Universiteit Utrecht

Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.

Krylov subspace methods offer good possibilities for the solution of

large sparse linear systems of equations.For general systems, some of

the popular methods often show an irregular type of convergence

behavior and one may wonder whether that could be improved or not. Many

suggestions have been made for improvement and the question arises

whether these corrections are cosmetic or not. There is also the

question whether the irregularity shows inherent numerical instability.

In such cases one should take extra care in the application of

smoothing techniques. We will discuss strategies that work well and

strategies that might have been expected to work well.

All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.