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).