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24-27 Jun 2024 Metz (France)
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Program
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Mon. 24 | Tue. 25 | Wed. 26 | Thu. 27 |
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09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
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9:00 - 10:00 (1h)
M. Ballesteros: Mathematical aspects of non-relativistic quantum electrodynamics
 Amphi
In this mini-course we study the so-called non relativistic quantum electrodynamics (NR QED). NR QED describes non relativistic particles coupled to the quantized electromagnetic field and it is a low energy approximation of quantum electrodynamics. The course is intended to be accessible to students that have no background on the mathematical formalism of quantum field theory. For this reason, basic mathematical tools will be presented. We will cover, for example, operators defined on Fock spaces (including creation and annihilation operators) and we will state and prove some basic properties and results. In the last part of the course, we will present more advanced topics such as resonances.
10:00 - 10:30 (30min)
Coffee break
Coffee Room
10:30 - 11:30 (1h)
M. Grillakis: On the Evolution of interacting Bosons
Amphi
I will consider the time evolution of N interacting Bosons (two body interaction) via a potential of the form N^{3β}V(N^β|x − y|) where 0 < β ≤ 1. I will derive what is called the Hartree-Fock-Bogoliubov approximation which results in a coupled system of Schrödinger type equations. These equations describe the coupling between the mean field (Nonlinear cubic Schrödinger equation) with the wave function of a pair of particles. Next I will explain some of the analysis needed to establish global in time existence of the limit equation (as N → ∞) of the coupled system. This part involves establishing Strichartz type estimates independent of N where N is the number of particles, since we are interested in the limit N → ∞ of the equations. This work is in collaboration with Matei Machedon, Zehua Zhang, Jacky Chong as well as some recent seminal work of X. Huang.
11:30 - 12:30 (1h)
S. Cenatiempo: Quantum Mechanics at our scale: a mathematical challenge
Amphi
While the theory of quantum mechanics describes interactions between constituents of matter at the microscopic scale, the effects of these interactions may lead to fascinating quantum mechanics phenomena at a macroscopic scale. The discovery of such phenomena, including superfluidity and superconductivity as two pioneering examples, has led to extraordinary developments in realizing new materials that exploit the principles of quantum mechanics to exhibit innovative behaviours. In these lectures, we will discuss the challenge of developing mathematical models that rigorously describe how these macroscopic effects emerge from the microscopic scale, focusing on the paradigmatic example of the interacting Bose gas.
12:30 - 14:30 (2h)
Lunch
Restaurant Universitaire
14:30 - 15:30 (1h)
M. Falconi: Scaling limits in quantum models of particle-field interaction
Amphi
In these lectures I will review a series of recent results on the interaction between quantum particles and fields, in different scaling regimes (all somewhat related to Bohr's correspondence principle). From a mathematical perspective, infinite dimensional semiclassical analysis lies at the backbone of our results; we will review its main developments and connect them with the study of systems of quantum particles in interaction with a semiclassical field, as well as with the equations of classical electrodynamics.
15:30 - 16:30 (1h)
I. M. Sigal: Gross-Pitaevski equation
Amphi
I. M. Sigal: Gross-Pitaevski equation - The Gross-Pitaevski equation describes superfluids and Bose-Einstein condensates (Nobel prizes 1996 and 2001) in ``macroscopic'' regime. Exactly the same equation appeared in theories of plasma and surface waves where it is called the nonlinear Schrödinger equation (NLS). Mathematically, it is a key nonlinear Hamiltonian partial differential equation (PDE) and it has led to rich mathematics. In this lecture, I will give an introduction into the theory of the GPE, proceeding quickly to recent results and exciting open problems.
16:30 - 17:00 (30min)
Coffee break
Coffee Room
17:00 - 18:00 (1h)
J. Fröhlich: Why text-book Quantum Mechanics calls for a completion, and why the "ETH - Approach to Quantum Mechanics" may provide an adequate one. Cancelled
Amphi
Jürg Fröhlich
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9:00 - 10:00 (1h)
M. Ballesteros: Mathematical aspects of non-relativistic quantum electrodynamics
Amphi
In this mini-course we study the so-called non relativistic quantum electrodynamics (NR QED). NR QED describes non relativistic particles coupled to the quantized electromagnetic field and it is a low energy approximation of quantum electrodynamics. The course is intended to be accessible to students that have no background on the mathematical formalism of quantum field theory. For this reason, basic mathematical tools will be presented. We will cover, for example, operators defined on Fock spaces (including creation and annihilation operators) and we will state and prove some basic properties and results. In the last part of the course, we will present more advanced topics such as resonances.
10:00 - 10:30 (30min)
Coffee break
Coffee Room
10:30 - 11:30 (1h)
M. Grillakis: On the Evolution of interacting Bosons
Amphi
I will consider the time evolution of N interacting Bosons (two body interaction) via a potential of the form N^{3β}V(N^β|x − y|) where 0 < β ≤ 1. I will derive what is called the Hartree-Fock-Bogoliubov approximation which results in a coupled system of Schrödinger type equations. These equations describe the coupling between the mean field (Nonlinear cubic Schrödinger equation) with the wave function of a pair of particles. Next I will explain some of the analysis needed to establish global in time existence of the limit equation (as N → ∞) of the coupled system. This part involves establishing Strichartz type estimates independent of N where N is the number of particles, since we are interested in the limit N → ∞ of the equations. This work is in collaboration with Matei Machedon, Zehua Zhang, Jacky Chong as well as some recent seminal work of X. Huang.
11:30 - 12:30 (1h)
S. Cenatiempo: Quantum Mechanics at our scale: a mathematical challenge
Amphi
While the theory of quantum mechanics describes interactions between constituents of matter at the microscopic scale, the effects of these interactions may lead to fascinating quantum mechanics phenomena at a macroscopic scale. The discovery of such phenomena, including superfluidity and superconductivity as two pioneering examples, has led to extraordinary developments in realizing new materials that exploit the principles of quantum mechanics to exhibit innovative behaviours. In these lectures, we will discuss the challenge of developing mathematical models that rigorously describe how these macroscopic effects emerge from the microscopic scale, focusing on the paradigmatic example of the interacting Bose gas.
12:30 - 14:30 (2h)
Lunch
Restaurant Universitaire
14:30 - 15:30 (1h)
M. Falconi: Scaling limits in quantum models of particle-field interaction
Amphi
In these lectures I will review a series of recent results on the interaction between quantum particles and fields, in different scaling regimes (all somewhat related to Bohr's correspondence principle). From a mathematical perspective, infinite dimensional semiclassical analysis lies at the backbone of our results; we will review its main developments and connect them with the study of systems of quantum particles in interaction with a semiclassical field, as well as with the equations of classical electrodynamics.
15:30 - 16:30 (1h)
I. M. Sigal: Ginzburg-Landau equations
Amphi
The Ginzburg-Landau equations (Nobel prize 2003) is perhaps the most successful PDE system in condensed matter physics. Originating in the theory superconductivity (Nobel prizes 1972 and 2003), it was extended to various other areas of condensed matter physics, such as theories of liquid crystals and fractional Hall effect (Nobel prizes 1991 and 1998). Independently, these equations appeared in particle physics providing the simplest U(1)-gauge theory. Their non-abelian generalization, the Yang- Mills-Higgs equations (Nobel prizes 1999, 2013) is a fundamental ingredient of the standard model of particle physics. GLE have also a natural geometrical interpretation coupling the section and connections of line bundles. In this lecture, I describe the basic properties of the GLE, its key solutions (magnetic vortices and vortex lattices), discuss recent results and say a few words on the extension of this equation to Riemann surfaces.
16:30 - 18:00 (1h30)
Posters
Poster room
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9:00 - 10:00 (1h)
M. Ballesteros: Mathematical aspects of non-relativistic quantum electrodynamics
Amphi
In this mini-course we study the so-called non relativistic quantum electrodynamics (NR QED). NR QED describes non relativistic particles coupled to the quantized electromagnetic field and it is a low energy approximation of quantum electrodynamics. The course is intended to be accessible to students that have no background on the mathematical formalism of quantum field theory. For this reason, basic mathematical tools will be presented. We will cover, for example, operators defined on Fock spaces (including creation and annihilation operators) and we will state and prove some basic properties and results. In the last part of the course, we will present more advanced topics such as resonances.
10:00 - 10:30 (30min)
Coffee break
Coffee Room
10:30 - 11:30 (1h)
M. Grillakis: On the Evolution of interacting Bosons
Amphi
I will consider the time evolution of N interacting Bosons (two body interaction) via a potential of the form N^{3β}V(N^β|x − y|) where 0 < β ≤ 1. I will derive what is called the Hartree-Fock-Bogoliubov approximation which results in a coupled system of Schrödinger type equations. These equations describe the coupling between the mean field (Nonlinear cubic Schrödinger equation) with the wave function of a pair of particles. Next I will explain some of the analysis needed to establish global in time existence of the limit equation (as N → ∞) of the coupled system. This part involves establishing Strichartz type estimates independent of N where N is the number of particles, since we are interested in the limit N → ∞ of the equations. This work is in collaboration with Matei Machedon, Zehua Zhang, Jacky Chong as well as some recent seminal work of X. Huang.
11:30 - 12:30 (1h)
S. Cenatiempo: Quantum Mechanics at our scale: a mathematical challenge
Amphi
While the theory of quantum mechanics describes interactions between constituents of matter at the microscopic scale, the effects of these interactions may lead to fascinating quantum mechanics phenomena at a macroscopic scale. The discovery of such phenomena, including superfluidity and superconductivity as two pioneering examples, has led to extraordinary developments in realizing new materials that exploit the principles of quantum mechanics to exhibit innovative behaviours. In these lectures, we will discuss the challenge of developing mathematical models that rigorously describe how these macroscopic effects emerge from the microscopic scale, focusing on the paradigmatic example of the interacting Bose gas.
12:30 - 13:45 (1h15)
Lunch
Restaurant Universitaire
14:45 - 18:00 (3h15)
Boat trip on the Moselle river and hike at Saint Quentin Mount
We will take a boat to reach the bottom of Mont Saint Quentin, do a hike of about 1:30, and come back to the town center with a bus.
The meeting point at the boat stop Ponton Metz’O Moyen Pont (Below rue de la garde).
Please be punctual, the boat will not wait for us.
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9:30 - 10:30 (1h)
M. Falconi: Scaling limits in quantum models of particle-field interaction
Amphi
In these lectures I will review a series of recent results on the interaction between quantum particles and fields, in different scaling regimes (all somewhat related to Bohr's correspondence principle). From a mathematical perspective, infinite dimensional semiclassical analysis lies at the backbone of our results; we will review its main developments and connect them with the study of systems of quantum particles in interaction with a semiclassical field, as well as with the equations of classical electrodynamics.
10:30 - 11:00 (30min)
Coffee break
Coffee Room
11:00 - 12:00 (1h)
I. M. Sigal: Kohn-Sham equation
Amphi
The Kohn-Sham equation (KSE) is a key equation of density functional theory (Nobel prize 1998), providing effective description of electronic structure of matter. Nowadays, most of the computations in quantum physics and quantum chemistry involving electrons is done using this equation. The mathematical theory for the KSE is presently under construction. In this lecture, I will describe the KSE, its main properties and the key existing results, outline some approaches and present open problems.
12:00 - 13:00 (1h)
J. Fröhlich: Infinitesimal perturbation theory and quantum-Poisson processes: Cancelled
Amphi
Jürg Fröhlich
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