SCIENTIFIC PROGRAMS AND ACTIVITIES

October 20, 2018

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July-December 2014
Thematic Program on
Variational Problems in Physics, Economics and Geometry

October 2-3
Minischool on Variational Problems in Physics

Organizing Committee
Luigi Ambrosio (Pisa)
Bob Jerrard (Toronto)
Felix Otto (Leipzig)

Mary Pugh (Toronto)
Robert Seiringer (Montreal)

MINISCHOOL PROGRAMME

Thursday, October 2

9:00-10:00

Robert Seiringer (Institute for Science and Technology, Austria) SLIDES
Structure of the excitation spectrum for many-body quantum systems 1

10:00-10:30

Coffee break

10:30-12:00

Luigi Ambrosio (Scuola Normale Superiore di Pisa) SLIDES
A crash course on Gamma convergence

12:00-2:00

Lunch

2:00-3:00

Robert Jerrard (University of Toronto) SLIDES
Variational methods for effective dynamics

3:00-3:30

Tea break
3:30-5:00
Robert Seiringer (Institute for Science and Technology, Austria) SLIDES
Structure of the excitation spectrum for many-body quantum systems 2
Friday, October 3
9:00-10:00

Felix Otto (Max-Planck Institute for Mathematics in the Natural Sciences at Leipzig)

10:00-10:30

Coffee break
10:30-12:00
Robert Jerrard (University of Toronto) SLIDES
Variational methods for effective dynamics
12:00-2:00
Lunch

2:00-3:00

Luigi Ambrosio (Scuola Normale Superiore di Pisa) SLIDES
A crash course on Gamma convergence

3:00-3:30

Tea break
3:30-5:00

Felix Otto (Max-Planck Institute for Mathematics in the Natural Sciences at Leipzig)

 

PROVISIONAL SPEAKERS, TITLE AND ABSTRACT INFO FOR MINISCHOOL

Luigi Ambrosio (Scuola Normale Superiore di Pisa)
A crash course on Gamma convergence

The theory of Gamma-convergence, introduced by E.De Giorgi in the '70, is a powerful mathematical tool that allows to study the limiting behaviour of variational problems. The great flexibility of this concept allows to treat in a rigorous manner many convergence problems: sharp phase transition models as limits of diffuse phase transition models, the derivation of continuum laws by discrete (atomistic) ones, homogenization, dimension reduction, singular perturbation, etc. In the course the basic concepts of the theory will be presented and then a series of examples will be given, mostly in the 1-dimensional case to avoid technical issues. If time allows, also the convergence of evolution problems (gradient flows) will be illustrated. The course will be based on A.Braides' monograph "Gamma-convergence: a beginner's guide" and, for the evolution problems, on its more recent monograph.
Reference: A. Braides. Gamma-convergence for Beginners.
Oxford University Press, 2002

Robert Jerrard (University of Toronto)
Variational methods for effective dynamics

in many problems, one wants to show that the evolution of a complicated physical system can be reduced, at least under certain circumstances, to a lower-dimensional problem. For example, one might want to prove that 2d fluids, under suitable conditions, possess objects one may call "point vortices"; that the evolution of the fluid is largely determined once one knows how these point vortices evolve; and that one can derive an equation governing point vortex dynamics. This course will examine, largely by consideration of several concrete model problems, ways in which techniques from the calculus of variations can be useful in problems of this type. We will focus on problems of hyperbolic and Schroedinger type, for which it is not a priori clear that variational methods are relevant or useful.

Felix Otto (Max-Planck Institute for Mathematics in the Natural Sciences at Leipzig)
TBA:

Mary Pugh* (University of Toronto)
TBA:

Robert Seiringer (Institute for Science and Technology, Austria)
Structure of the excitation spectrum for many-body quantum systems

Many questions concerning models in quantum mechanics require a detailed analysis of the spectrum of the corresponding Hamiltonian, a linear operator on a suitable Hilbert space. Of particular relevance for an understanding of the low-temperature properties of a system is the structure of the excitation spectrum, which is the part of the spectrum close to the spectral bottom. We present recent progress on this question for bosonic many-body quantum systems with weak two-body interactions. Such system are currently of great interest, due to their experimental realization in ultra-cold atomic gases. We investigate the accuracy of the Bogoliubov approximations, which predicts that the low-energy spectrum is made up of sums of elementary excitations, with linear dispersion law at low momentum. The latter property is crucial for the superfluid behavior the system.

* to be confirmed

 

 

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