A permutation group is called sharply n-transitive if it acts 

freely and transitively on the set of ordered n-tuples of distinct 

points. The investigation of such permutation groups is a classical 

branch of group theory; it led Emile Mathieu to the discovery of the 

smallest finite simple sporadic groups in the 1860's. In this talk I 

will discuss the case where the permutation group is assumed to be a 

locally compact transformation group, and explain how this set-up is 

related to Gromov hyperbolicity and to arithmetic lattices in products 

of trees.

Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$.  The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport.  These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic.  Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension.  We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation.  Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns.  Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics.  This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).

The  study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics.  This makes a very close connection between closed geodesics and the topology of loop spaces.  

Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999.  In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed  geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.

It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.

I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$  of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.

Computational models of the heart have been primarily developed to simulate, analyse and understand experimental measurements. Increasingly biophysical models are being used to understand cardiac disease and pathologies in patients. This shift from laboratory to clinical contexts requires the development of new modelling frameworks to simulate pathological states that invalidate assumptions in existing modelling frameworks, work flows to integrate multiple data sets to constrain model parameters and an understanding of the clinical questions that models can answer. We report on the development and application of biophysical modelling frameworks representing the cardiac electrical and mechanical systems, which are currently being customised for modelling cardiac pathologies.

Genomic sequences contain rich evolutionary information about functional constraints on macromolecules such as proteins. This information can be efficiently mined to detect evolutionary couplings between residues in proteins and address the long-standing challenge to compute protein three-dimensional structures from amino acid sequences. Substantial progress on this problem has become possible because of the explosive growth in available sequences and the application of global statistical methods. In addition to three-dimensional structure, the improved analysis of covariation helps identify functional residues involved in ligand binding, protein-complex formation and conformational changes. We expect computation of covariation patterns to complement experimental structural biology in elucidating the full spectrum of protein structures, their functional interactions and evolutionary dynamics. Use the http://evfold.org  server to compute EVcouplings and to predict 3D structure for large sequence families. References:  http://bit.ly/tob48p - Protein 3D Structure from high-throughput sequencing;  http://bit.ly/1DSqANO - 3D structure of transmembrane proteins from evolutionary constraints; http://bit.ly/1zyYpE7 - Sequence co-evolution gives 3D contacts and structures of protein complexes.

We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.