L3

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Fix a positive integer $k$, and consider the class of all graphs which do not have $k+1$  vertex-disjoint cycles.  A classical result of Erdos and P\'{o}sa says that each such graph $G$ contains a blocker of size at most $f(k)$.  Here a {\em blocker} is a set $B$ of vertices such that $G-B$ has no cycles.

 

We give a minor extension of this result, and deduce that almost all such labelled graphs on vertex set $1,\ldots,n$ have a blocker of size $k$.  This yields an asymptotic counting formula for such graphs; and allows us to deduce further properties of a graph $R_n$ taken uniformly at random from the class: we see for example that the probability that $R_n$ is connected tends to a specified limit as $n \to \infty$.

 

There are corresponding results when we consider unlabelled graphs with few disjoint cycles. We consider also variants of the problem involving for example disjoint long cycles.

 

This is joint work with Valentas Kurauskas and Mihyun Kang.

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Gupta et al. introduced a very general multi-commodity flow problem in which the cost of a given flow solution on a graph $G=(V,E)$ is calculated by first computing the link loads via a load-function l, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function.

 

We show the existence of an oblivious routing scheme with competitive ratio $O(\log n)$ and a lower bound of $\Omega(\log n/\logl\og n)$ for this model when the aggregation function agg is an $L_p$-norm.

 

Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics and the work on minimum congestion oblivious. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the $L_p$-norm of the link loads. The embedding techniques of Bartal and Fakcharoenphol et al. [FRT03] can be viewed as solving this problem in the $L_1$-norm while the result on congestion minmizing oblivious routing solves it for $L_\infty$. We give a single proof that shows the existence of a good tree-based oblivious routing for any $L_p$-norm.

One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability $p$, then long range connections appear if and only if $p>1/2$.  The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.

Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.

Lattices in semisimple Lie groups have been studied from the point of view of number theory, algebraic groups, topology and geometry, and geometric group theory. The Fragestellung of one line of investigation is to what extent the properties of the lattice determine, and are determined by, the properties of the group. This talk reviews a number of results about lattices, and in particular looks at Mostow--Margulis rigidity.