Page 76 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 76
EMS PRIZE WINNERS
Excitation spectrum of dilute trapped Bose gases
Phan Thanh Nam, nam@math.lmu.de
LMU Munich, Germany
We will discuss the low-lying eigenvalues of trapped Bose gases in the Gross-Pitaevskii regime.
In particular, we will derive a nonlinear correction to the Bogoliubov approximation, thus cap-
turing precisely the two-body scattering process of the particles. This extends a result of Boc-
cato, Brennecke, Cenatiempo and Schlein on the homogeneous Bose gas. The talk is based on
joint work with Arnaud Triay.
From branching singularities of minimal surfaces to non-smoothness
points on an ice-water interface
Joaquim Serra, joaquim.serra@math.ethz.ch
ETH Zurich, Switzerland
Stefan’s problem, dating back to the XIX century, aims to describe the evolution of a block of
ice melting in water. Its mathematical analysis experienced few progress until the 1970’s, when
Duvaut reformulated it as the gradient flow of a nice convex functional. In 1977, Caffarelli
proved that the ice-water interface is an smooth surface outside of a certain closed set: the so-
called singular set. This was as huge breakthrough. However, methods available back in the
1970’s did not allow for a fine description of the structure of the singular set.
During the following 20 years, Almgren developed his theory of branching singularities
of minimal surfaces, and in 2008 these methods were applied to the not-so-well-known “thin
obstacle problem”. This induced, in recent years, a fruitful use of Almgren’s methods to study
singularities in Stefan’s problem. But even with these powerful new tools in hand, we were only
halfway to obtaining a fully satisfying description of the non-smoothness points on an ice-water
interface...
Elliptic curves and modularity
Jack Thorne, jat58@cam.ac.uk
University of Cambridge, United Kingdom
The modularity conjecture for elliptic curves is most famous in its formulation for elliptic curves
over the rational numbers – indeed, Wiles proved the modularity of semistable elliptic curves
over the rationals as part of his proof of Fermat’s Last Theorem.
Recently it has become possible to attack the problem of modularity of elliptic curves over
more general number fields. I will explain what this means, what we know, and what kinds of
consequences we can expect for the solution of Diophantine equations.
74
Excitation spectrum of dilute trapped Bose gases
Phan Thanh Nam, nam@math.lmu.de
LMU Munich, Germany
We will discuss the low-lying eigenvalues of trapped Bose gases in the Gross-Pitaevskii regime.
In particular, we will derive a nonlinear correction to the Bogoliubov approximation, thus cap-
turing precisely the two-body scattering process of the particles. This extends a result of Boc-
cato, Brennecke, Cenatiempo and Schlein on the homogeneous Bose gas. The talk is based on
joint work with Arnaud Triay.
From branching singularities of minimal surfaces to non-smoothness
points on an ice-water interface
Joaquim Serra, joaquim.serra@math.ethz.ch
ETH Zurich, Switzerland
Stefan’s problem, dating back to the XIX century, aims to describe the evolution of a block of
ice melting in water. Its mathematical analysis experienced few progress until the 1970’s, when
Duvaut reformulated it as the gradient flow of a nice convex functional. In 1977, Caffarelli
proved that the ice-water interface is an smooth surface outside of a certain closed set: the so-
called singular set. This was as huge breakthrough. However, methods available back in the
1970’s did not allow for a fine description of the structure of the singular set.
During the following 20 years, Almgren developed his theory of branching singularities
of minimal surfaces, and in 2008 these methods were applied to the not-so-well-known “thin
obstacle problem”. This induced, in recent years, a fruitful use of Almgren’s methods to study
singularities in Stefan’s problem. But even with these powerful new tools in hand, we were only
halfway to obtaining a fully satisfying description of the non-smoothness points on an ice-water
interface...
Elliptic curves and modularity
Jack Thorne, jat58@cam.ac.uk
University of Cambridge, United Kingdom
The modularity conjecture for elliptic curves is most famous in its formulation for elliptic curves
over the rational numbers – indeed, Wiles proved the modularity of semistable elliptic curves
over the rationals as part of his proof of Fermat’s Last Theorem.
Recently it has become possible to attack the problem of modularity of elliptic curves over
more general number fields. I will explain what this means, what we know, and what kinds of
consequences we can expect for the solution of Diophantine equations.
74
