24/11/2022: Luca Fresta @ UniMi

November 24, 2022, 15:45
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Luca Fresta (Universität Bonn)

Stochastic Analysis of Subcritical Euclidean Fermionic Field Theories

In my talk, I will introduce a forward-backward stochastic differential equation
which provides a stochastic quantisation of subcritical Grassmann measures. The method is inspired by the so-called continuous renormalisation group, but
avoids the technical difficulties encountered in the direct study of the Polchinski’s flow equation for the effective potentials. If time permits, I will also show how to prove the exponential decay of correlations by a coupling method.
Work in collaboration with De Vecchi and Gubinelli.

21/11/2022: Nikolai Leopold @ UniMi

November 21, 2022, 11:15
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Nikolai Leopold (Universität Basel)

Norm approximations for the Fröhlich dynamics

In this talk I will discuss recent results about the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. For large particle number and initial data in which the particles are in a Bose-Einstein condensate and the excitations of the phonon field are in a coherent state I will show that the time evolved many-body state can be approximated in norm by an effective dynamics. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics.
If time permits I will, in addition, present a joint work with D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer about the Fröhlich model in the strong coupling limit and compare the Bogoliubov dynamics in the strong coupling and mean-field regime.

20/12/2022: Thematic Day @ Como

December 20, 2022, 10:30 - 15:30

Program (to be confirmed)

    • 10:00 – Andrea Mantile
    • 10:45 – Coffee Break
    • 11:15 – Cristina Caraci
    • 12:00 – Markus Lange
    • 12:45 – Lunch Break
    • 14:45 – Hynek Kovarik
    • 15:30 – Annalisa Panati
    • 16:15 – Closing

    Titles and Abstracts

    26/10/2022: Jérémy Faupin @ PoliMi

    October 26, 2022, 11:15 (new time this semester)
    Aula Seminario - III piano
    Politecnico di Milano
    Edificio 14 (Nave), Campus Leonardo
    P.zza da Vinci 32, Milano, Italy

    SPEAKER: Jérémy Faupin (U de Lorraine @Metz)

    Quasi-classical ground states in non-relativistic QED and related models

    We will consider in this talk a non-relativistic particle bound by an external
    potential and coupled to a quantized radiation field. This physical system is
    mathematically described by a Pauli-Fierz Hamiltonian. We will study the energy
    functional of product states of the form u⊗Ψ_f, where u is a normalized state
    for the non-relativistic particle and Ψ_f is a coherent state in Fock space for
    the field. This gives the energy of a Klein-Gordon-Schrödinger system in the
    case of a spinless particle linearly coupled to a scalar field, or the energy
    of a Maxwell-Schrödinger system in the case of an electron coupled to the
    photon field. In both cases, we will discuss results concerning the existence
    and uniqueness of a ground state, under general conditions on the external
    potential and the coupling form factor. In particular, neither an ultraviolet
    cutoff nor an infrared cutoff needs to be imposed. We will also discuss the
    convergence in the ultraviolet limit and the second-order asymptotic expansion
    in the coupling constant of the ground state energy.
    This is joint work with J. Payet and S. Breteaux.

    30/5/2022: Laurent Lafleche @ UniMi

    May 30, 2022, 16:30 (non-standard time)
    Sala di Rappresentanza
    Mathematics Department, University of Milan
    Via Cesare Saldini 50, Milano, Italy

    SPEAKER: Laurent Lafleche (U Texas at Austin)

    Semiclassical regularity and mean-field limit
    with singular potentials

    In this talk I will present several techniques and concepts used in the context of the mean-field and the classical limit allowing to go from the N -body Schrödinger equation with singular potential to the Hartree–Fock and Vlasov equations, linked to works in collaboration with Chiara Saffirio and Jacky Chong. At the level of the Vlasov–Poisson equation, typical mean-field techniques from quantum mechanics for pure states can be translated to a weak-strong stability estimate in L^1 for the Vlasov equation. Another weak-strong stability can be obtained for the difference of the square roots of the solutions in L^2. They allow to better understand the mean-field and semiclassical estimates. These estimates are weak-strong in the sense that they require only the regularity of one of the solutions. This requires the propagation of a semi-classical notion of regularity uniformly in N and h. A typical obstacle is the lack of positivity of the Wigner transform and its few conserved quantities. A solution to this problem is to consider operators as the right generalization of the phase space distribution, and a quantum analogue of Sobolev spaces defined using Schatten norms. The advantage of these techniques is that they allow to obtain regularity estimates without higher order error terms.

    16/5/2022: Robin Reuvers @ UniMi

    May 16, 2022, 16:30 (non-standard time)
    Sala di Rappresentanza
    Mathematics Department, University of Milan
    Via Cesare Saldini 50, Milano, Italy

    SPEAKER: Robin Reuvers (U Roma Tre)

    Ground state energy of dilute Bose gases in 1D

    In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact, Bethe ansatz solution, the model and its generalizations soon became popular objects of study in mathematical physics. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, the Lieb-Liniger model found experimental use and became even better known.

    In the meantime, Lieb and collaborators had moved on, and were rigorously studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to acquire, and a popular goal was the rigorous derivation of the ground state energy of gases of bosons in various settings in 2 and 3D. Many of the results focused on the dilute limit, in which the density of the boson gas is very low.

    Somehow, Bose gases in 1D were excluded from this development. Of course, the original Lieb-Liniger model provided a solvable example, but we can nevertheless use insights from the 2 and 3D approaches to prove new results about the ground state energy of dilute Bose gases in 1D.

    In the talk, I will review the developments above, and explain the new results.