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.

In this talk, we discuss the development of fast iterative solvers for matrix systems arising from various constrained optimization problems. In particular, we seek to exploit the saddle point structure of these problems to construct powerful preconditioners for the resulting systems, using appropriate approximations of the (1,1)-block and Schur complement.

The problems we consider arise from two well-studied subject areas within computational optimization. Specifically, we investigate the
numerical solution of PDE-constrained optimization problems, and the interior point method (IPM) solution of linear/quadratic programming
problems. Indeed a particular focus in this talk is the interior point method solution of PDE-constrained optimization problems with
additional inequality constraints on the state and control variables.

We present a range of optimization problems which we seek to solve using our methodology, and examine the theoretical and practical
convergence properties of our iterative methods for these problems.
 

A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

Ice streams are narrow bands of rapidly sliding ice within an otherwise

slowly flowing continental ice sheet. Unlike the rest of the ice sheet,

which flows as a typical viscous gravity current, ice streams experience

weak friction at their base and behave more like viscous 'free films' or

membranes. The reason for the weak friction is the presence of liquid

water at high pressure at the base of the ice; the water is in turn

generated as a result of dissipation of heat by the flow of the ice

stream. I will explain briefly how this positive feedback can explain the

observed (or inferred, as the time scales are rather long) oscillatory

behaviour of ice streams as a relaxation oscillation. A key parameter in

simple models for such ice stream 'surges' is the width of an ice stream.

Relatively little is understood about what controls how the width of an

ice stream evolves in time. I will focus on this problem for most of the

talk, showing how intense heat dissipation in the margins of an ice stream

combined with large heat fluxes associated with a switch in thermal

boundary conditions may control the rate at which the margin of an ice

stream migrates. The relevant mathematics involves a somewhat non-standard

contact problem, in which a scalar parameter must be chosen to control the

location of the contact region. I will demonstrate how the problem can be

solved using the Wiener-Hopf method, and show recent extensions of this

work to more realistic physics using a finite element discretization.