L6

Phylogenies, or evolutionary histories, play a central role in modern biology, illustrating the interrelationships between species, and also aiding the prediction of structural, physiological, and biochemical properties. The reconstruction of the underlying evolutionary history from a set of morphological characters or biomolecular sequences is difficult since the optimality criteria favored by biologists are NP-hard, and the space of possible answers is huge. Phylogenies are often modeled by trees with n leaves, and the number of possible phylogenetic trees is $(2n-5)!!$. Due to the hardness and the large number of possible answers, clever searching techniques and heuristics are used to estimate the underlying tree.

We explore the combinatorial structure of the underlying space of trees, under different metrics, in particular the nearest-neighbor-interchange (NNI), subtree- prune-and-regraft (SPR), tree-bisection-and-reconnection (TBR), and Robinson-Foulds (RF) distances.  Further, we examine the interplay between the metric chosen and the difficulty of the search for the optimal tree.

How many $H$-free graphs are there on $n$ vertices? What is the typical structure of such a graph $G$? And how do these answers change if we restrict the number of edges of $G$? In this talk I will describe some recent progress on these basic and classical questions, focusing on the cases $H=K_{r+1}$ and $H=C_{2k}$. The key tools are the hypergraph container method, the Janson inequalities, and some new "balanced" supersaturation results. The techniques are quite general, and can be used to study similar questions about objects such sum-free sets, antichains and metric spaces.

I will mention joint work with a number of different coauthors, including Jozsi Balogh, Wojciech Samotij, David Saxton, Lutz Warnke and Mauricio Collares Neto. 

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

Let $G(n,d)$ be a random $d$-regular graph on $n$ vertices. In 2004 Kim and Vu showed that if $d$ grows faster than $\log n$ as $n$ tends to infinity, then one can define a joint distribution of $G(n,d)$ and two binomial random graphs $G(n,p_1)$ and $G(n,p_2)$ -- both of which have asymptotic expected degree $d$ -- such that with high probability $G(n,d)$ is a supergraph of $G(n,p_1)$ and a subgraph of $G(n,p_2)$. The motivation for such a coupling is to deduce monotone properties (like Hamiltonicity) of $G(n,d)$ from the simpler model $G(n,p)$. We present our work with A. Dudek, A. Frieze and A. Rucinski on the Kim-Vu conjecture and its hypergraph counterpart.

Representations of free loop groups possess an operation, akin to
tensor product, under which they form a braided tensor category. I
will discuss a similar operation, which is present on the category of
representations of the based loop groups, and which equips it with the
structure of a monoidal cateogory. Finally, I will present a recent
result, according to which the Drinfel'd centre of the category of
representations of a based loop group is equivalent to the category of
representations of the corresponding free loop group.

*/ /*-->*/ Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence  of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to  the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs.  We will then, explain applications of these results to the geometry of Coxeter groups.  Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.