Vorträge in der Woche 18.11.2019 bis 24.11.2019

Dienstag, 19.11.2019: Kazhdan Property (T), Expanders and the Ruziewicz problem II

Anton Deitmar

Mittwoch, 20.11.2019: 3264 Conics in a Second

Sascha Timme (TU Berlin)

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical nonlinear algebra determines these solutions for any given instance. This talk illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a wonderful piece of 19th century geometry, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics and data science, with focus on the maximum likelihood estimation for linear Gaussian covariance models.

Donnerstag, 21.11.2019: Justification of the Korteweg-de Vries and the Nonlinear Schrödinger ap- proximation for the two-dimensional water wave equations

Prof. Dr. Wolf-Patrick Düll (Universität Stuttgart)

We consider the evolution system for two-dimensional surface water waves in an in- finitely long canal of finite depth. Since the full system is too complicated for a direct analysis of the qualitative behavior of its solutions, it is important to approximate the system in different parameter regimes by suitable reduced model equations whose solutions have similar but more easily accessible qualitative properties. The most famous nonlinear reduced models are the Korteweg-de Vries (KdV) equa- tion for the approximate description of the dynamics of soliton-like solutions and the Nonlinear Schrödinger (NLS) equation for the approximate description of the dynamics of wave packet-like solutions. To understand to which extent these approx- imations yield correct predictions of the qualitative behavior of the original system it is important to justify the validity of these approximations by estimates of the approximation errors in the physically relevant length and time scales. In this talk, we give an overview on the KdV and the NLS approximation and their justifications. Special emphasis will be put on the most challenging case, namely, the proof of error estimates for the NLS approximation being valid for surface water waves with and without surface tension. These estimates are obtained by parametriz- ing the two-dimensional surface waves by arc length, which enables us to derive error bounds that are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.

Donnerstag, 21.11.2019: Relativistic Collapse Theory With Interaction

Prof. Roderich Tumulka (Tübingen)

I have recently managed to solve a problem that had been open for 15 years. In 2004, I had developed the first seriously relativistic model of spontaneous wave function collapse in quantum mechanics, a modification of the non-relativistic Ghirardi-Rimini-Weber model, but only for non-interacting particles. The non-interacting case is already of interest because it involves quantum non-locality, and to develop any model that is both fully relativistic and non-local was a challenge. However, the problem remained open to define a generalization of the model for interacting particles. In my talk, after explaining the Ghirardi-Rimini-Weber model and the non-interacting relativistic version, I will explain how to define the interacting version, where the difficulties lie, how they can be dealt with, and why the model works. The model introduces collapses in between episodes of evolution by unitary operators that are assumed to be given, incorporate the interaction, and lead in Tomonaga-Schwinger style from any one spacelike hypersurface to any other.