Analytical Methods in Quantum and Continuum Mechanics

Winter School In Turin November 29 – December 3, 2021.

Programme

Courses

Prof. Marco Cicalese (Technische Universität München)

Title: Topological singularities in classical spin systems: a variational perspective.

Abstract: Typical low–energy states of lattice spin systems show the emergence of com- plex structures at large scales. The analysis of such structures, their fine geometry and energetic behaviour is the main goal of this course. Due to the multiscale nature of spin systems, their rigorous mathematical study poses a number of very challenging problems. Their analysis often requires the combination of different analytical, stochastic, geometric, and combinatorial techniques. This course aims at introducing a possible variational approach to the analysis of some energy driven lattice systems, mainly focusing on those energy scalings which lead to the emergence of topological singularities. After a quick review of the necessary preliminary results from Geometric Measure Theory and the Direct Methods in the Calculus of Variations, I will discuss the large–scale behaviour of the ferromagnetic and anti–ferromagnetic xy systems on (possibly geometrically frustrated) lattices.

Prof. Andrea Malchiodi (SNS Pisa)

Title: Variational theory of Liouville equations.

Abstract: Liouville equations play a fundamental role in Geometry when prescribing the Gaussian curvature of a (possibly singular) surface, as well as in models from Mathemat- ical Physics that describe stationary Euler flows, Electro-weak, Chern-Simons models of superconductivity and String Theory. We will attack the existence issue by exploiting the variational structure of the problem. In general, global minimizers may not exist, and we will focus on the existence of saddle-type solutions constructed via min-max theory. A crucial tool in doing this will be to find suitable improvements of the Moser–Trudinger inequality by a fine analysis of the distribution of volume over the surface. Some appli- cations to Functional Determinants in spectral theory will be also described.

Prof. Nicola Visciglia (Università di Pisa)

Title: The nonlinear Schrödinger equation with spatial white noise.

Abstract: We discuss the existence and uniqueness of global solutions almost surely for NLS posed on T2 and perturbed by a spatial white noise. The main idea is to use a gauge transform, first used by M. Hairer in the parabolic setting, in order to work in a more regular framework and then to renormalize the corresponding stochastic objects. Once this probabilistic procedure is concluded we shall show how to get deterministic uniform a priori bounds on the corresponding solutions and how to pass to the limit. In particular we shall consider the cubic and fourth order nonlinearity. The cubic case was first achieved by A. Debbusche and H. Weber, the quartic case is treated in a joint work of the speaker with N. Tzvetkov.