Program

Title and Abstracts

Christian Brennecke (University of Bonn)

Title: A Short Proof of BEC in the GP Regime and Beyond

Abstract: In this talk, I present a new proof of BEC for the ground state of $N$ bosons moving in the three-dimensional unit torus and interacting through a pair potential with scattering length of order $N^{\kappa-1}$, for a parameter $kappa \in [0,1/20)$. The proof is based on an application of the Schur complement formula and mild a priori bounds on high momentum occupation number operators in the ground state. It improves and significantly simplifies previous results obtained jointly with Adhikari-Schlein; in particular, we avoid the use of operator exponentials which are quartic in creation and annihilation operators. The talk is based on joint work with M. Brooks, C. Caraci and J. Oldenburg.

Li Chen (University of Mannheim)

Title: Mean Field Control Problem for two dimensional Keller-Segel System

Abstract: In this talk, I will show that by using the mean field limit theory the optimal control of two dimensional Keller-Segel system can be obtained as a limit of the corresponding control problem of the interacting stochastic particle system.   In these problems,  the cost functions are chosen in the same structure which includes the mean field effect.   The difficulty arises from the singularity of the fundamental solution of the two dimensional Poisson equation.   Therefore, we start with the control problem of a smoothed version of particle system.   In the compactness argument,  Gamma convergence technique has been used.  The important technique step is to show that the compactness of the cost functions implies strong convergence of the mean field limit.   It is achieved by using a combination of relative entropy method and the convergence in probability of the trajectory.  The talk is based on the work together with Yucheng Wang and Zhao Wang.

Yonggeun Cho (Jeonbuk National University)

Title: Scattering of Dirac equations with self interaction

Abstract: In order to describe the self-field effects the Hartree type Dirac equations has been studied as a toy model of Dirac-Klein-Gordon or Maxwell-Klein-Gordon system. Recently, it turned out that the self-field effect is linear under Yukawa interaction and nonlinear under Coulomb interaction. They can be rephrased as linear scattering and nonlinear scattering to be discussed in this talk.

Martin Ravn Christiansen (LMU Munich)

Title: The Correlation Energy of a Fermi Gas

Abstract: We consider the correlation energy of a Fermi gas on a torus as the particle number N goes to infinity, with the interaction potential scaled by a factor proportional to N^{-1/3}.

In the second-quantized picture, the Hamiltonian of such a system can be written in the form of a quadratic Hamiltonian with respect to certain "quasi-bosonic" operators, which can formally be diagonalized to obtain a "bosonic contribution" to the correlation energy.

In this talk we will see how this formal argument can be realized in terms of a purely algebraic factorization of the Hamiltonian, which directly yields a lower bound on the correlation energy.

Based on joint work with Christian Hainzl and Phan Thành Nam.

Michele Correggi (Polytechnic University of Milan)

Title: Schroedinger and Pauli Operators with Aharonov-Bohm Fluxes

Abstract: We present a review of recent and earlier results concerning Schroedinger and Pauli operators with singular magnetic fields of Aharonov-Bohm (AB) type, i.e., concentrated at isolated points. We focus first on Schroedinger and Pauli operators with a single AB flux: we classify all the self-adjoint realizations and thoroughly discuss all the main spectral and scattering properties. Next, we analyse Schroedinger operators with many fluxes and, finally, we discuss the homogenization limit of infinitely many fluxes of rescaled intensities in a given bounded domain.

Søren Fournais (University of Copenhagen)

Title: The free energy of the Bose gas at low temperature

Abstract: In this talk, I will report on recent work (joint with Junge, Girardot, Morin, Oliviera and Triay) on the calculation of the free energy in the thermodynamic limit of the interacting Bose gas in the case of hard sphere interactions. The temperature is allowed to be of order $\rho a$, with $\rho$ being the density of particles and $a$ the scattering length of the interaction. This makes the entropy part of the free energy of the same order as  the Lee-Huang-Yang term in the ground state energy.

Our proof combines the Neumann localization technique employed by Hainzl, Nam and Triay in recent work, with the localization in momentum space of Fournais and Solovej.

Emanuela Giacomelli (LMU Munich)

Title: Correlation energy for the low density Fermi gas

Abstract: In recent decades, the study of many-body systems has been an active area of research in both physics and mathematics. In this talk we consider a system of N interacting fermions with spin 1/2 confined in a box in the dilute regime. We are interested in studying the correlation energy, defined as the difference between the ground state energy and that of the free Fermi gas. We will discuss how to derive an asymptotics for the correlation energy in the thermodynamic limit, where the number of particles and the size of the box are sent to infinity while keeping the density fixed. In particular, we will present a result on an upper bound for the correlation energy that is consistent with the well-known Huang-Yang formula of 1957.

Alessandro Giuliani (Roma Tre University)

Title: Universality of the critical conductivity of the Haldane-Hubbard model

Abstract:  The Haldane model is a standard tight binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar, magnetic field. We consider its interacting version and study in particular the critical case at the transition between the trivial and the "topological" insulating phases. In previous works, we proved the quantization of the critical longitudinal conductivity for weak enough interaction strength. We now report a recent extension of the result to the critical transverse conductivity, which turns out to be quantized at half-integer values, irrespective of the interaction strength. Proofs are combination of constructive Renormalization Group methods and exact lattice Ward Identities. Based on joint works with V. Mastropietro, M. Porta, I. Jauslin, S. Fabbri, R. Reuvers

Jin Woo Jang (POSTECH)

Title: On the Relativistic Boltzmann Equation with Long-Range Interactions

Abstract: This talk will discuss three recent, interrelated results concerning the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, it is essential to have a uniform positive lower bound of the Jacobian determinant for the change of variables from $v \mapsto v'$ for the widely used "cancellation lemma". Firstly, in collaboration with James Chapman and Robert M. Strain, we calculate this very complex ten-variable Jacobian determinant in the special relativistic situation and illustrate some numerical results showing that it has many distinct points where it is machine zero. Secondly, with R. M. Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well-known change of variables p \to p' is not well defined in the special relativistic context. Finally, we will present our recent proof, also with R. M. Strain, of global-in-time existence and uniqueness of the solutions near the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff and its asymptotic stability. We work in the case of a spatially periodic box. We assume the generic hard-interaction and mildly-soft-interaction conditions on the collision kernel derived by Dudyński and Ekiel-Jeżewska (1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator.

Seunghyeok Kim (Hanyang University)

Title: Quantitative stability for sharp inequalities and critical points of their Euler-Lagrange equations

Abstract: The quantitative stability for sharp inequalities in analysis and geometry is a fascinating subject that has attracted many researchers for decades.

Since the seminar works of Brezis and Lieb (1985), many results have appeared dealing with the properties of the Sobolev inequalities, their variants, harmonic maps, etc.

In contrast, the quantitative stability for critical points of the Euler-Lagrange equations induced by sharp inequalities has been less understood.

Nonetheless, it was completely analyzed in a Hilbertian Sobolev setting recently, thanks to the contributions of Ciraolo et al. (2018), Figalli and Glaudo (2020), and Deng et al. (2021+).

In this talk, we explain the development of the study on quantitative stability and introduce my recent works in this direction, collaborated with H. Chen (Hanyang U.) for the Yamabe problem and J. Wei (UBC, Canada) for the fractional Lane-Emden equation of all possible orders.

Soonsik Kwon (KAIST)

Title: Blow-up dynamics of the self-dual Chern-Simons-Schrödigner equations under equivariant symmetry.

Abstract: The self-dual Chern-Simons-Schrödinger (CSS) equation is viewed as a gauged version of the 2D cubic nonlinear Schrödinger (NLS) equation. It admits static solutions and explicit blow-up solutions via the pseudoconformal transformation. I will discuss recent progress on its blow-up dynamics, from constructions, instability mechanism, and rigidity of blow-up rate. I will highlight remarkable features of (CSS) dynamics, which differ from related models like 2D cubic NLS, Schrödinger maps, wave maps, and others. This talk is based on joint works with Kihyun Kim and Sung-Jin Oh.

Marius Lemm (University of Tübingen)

Title: Propagation bounds for interacting lattice bosons

Abstract: Lieb-Robinson bounds control the propagation speed of correlations in quantum many-body dynamics, usually of lattice systems. They have found decisive applications in condensed-matter physics and quantum information theory. The standard proofs of Lieb-Robinson bounds break down for Hamiltonians with unbounded interactions, which occur in practice for lattice bosons. This talk will review the recent and ongoing progress in deriving propagation bounds for the many-body dynamics generated by a wide class of Bose-Hubbard Hamiltonians. We will discuss the notably different behavior of particle propagation and information propagation in certain settings. While some questions have been resolved, fundamental open problems remain, e.g., to what extent a closed systems of lattice bosons is capable of accelerated information propagation.

Mathieu Lewin (CNRS and Paris Dauphine University - PSL)

Title: The Gross-Pitaevskii equation for supersolids

Abstract:: I will discuss the Gross-Pitaevskii equation for bounded functions not tending to 0 at infinity, that describes the condensate part of an infinite Bose gas. The interaction potential is assumed to be superstable and short range. It can but need not be proportional to a delta (Ginzburg-Landau theory). The main question I will address is whether the solution is constant (describing a superfluid), of varying everywhere in space (describing a supersolid), depending on the value of the average density of particles in the system. When the interaction potential is not of positive-type, the existence of a unique phase transition is proved. A big unsolved conjecture is that the solution is periodic at high density. Joint work with Phan Thanh Nam (Munich).

Domenico Monaco (Sapienza University of Rome)

Title: Purely linear response of the quantum Hall current to space-adiabatic perturbations

Abstract: Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect. This is joint work with Giovanna Marcelli [Lett. Math. Phys 112 (2022) / arXiv:2112.03071].

Marcello Porta (SISSA)

Title: Dynamics of mean-field Fermi systems with nonzero pairing

Abstract: I will discuss the dynamics of many-body Fermi gases, in the mean-field regime. I will consider a class of initial data which are close enough to quasi-free states, with a non-zero pairing matrix. Assuming a suitable semiclassical structure for the initial datum, expected to hold at low enough energy and that we can establish for translation-invariant states, I will present a theorem that shows that the many-body evolution of the system can be well approximated by the Hartree-Fock-Bogoliubov equation, a non-linear effective evolution equation describing the coupled dynamics of the reduced one-particle density matrix and of the pairing matrix. Joint work with Stefano Marcantoni (Nice) and Julien Sabin (Rennes).

Robert Seiringer (ISTA)

Title: Energy-momentum spectrum and effective mass of a strongly coupled polaron

Abstract:  We explain recent bounds on the quantum corrections to the (classical) Pekar approximation of the ground state energy of the Fröhlich polaron model in the strong coupling limit, and their consequence on the existence of excited states and the polaron's effective mass.

Jan Philip Solovej (University of Copenhagen)

Title: The mathematics of the periodic table of the elements

Abstract: Despite its name the periodic table of the elements is not particularly periodic. It does nevertheless have the feature that elements in the same group have similar chemical properties. The noble gases He, Ne, Ar, Kr, … for example all have very weak chemical reactions. Elements with similar properties form the columns in the periodic table.  The Aufbau principle (or Madelung rule) from chemistry  gives a phenomenological explanation for the structure of the periodic table. In this talk I will discuss two mathematical results relating to the periodic table. To make general mathematical statements I will allow myself to consider atoms with atomic number much larger than those that exist in nature. The non-relativistic quantum mechanical description of atoms can be studied mathematically for arbitrarily large atomic number. I will ask the following two questions. Does the periodic structure persist for arbitrarily large atoms? Is the Aufbau principle still valid for large atoms? The answer to the second question is no. The Aufbau Principle fails for large atoms. The first question is more difficult and I do not know the answer in a full many-body quantum description of the atom. I will, however, show that in a certain mean field model of atoms there is an exact periodic behavior in the limit of infinitely large atoms.  This is based on work with Bjerg, Fournais, and Hearnshaw.

Arnaud Triay (LMU Munich)

Title: On the energy of dilute Bose gases

Abstract: I will give an overview on some recent developments on the analysis of dilute bosonic systems with focus on the description of their correlation structure.

Bose-Einstein condensation is one of the most successful predictions of quantum mechanics and has been extensively studied both theoretically and experimentally. In the thermodynamic limit, the energy of the dilute Bose gas is given by the celebrated Lee-Huang-Yang formula as expansion in terms of the density of particles and the scattering length of the interaction potential. While rigorously establishing condensation in this regime is still an open question, the formula has been both experimentally observed and rigorously justified (Fournais-Solovej 2020). We will explain the crucial role of the particles outside the condensate in the physical properties of the system.