Interviews for RGGC 4-5 May 2016



Matt Tointon, May 4th, 11:00 (A1.01) "Extracting algebraic structure from combinatorial hypotheses."

I will briefly discuss my work in two distinct but related areas: 1) Approximate groups. 2) Harmonic functions and random walks on groups. These fields have significant interactions with a broad range of other mathematical disciplines, from number theory and combinatorics to geometric group theory and differential geometry. Some results that have been particularly fruitful in applications are so-called "inverse theorems". In such theorems one typically starts with a subset of a group that obeys some combinatorial hypothesis, and then seeks to extract a much more explicit algebraic description of that subset. My talk will focus on this kind of result.

Timo Hirscher, May 5th, 11:45 (B3.02) "Consensus formation in the Deffuant model"

In this presentation, I will talk about a mathematical model used in the context of social interaction in large groups, introduced by Deffuant et al. in 2000. Interactions take place in random pairwise encounters and are governed by a bounded confidence restriction: Two agents will only discuss and approach a compromise if their opinions do not differ too much . Results about the long-term behavior of the model on different infinite networks will be discussed, as well as a related problem from combinatorial optimization.

Christoph Koch, May 5th, 12:15 (B3.02) "Phase transition phenomena in random graphs and hypergraphs"

Loosely speaking, a random graph exhibits a phase transition if there > exists a critical value for some parameter (e.g.\ the edge density) that > triggers a sudden drastic change in the graph structure whenever the > parameter exceeds this critical value. Arguably the most well-studied > phase transition phenomena concern the evolution of connected components > in the binomial random graph $G(n,p)$. These comprise in particular the > emergence of the giant component, and the threshold for connectedness. > Similarly such a critical value, or threshold, also exists for various > (percolation) processes on random (hyper-)graphs. > > In the first part of the talk we present a the notion of high-order > connectedness in hypergraphs: Given integers $1\le j $k$-uniform hypergraph $H$ two $j$-sets (sets of $j$-vertices) are > $j$-connected if there is a walk from one to the other such that any two > consecutive edges overlap in at least $j$ vertices. Then a $j$-component > is a maximal collection of pairwise $j$-connected $j$-sets. (The case > $k=2$ and $j=1$ corresponds to connectedness in graphs.) We describe > the evolution of $j$-connected components in the $k$-uniform binomial > random hypergraph $H^k(n,p)$. In particular we determine the size of the > giant $j$-component right after its emergence, and show that the > hypergraph becomes $j$-connected precisely when the last isolated > $j$-set disappears. The analysis of high-order connected components is > significantly more involved than studying their vertex-connected > counterparts ($j=1$), and is based on a powerful new tool---the > \emph{smoothness lemma}. > > In the second part of the talk we briefly introduce a novel > inhomogeneous random graph model with underlying geometry (GIRG). It was > recently introduced by Bringmann, Keusch, and Lengler showing that it > possess many properties common to real-world networks. We analyse > bootstrap percolation on GIRGs with a geometric flavour: we establish > the threshold for a large scale outbreak originating from a (small) > \emph{local} region in terms of the initial infection rate. This extends > a previous result by E. Candellero and N. Fountoulakis on hyperbolic > random graphs (which can be embedded into the GIRG model). > > These results are joint work with O. Cooley (TU Graz) and M. Kang (TU > Graz), as well as J. Lengler (ETH Zurich).