This volume includes expanded lecture notes from the Mathemata Summer School and Festum Pi 2024 conference, held in Chania, Crete, in July 2024. The lectures focus on advanced analysis techniques for tackling problems in mathematical physics, covering recent advancements in the study of phase transitions, interacting particle systems, kinetic theory, qualitative properties of degenerate Vlasov equations, optimal transport methods for parabolic diffusion equations, and Ricci flow. This volume serves as a valuable resource for researchers exploring modern developments in mathematical analysis. Additionally, it includes two lectures on music theory, reflecting the spirit of Festum Pi 2024.

Lecture notes by world leading experts in mathematical analysis Covers recent advancements in mathematical physics Accessible to beginning researchers in the field

Autorentext

Cédric Villani is a french mathematician (Université de Lyon) very much involved in public debate. His research in mathematical physics, at the crosspoint between probability, analysis and geometry, and particularly on the equations of gases and plasmas, won him the Fields medal in 2010, among other awards. His passion at writing synthesis books and lecture notes won him the Doob Prize of the AMS for his reference book « Optimal transport, old and new ». Involved in politics and public debates, he has done public lectures by the thousands, and sat in Parliament for five years. He is currently the president of the Fondation de l’écologie politique.

Nicholas D. Alikakos received his PhD from Brown University in 1978.  He subsequently served on the faculties of the University of Tennessee, Purdue University, and the University of Crete, before being appointed Professor at the University of Athens in 1993. He has also held short-term visiting positions at several universities and research centres, including Heriot-Watt University, Stanford University, the Courant Institute (New York), Université Pierre et Marie Curie (Paris), and the Max-Planck Institute (Leipzig). His principal contributions are in the theory of reaction-diffusion equations, in diffused vs. sharp interface dynamics, in singular perturbations, and in the calculus of variations.
In his spare time, he relaxes by playing the piano.

Inhalt

Foreword (Dionysios Dervis-Bournias).-- Introduction (Nicholas Alikakos & Cédric Villani).- Part I: Modelling and derivation.- 1. Graph and Mean-Field Limits for Interacting Particles Systems (Nathalie Ayi).- 2. Four Lectures on The Mathematical Theory of (multi) Phase Transitions (Nicholas Alikakos).- Part II: Analysis Tools.- 3. Introduction to Quantitative De Giorgi Methods (Giovanni Brigati & Clément Mouhot).- 4. Optimal transport methods for parabolic diffusion equations: the JKO scheme (Filippo Santambrogio).- 5. Resonances and computations (Yvan Bruned, Frédéric Rousset & Katharina Schratz).- Part III: Advances in specific programs.- 6. Splitting maps in Type I Ricci flows (Panagiotis Gianniotis).- 7. Fisher Information in Kinetic Theory (Cédric Villani).- Part IV: Music and mathematics, a gentle bridge.- 8. Microtones, 31-Tet and Just Intonation (Franck Jedrzejewski).- 9. From bird feather to digits: Some thoughts about Olivier Messiaen (Claude Ledoux).