This present book collects a distinguished selection of contributions by scholars who participated as speakers or as visiting scientists in the intensive programme Puglia Summer Trimester 2023 took place in Bari, Italy, from April to July 2023, and also includes contributions by further scholars who are expert in related fields. The programme was structured around a series of main meetings, including a general conference and a summer school, supplemented by the local presence and activities of an amount of visiting scientists. Additionally, efforts were made to disseminate and popularise mathematics among schools and the general public, with the aim of extending the programme's impact beyond the immediate academic sphere. Each chapter, in the form of retrospective reviews, overviews on recent developments, announcements and comments of new results, as well as outlooks on future perspectives, represents some of the main scientific instances of the trimester in Bari. The trimester was actually focussed on a spectrum of mathematical problems, directly stemming or inspired from a variety of physical domains, involving singular modelling, asymptotic and emergent phenomena, singular interactions, non-trivial limit effects. Natural backgrounds are quantum physics, cold atom physics, soft matter physics, with methods and tools, suitably adapted to such singular settings, spanning across operator and spectral theory, functional analysis, probability, differential geometry, partial differential equations, and numerical analysis.
Globally integrable quantum systems and their perturbations.- On
two-dimensional Dirac operators with $\delta$-shell interactions supported on
unbounded curves with straight ends.- Attractor Subspace and Decoherence-Free
Algebra of Quantum Dynamics.- Algebraic localization of generalized Wannier
bases implies Roe triviality in any dimension.- Hearing the boundary
conditions of the one-dimensional Dirac operator Bosonized Momentum
Distribution of a Fermi Gas via Friedrichs Diagrams.- Self-adjointness and
Domain of Generalized SpinBoson Models with Mild Ultraviolet Divergences.-
Random Linear Systems with Quadratic Constraints: from Random Matrix Theory
to replicas and back.- New analytical and geometrical aspects on
Trudinger-Moser type inequality in 2D.- Resolvent limits of exterior boundary
value problems and singular perturbation of Laplace operator in 3D.- The
Search for NLS Ground States on a hybrid domain: motivations, methods, and
results.- From microscopic to macroscopic: the large number dynamics of
agents and cells, possibly interacting with a chemical background.- Open
problems and perspectives on solving Friedrichs systems by Krylov
approximation.- Singularity: a Seventh Memo.
Biagio Cassano is an associate professor at the University of Campania "L. Vanvitelli". His activity is set in the field of mathematical analysis and focuses on harmonic analysis and the study of partial differential equations. In particular, his research explores the properties of the equations of (non-relativistic and relativistic) quantum mechanics, possibly in presence of singular terms or in critical conditions.
Fabio Deelan Cunden is an assistant professor at the University of Bari, Italy. He works in mathematical physics and probability, in particular random matrices and their applications.
Matteo Gallone is a researcher in mathematical physics at the Mathematics Department of SISSA (Trieste). His research interests span theoretical physics, mathematical physics, and functional analysis. Alongside his academic work, he is passionate about science communication and actively engages in outreach to make complex scientific concepts accessible to a general audience. His work has explored operator-theoretic problems and nonlinear partial differential equations arising from quantum mechanical models. He has also investigated topics such as thermalisation and pre-thermalisation in classical and quantum systems, with a particular focus on their connections to hydrodynamics and turbulence. Recently, his research has extended to critical phenomena in rigorous statistical mechanics, applying techniques from the constructive renormalisation group to study these issues.
Marilena Ligabò is an associate professor at the University of Bari. Her research activity mainly concerns the mathematical aspects of quantum mechanics such as spectral analysis, the approximation of quantum dynamics and the semiclassical limit. Additional research interests include tomography, random matrix theory and the study of PDEs for elasticity and thermo-elasticity models.
Alessandro Michelangeli is an Alexander von Humboldt Senior Researcher at the Institute for Applied Mathematics and at the Hausdorff Center for Mathematics, Bonn, an associated member of the Trieste Institute for Theoretical Quantum Technologies, Trieste, and professor of mathematics and natural sciences at the Prince Mohammad Bin Fahd University, Al-Khobar, and David Flanagan professor of mathematics at the American University Bulgaria, Blagoevgrad. He previously held positions at the LMU Munich and the SISSA Trieste, as well as visiting positions at the CIRM Trento, the DPMMS Cambridge, and the departments of mathematics and of physics at Bilkent University, Ankara. His research is at the interface of analysis, mathematical physics, and theoretical physics. He is primarily involved in functional analysis, operator theory, and nonlinear partial differential equations.
