Vorträge in der Woche 15.01.2018 bis 21.01.2018

Dienstag, 16.01.2018: Cuspidal trace formula and Weyl's law

Stefan Köberle

Donnerstag, 18.01.2018: The numerical range is a $(1+\sqrt{2})$-spectral set

Cesar Palencia (Universität Valladolid)

Donnerstag, 18.01.2018: How Electrons Spin

Dr. Charles Sebens (University of California at San Diego)

There are a number of reasons to think that the electron cannot truly be spinning. Given how small the electron is generally taken to be, it would have to rotate superluminally to have the right angular momentum and magnetic moment. Also, the electron’s gyromagnetic ratio is twice the value one would expect for an ordinary classical rotating charged body. These obstacles can be overcome by examining the flow of mass and charge in the Dirac field (which gives the classical state of the electron). Superluminal velocities are avoided because the electron’s mass and charge are spread over sufficiently large distances that neither the velocity of mass flow nor the velocity of charge flow need to exceed the speed of light. The electron’s gyromagnetic ratio is twice the expected value because its charge rotates twice as fast as its mass.

Freitag, 19.01.2018: Analysis and stochastics on and off fractals

Prof. Dr. Uta Freiberg (Universität Stuttgart)

Fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, there is no canonical notion of "derivative" on such sets, as they are to "rough" to posses a tangent space. We give an overview over the several theories to overcome this problem, starting from the very first attemps made in the 1980's until the most recent approaches, with a special emphasis to random models.

Freitag, 19.01.2018: Spectral Asymptotics for Krein-Feller-Operators with respect to Random Recursive Cantor Measures

Lenon Minorics (Universität Stuttgart)

We study the limit behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second order differential operators d/dmu d/dx, where mu is a finite atomless Borel measure on some compact interval [a,b]. We firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we give the spectral asymptotics for so called random recursive Cantor measures. Finally, we compare the results for random recursive and random homogeneous Cantor measures.

Freitag, 19.01.2018: Spectral asymptotics on the Hanoi attractor

Elias Hauser (Universität Stuttgart)

The Hanoi attractor (or Stretched Sierpinnski Gasket) is an example of a non self similar fractal that still exhibits a lot of symmetry. The existence of various symmetric resistance forms on the Hanoi attractor was shown in 2016 by Alonso-Ruiz, Freiberg and Kigami. To get self adjoint operators from these resistance forms we have to choose a locally finite measure. The goal of this work is to calculate the leading term for the asymptotics of the eigenvalue counting function from these operators. Furthermore we try to get a refinement of the asymptotics, including periodic behaviour and the existence of localized eigenfunctions.

Freitag, 19.01.2018: Networks or how the proteins became friends

Klemens Taglieber (Universität Stuttgart)

The modelling and anlysis of random graphs gives us the opportunity to investigate and understand real world networks. The first random graphs were defined by Paul Erdös and Alfred Renyi between 1959 and 1960. Up until now there are new models developed. In 1999 Albert-Laszlo Barabasi and Reka Albert constructed a model which is now used to model most real world networks. In biology random graphs are more and more used to study different mechanisms such as the evolution of proteins or the spreading of diseases. When looking at a network representing the similarity of proteins the question arises if it is possible to find models which can be used to predict unknown or not yet existing proteins. In this master thesis we will first look at different properties of random graphs and their subgraphs in order to find a fitting model for such protein networks. These networks result through connecting closely related proteins. Amongst other things we want to know if we are able to model subgraphs in the same way as the graphs they descend from. Furthermore we are interested in the differences that occur if one constructs the subgraphs differently. A general interest lies in the interpretation of graphs as electrical networks and the connections between graph theory and the theory of electrical networks.

Freitag, 19.01.2018: Analysis on self-similar graphs

Konstantinos Tsougkas (University Uppsala)

In recent years fractals have become a widely studied topic in Mathematics and analysis on self-similar fractals has been subsequently developed through both a probabilistic and analytic perspective. In this talk we will give a brief introduction to the topic and then take the approach of analysis on sequences of "self-similar graphs". We will define the Laplace operator on graphs and study properties of harmonic functions as well as their probabilistic interpretation through random walks and electrical networks. In the end, we will mention a probabilistic fractal model of the unit interval, connections with the Riemann zeta function and regularized determinants.