Schedule, Titles and Abstracts 

• Lecturer: Riccardo Adami (Politecnico di Torino) 

Title: Mathematics for Atomtronics: standing waves for the Nonlinear Schrödinger equation on graphs and hybrids

Abstract: Whenever in a physical experiment a ramified structure appears, for instance in propagation of signals or in a circuit of quantum wires, it proves useful to approximate such a structure with a network, that can be composed of several one-dimensional parts connected at some points, er even of parts of different dimensions. It becomes then important to define and study a suitable dynamics on such structures. Under this respect, we illustrate a research line concerning the existence of Ground States for the focusing nonlinear Schroedinger equation on exotic domains, specifically on quantum graphs and hybrids. In such a context, a Ground State is defined as a minimizer of the energy at fixed mass. For quantum graphs, we discuss the influence of the topology and the metric on the formation of ground states. For hybrids, we give some recent results and conjectures on simple models. The nonlinearity we discuss is subcritical and critical for the graphs, subcritical only for the hybrids. As a particular case, we analyze the example of periodic grids, that interpolate from one and two-dimensional behaviour. The problem of the Ground States for the NLS has several applications in physics, in particular on modeling exotic traps for Bose-Einstein Condensates. This is a joint project with F. Boni (Napoli Federico II), R. Carlone (Napoli Federico II),  S. Dovetta (PoliTo), E. Serra (PoliTo), L. Tentarelli (PoliTo), P. Tilli (PoliTo).

• Lecturer: Nilanjana Datta (University of Cambridge) 

Title: Entropies: mathematical properties and operational interpretations in Information Theory 

Abstract: Entropies play a fundamental role in both classical and quantum information theory. Expressions for optimal rates of various information theoretic tasks are given in terms of entropic quantities. Entropies also satisfy a host of interesting mathematical properties. In this lecture series, different families of entropies that are relevant in information theory will be introduced and their properties and operational interpretations will be elucidated. The proof of Shannon's Source Coding theorem, which establishes the Shannon entropy as the optimal rate of compression of information emitted by an information source, will be explained. The lecture series will end with a detailed discussion of a novel, universal proof of uniform continuity bounds of various classical and quantum entropies, employing the elegant notion of majorization flow.

• Lecturer: Pierpaolo Vivo (King's College, London) 

Title: Optimization problems on the sphere: from random matrices to replicas and back

Abstract: I will consider the problem of computing the best approximation for the solution of a large random system of M linear equations in N unknowns, subject to the constraint that the solution must live on the N-spere, ||x||^2 = N. The coefficients and the known parameters of the system are taken as random variables. This framework is closely related to the so-called “oblique Procrustes problem”, and was considered in a series of papers by Y. Fyodorov and R. Tublin. The problem will be analysed from two different viewpoints: first, I will consider the statistics of the Lagrange multipliers used to enforce the spherical constraint, which in turn leads to considering the “system’s satisfiability” (the minimal attainable discrepancy between the left and the right hand side of the linear system) and the associated solutions’ landscape. Then, I will recover the main result — namely, the existence of a “satisfiability threshold” in the parameter \alpha = M/N — using the heuristic replica method from the field of disordered systems in physics.