We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that
maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This
is joint work with Henrik Matthiesen.

We discuss mathematical questions that play a fundamental role in quantitative analysis of incompressible viscous fluids and other incompressible media. Reliable verification of the quality of approximate solutions requires explicit and computable estimates of the distance to the corresponding generalized solution. In the context of this problem, one of the most essential questions is how to estimate the distance (measured in terms of the gradient norm) to the set of divergence free fields. It is closely related to the so-called inf-sup (LBB) condition or stability lemma for the Stokes problem and requires estimates of the LBB constant. We discuss methods of getting computable bounds of the constant and espective estimates of the distance to exact solutions of the Stokes, generalized Oseen, and Navier-Stokes problems.

We present a framework for the design, analysis and application of computational multiscale methods for slow-fast high-dimensional stochastic processes. We call these processes "microscopic'', and assume existence of an approximate "macroscopic'' model that captures the slow behaviour of a selected set of macroscopic state variables. The methodology combines short bursts of microscopic simulation with extrapolation at the macroscopic level. The methodology requires the careful study of a few key algorithmic ingredients. First, we need to properly initialise the microscopic system, based on a given macroscopic state and (possibly) a prior microscopic state that contains additional information about the system. Second, we need to control the variance of the noise that originates from the microscopic Monte Carlo simulation. Third, we need to analyse stability of the extrapolation step. We will discuss these aspects on two types of model problems -- scale-separated SDEs and kinetic equations -- and show the efficacity of the resulting methods in diverse applications, ranging from tumor growth to fusion energy.

Inelastic surface growth associated with continuous creation of incompatibility on the boundary of an evolving body is behind a variety of both natural processes (embryonic development,  tree growth) and technological processes (dam construction, 3D printing). Despite the ubiquity of such processes, the mechanical aspects of surface growth are still not fully understood. In this talk we present  a new approach to surface growth that allows one to address inelastic effects,  path dependence of the growth process and the resulting geometric frustration. In particular, we show that incompatibility developed during deposition can be fine-tuned to ensure a particular behaviour of the system in physiological (or working) conditions. As an illustration, we compute an explicit deposition protocol aimed at "printing" arteries, that guarantees the attainment of desired stress distributions in physiological conditions. Another illustration is the growth starategy for explosive plants, allowing a complete release of residual elastic energy with a single cut.

Fibrillation is a chaotic, turbulent state for the electrical signal fronts in the heart. In the ventricle it is fatal if not treated promptly. The standard treatment is by an electrical shock to reset the cardiac state to a normal one and allow resumption of a normal heart beat.

The fibrillation wave fronts are organized into scroll waves, more or less analogous to a vortex tube in fluid turbulence. The centerline of this 3D rotating object is called a filament, and it is the organizing center of the scroll wave.

The electrical shock, when turned on or off, creates charges at the conductivity discontinuities of the cardiac tissue. These charges are called virtual electrodes. They charge the region near the discontinuity, and give rise to wave fronts that grow through the heart, to effect the defibrillation. There are many theories, or proposed mechanisms, to specify the details of this process. The main experimental data is through signals on the outer surface of the heart, so that simulations are important to attempt to reconstruct the electrical dynamics within the interior of the heart tissue. The primary electrical conduction discontinuities are at the cardiac surface. Secondary discontinuities, and the source of some differences of opinion, are conduction discontinuities at blood vessel walls.

In this lecture, we will present causal mechanisms for the success of the virtual electrodes, partially overlapping, together with simulation and biological evidence for or against some of these.

The role of small blood vessels has been one area of disagreement. To assess the role of small blood vessels accurately, many details of the modeling have been emphasized, including the thickness and electrical properties of the blood vessel walls, the accuracy of the biological data on the vessels, and their distribution though the heart. While all of these factors do contribute to the answer, our main conclusion is that the concentration of the blood vessels on the exterior surface of the heart and their relative wide separation within the interior of the heart is the factor most strongly limiting the significant participation of small blood vessels in the defibrillation process.

Since the pioneering work of Hodgkin and Huxley , we know that electrical signals propagate along a nerve fiber via ions that flow in and out of the fiber, generating a current. The voltages these currents generate are subject to a diffusion equation, which is a reduced form of the Maxwell equation. The result is a reaction (electrical currents specified by an ODE) coupled to a diffusion equation, hence the term reaction diffusion equation.

The heart is composed of nerve fibers, wound in an ascending spiral fashion along the heart chamber. Modeling not individual nerve fibers, but many within a single mesh block, leads to partial differential equation coupled to the reaction ODE.

As with the nerve fiber equation, these cardiac electrical equations allow a propagating wave front, which normally moves from the bottom to the top of the heart, giving rise to contractions and a normal heart beat, to accomplish the pumping of blood.

The equations are only borderline stable and also allow a chaotic, turbulent type wave front motion called fibrillation.

In this lecture, we will explain the 1D traveling wave solution, the 3D normal wave front motion and the chaotic state.

The chaotic state is easiest to understand in 2D, where it consists of spiral waves rotating about a center. The 3D version of this wave motion is called a scroll wave, resembling a fluid vortex tube.

In simplified models of reaction diffusion equations, we can explain much of this phenomena in an analytically understandable fashion, as a sequence of period doubling transitions along the path to chaos, reminiscent of the laminar to turbulent transition.