8/2/2023. In this note we will introduce the concept of character variety in the simple example of a torus with one disk removed. We will show how this is a surface in C^{3} defined by a cubic polynomial, called the Markov cubic. We will show the relation between the
(Instituut-Lorentz, Leiden). Received on 4/12/2022. Quantum mechanics allows for correlations that cannot be explained within two principles that are fundamental to our intuitive understanding of the world: locality and realism [1]. This led to Einstein and collaborators to wonder whether quantum mechanics was a complete theory for the description of reality [2]. Albeit these discussions
(DMAT, IMTech). Received on 21 November, 2022. Motivation: the conjecture of Birch and Swinnerton-Dyer In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute proposed seven challenges and one of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the sixties. A large portion of my research is devoted
(DMAT, IMTech, CRM). Received 23 July, 2022. The paper [7] (see also [8]) solves a version of the Arnold diffusion problem for Hamiltonian systems that remained open since the 1960s. Hamiltonian systems appear naturally as models of many systems with negligible friction. We could mention, for instance, the models in Celestial Mechanics or the models
