Lectures
Courses
Courses
April 4 - 22
April 4 - 22
ABSTRACT: Linear quantum field theory (QFT) has a wide range of applications in physics. It can be used to model quantum fields in the presence of external sources or on a curved spacetime. The common wisdom says that linear QFT is quite straightforward and not very interesting. In my lectures I will try to convince the audience that this is not true.
The mathematical status of nonlinear QFT, such as QED or the Standard model, is (and, probably, will always be) quite problematic, because one is forced to use perturbative techniques, which are useful only in the presence of a small parameter. Linear QFT can be well understood beyond perturbation theory. It often has surprising properties. It sheds light onto various aspects of full nonlinear QFT. It shows the power of advanced mathematics. Various relatively sophisticated mathematical concepts, such as of tempered distributions, boundary conditions for differential operators, spectral shift function, Krein spaces, scales of Hilbertizable spaces, wave front sets, are very useful in linear QFT.
Schedule and futher info
April 19 - May 6
April 19 - May 6
ABSTRACT: Quantum trajectory models time evolution of a quantum system including a particular measurement strategy. In a quantum trajectory theory, the state of the system is a stochastic process depending on the measurement outputs. In this course, I will introduce quantum trajectories, describe their theory, and mention some experimental applications and mathematical open problems. The only prerequisite to follow the main part of the course is a basic knowledge of linear algebra and Markov processes, though a knowledge of quantum mechanics might be useful to follow some examples and applications.
Schedule and futher info
Mini-courses
Mini-courses
April 4 - 8
April 4 - 8
ABSTRACT: The relationship between classical and quantum field theories can be as problematic as it is illuminating. For instance, it is well known that divergences occur in different levels of such theories and it is hard to cure or even to reconcile such difficulties. Nevertheless, I will highlight in this course a mathematical framework in which it is possible to rigorously study the classical limits of certain relevant QFTs. Thus, recovering several examples of fundamental nonlinear PDEs (Schrödinger-Klein-Gordon, Newton-Maxwell, Hartree) as a scaling limit of these quantum theories. In particular, I will focus on a conceptual method based on Wigner measures and Liouville equations that has been developed over the last fifteen years.
Schedule and futher info
March 28 - 30
March 28 - 30
ABSTRACT: Understanding low-energy properties of the weakly interacting Bose gas has a long history in theoretical and mathematical physics. In 1947, assuming that the ground state exhibits Bose-Einstein condensation (BEC), Bogoliubov suggested a theory that provides expressions for the leading order contributions to the ground state energy and that predicts a linear excitation spectrum for dilute Bose gases in the thermodynamic limit. In this course, I discuss recently developed methods with A. Adhikari, C. Boccato, M. Caporaletti, S. Cenatiempo, B. Schlein and S. Schraven to establish BEC and Bogoliubov’s energy predictions in the Gross-Pitaevskii scaling regime (and slightly beyond it). I conclude the lectures with some open questions related to the thermodynamic limit.
Schedule and futher info
March 14 - 28
March 14 - 28
ABSTRACT: This is an introductory course exploring notions and results in the theory of operator algebras which are useful for quantum theory. The topics include: Observable algebras, states, trace-class operators, Gelfand-Naimark-Segal (GNS) representation, Kubo-Martin-Schwinger (KMS) equilibrium states, Tomita-Takesaki theory, standard form of von Neumann algebras, Araki-Woods representation of the infinite free Bose gas.
Schedule and futher info
May 16 - 20
May 16 - 20
ABSTRACT: The propagation of a (quantum) wave in a random medium and the derivation of the radiative transfer equation or the linear Boltzmann equation, is a fundamental problem which has received several types of mathematical answers in the last decades. Although those results are in accordance with the physical models and sometimes suffice for additional modelling work, they are still puzzling : either their proof technically diverge from the obvious heuristic arguments or they seem incomplete from a mathematical point of view. Several models, eg. translation invariant gaussian or poissonian potentials in the weak or low density limit, can be formulated in the framework of bosonic QFT where the number of “particles” is a translation of the probalistic “chaos decomposition”. The different frameworks allow a wide variety of techniques, which make this problem extremely rich from a mathematical point of view. A key point, although it is not the only one, is about accurate number estimates. By combining the endpoint Strichartz estimates by Keel and Tao with an adaptation of the proof of Cauchy-Kowalevski theorem, we obtained with Sébastien Breteaux in a work in progress, such accurate number estimates. They can be used to get refined a priori pieces of information for a general class of initial data and essentially reduce the problem to finite dimensional semiclassical and microlocal analysis issues.
Schedule and futher info
April 26 - May 6
April 26 - May 6
ABSTRACT: The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured (and, very recently, experimentally confirmed) that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles. These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof.
The cornerstone of our theoretical understanding is the Laughlin state, a well-educated ansatz for the ground state of 2D particles subjected to large magnetic fields and strong interactions. The two latter effects conspire to generate strong and very specific correlations between particles. We shall discuss mathematically the rigidity these correlations display in their response to perturbations. The main message is that trapping and disorder potentials can be taken into account by generating uncorrelated quasi-particles (Laughlin quasi-holes) on top of the Laughlin state.
Schedule and futher info
Talks
Talks
March 30, 4:30 pm, Sala Consiglio
March 30, 4:30 pm, Sala Consiglio
ABSTRACT: In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated with the Luttinger's model, with a special focus at the one-dimensional case. It is shown that the so-called thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. We also describe some results about the perturbation by potential and the related scattering theory. We will finish with some hints to the classic problem and the comparison between classical and quantum behavior. Joint work with: Vicente Lenz (Delft University of Technology).
May 18, 4:30 pm, Sala Consiglio
May 18, 4:30 pm, Sala Consiglio
ABSTRACT: After reviewing the known leading strong field asymptotics of the Dirichlet magnetic Laplcian, the focus will be on the case where the magnetic field is in a Sobolev space. The asymptotics are then derived via an averaging mechanism. The case of the Neumann boundary condition and other cases of less regular magnetic fields will be discussed too.
Aharonov-Bohm day
Aharonov-Bohm day
May 19, 10 am, Sala Consiglio
May 19, 10 am, Sala Consiglio
ABSTRACT: I shall introduce an eigenvalue problem for the stationary Aharonov-Bohm Schrödinger operator in the plane. I shall discuss the rate of variation of simple eigenvalues, as the Dirac delta-type singularity moves, with particular attention to the case of half-integer circulation. The talk collects several results obtained, starting from 2010, in collaboration with L. Abatangelo, V. Bonnaillie-Noël, V. Felli, M. Nys and S. Terracini.
May 19, 11 am, Sala Consiglio
May 19, 11 am, Sala Consiglio
ABSTRACT: I shall discuss connections between asymptotic behaviors of A-B eigenvalues and spectral minimal partitions as introduced in seminal works by Helffer, Hoffman-Ostenoff and Terracini and extremal points for the eigenbranches. These are also related to possible multiple eigenvalues for the A-B operator, which will be briefly discussed as well.
May 19, 2 pm, Sala Consiglio
May 19, 2 pm, Sala Consiglio
ABSTRACT: The Aharonov-Bohm effect relates to the phase shift experienced by non-relativistic charged quantum particles interacting with magnetic fields confined inside ideal solenoids. The prototypical model is described by a Schrödinger operator with a singular vector potential, yielding a finite magnetic flux concentrated along a line. In this seminar we first review classical results, using Von-Neumann theory and resolvent techniques to characterize all admissible one-body Hamiltonians as self-adjoint realizations of the said Schrödinger operator. Next, we discuss recent advances using quadratic form methods to treat related models, including magnetic perturbations and multiple fluxes configurations. We also mention connections to anyonic particles systems.Based on joint works with Michele Correggi (Politecnico di Milano).