The PhD lectures take place on Wednesday morning, usually at 11:30 Italian time.
November 6th, 2024
Benjamin Alvarez (Université de Toulon)
An introduction to non relativistic QED
This talk is dedicated to presenting the fundamental notions to adress non relativistic QED models. The general framework of quantum field theory will be presented together with the Hamiltonian of non relativistic QED. In addition, we aim to give a taste of important works that have been done in the field together with some open problems.
December 18th, 2024
Angelo Lucia (Universidad Complutense de Madrid)
Locality and spectral gaps in quantum spin systems
In this lecture, I will give a brief introduction to the mathematical framework used to describe quantum spin systems on lattices. The main goal will be to connect the study of models defined over finite volumes to an idealized infinite volume limit called the thermodynamic limit. A key role will be played by locality estimates known as Lieb-Robinson bounds. I will then focus on the spectral gap of the model, the difference between the two lowest energy levels, and I will explain how it is related to the problem of phase classification of quantum phases of matter.
January 15th, 2025
Benjamin Hinrichs (Universität Padeborn)
An Invitation to Path Measure Methods for Polaron Models
This talk gives an introduction to the connection between the Hamiltonian formulation of quantum mechanics and its probabilistic counterpart. We first discuss this in terms of Feynman-Kac formulas for Schrödinger operators together with some applications. Then, we move to their analogues for polaron models, i.e., models of quantum particles linearly coupled to a quantum field. We also sketch how the named applications extend to the polaron, e.g., when studying ground states or the effective mass.
March 12th, 2025
Andrew Rout (Univerisity of Rennes)
A (brief) introduction to Gibbs measures for the nonlinear Schrödinger equation
In this talk I will give an introduction to Gibbs measures for the nonlinear Schrödinger equation. The construction of global solutions to dispersive PDEs usually relies on the conservation of quantities like the energy and the mass. For lower regularity functions, these quantities are infinite, so cannot be used. Instead one introduces an invariant measure, which can act as a substitute for the conserved quantities.
I give the heuristic ideas for the construction the Gibbs measure, and also sketch the details of the rigorous construction. I will also discuss how to use the Gibbs measure to construct global solutions to the nonlinear Schrödinger equation.
April 9th, 2025
Konstantin Merz (Technische Universität Braunschweig)
The Ionization Conjecture in Quantum Mechanics and Density Functional Theory: An Introduction
Quantum mechanics accurately describes physics on atomic length scales. However, many fundamental questions about the structure of matter remain unanswered. A prominent example is the Ionization Conjecture. It asserts that an atom with nuclear charge Z can bind at most Z+1 electrons. Although this has been experimentally documented since the 1970s, a mathematical proof is not in sight. In this lecture, we introduce the Ionization Conjecture and review some landmark results and recent progress. Although the conjecture for the full many-body problem seems out of reach, significant progress has been made in certain effective single-particle models. These models are easier to study yet still capture key aspects of the full many-body system. Among these are density functional theories such as the classic Thomas–Fermi model, which effectively describe the energy and particle distribution of large atoms and molecules. Remarkably, despite describing the bulk of the electrons, these models also provide insights into the outermost electrons—the key objects in the Ionization Conjecture.
May 14th, 2025
Alessio Ranallo (Université de Genève)
An Invitation to AQFT
The search for a satisfactory set of axioms on which to base QFT dates back to the fifties and is still ongoing. The problem is of course not only formal, in fact it is of primary importance to address structural properties of QFTs: as shown by the successes of the constructive programme, new mathematics is discovered along the way; in this respect one could regard the Osterwalder-Schrader reconstruction theorem as a positive example of the importance of proposing a set of such axioms. In this talk we will focus on the Haag-Kastler axioms in Algebraic QFT (AQFT). We will review the basic mathematics underlying the Haag-Kastler approach and how scalar free fields are encoded in such a framework. Depending on time availability, we will discuss a topic that is central to understanding (part of) current research in the field: namely Tomita-Takesaki's modular theory.
