Vorträge in der Woche 21.11.2016 bis 27.11.2016

Dienstag, 22.11.2016: Poincaré Paarung auf hyperbolischen Flächen

Benedikt Otto

Donnerstag, 24.11.2016: Constructing polynomial systems with many positive solutions using tropical geometry

Dr. Boulos El-Hilany (Universität Tübingen)

The number of positive solutions of a system of two polynomials in two variables with a total of five distinct monomials defined in the field of real numbers cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. It is known that a system as above constructed using this method has at most 6 positive solutions. In this talk, we show that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, we constructed a system as above having 7 positive solutions.

Donnerstag, 24.11.2016: Connection between $p$-harmonic functions and the inverse mean curvature flow

Sascha Muhr (Universität Tübingen)

In this talk we will present Roger Mosers method for an alternative proof for the existence of solutions to the IMCF and outline a proof for the more general case of a complete and closed Riemannian manifold with bounded curvature. In the course of doing so we will also discuss why certain properties of $p$-harmonic functions, such as the Hölder-continuity of the gradient, will be preserved and give an outlook on further proceedings in the field.

Donnerstag, 24.11.2016: Randomness in Quantum Mechanics - or: "eth" in Quantum Theory

Prof. Jürg Fröhlich (ETH Zürich)

An approach to the quantum theory of events and of direct (projective, or von Neumann) measurements and of indirect (Kraus) measurements/observations developed in my group in Zurich is reviewed. Our aim has been to dispel some of the many confusions surrounding the foundations of quantum mechanics. Our approach is based on ideas, concepts and results from the theory of operator algebras. Some concrete mathematical problems arising in our endeavor are discussed.

Freitag, 25.11.2016: The Arrow of Time - Images of Irreversible Behavior

Prof. Jürg Fröhlich (ETH Zürich)

After a short introductory discussion of entropy, I present examples of irreversible phenomena that are reasonably well understood mathematically. - I will start with an analysis of the Second Law of Thermodynamics from the point of view of non-equilibrium statistical mechanics. - I will then describe results concerning a mathematical derivation of diffusive particle motion - "quantum Brownian motion" - from unitary quantum dynamics. - An analysis of a Hamiltonian model of friction (caused by emission of Cerenkov radiation) will follow next. - I will end up with some casual remarks on the fundamental irrever- sibility of evolution in quantum mechanics.