Past

We give a construction of Brownian motion in a Weyl chamber, by a

multidimensional generalisation of Pitman's theorem relating one

dimensional Brownian motion with the three dimensional Bessel

process. There are connections representation theory, especially to

Littelmann path model.

The heart can be described as an electrically driven mechanical pump. This

pump couldn't adapt to beat-by-beat changes in circulatory demand if there

was no feedback from the mechanical environment to the electrical control

processes. Cardiac mechano-electric feedback has been studied at various

levels of functional integration, from stretch-activated ion channels,

through mechanically induced changes in cardiac cells and tissue, to

clinically relevant observations in man, where mechanical stimulation of the

heart may either disturb or reinstate cardiac rhythmicity. The seminar will

illustrate the patho-physiological relevance of cardiac mechano-electric

feedback, introduce underlying mechanisms, and show the utility of iterating

between experimental research and mathematical modelling in studying this

phenomenon.

I shall give a brief overview of the partial regularity results for minima

of integral functionals and solutions to elliptic systems, concentrating my

attention on possible estimates for the Hausdorff dimension of the singular

sets; I shall also include more general variational objects called almost

minimizers or omega-minima. Open questions will be discussed at the end.

We consider Brownian motion on a manifold conditioned not to leave

the tubular neighborhood of a closed riemannian submanifold up

to some fixed finite time. For small tube radii, it behaves like the

intrinsic Brownian motion on the submanifold coupled to some

effective potential that depends on geometrical properties of

the submanifold and of the embedding. This characterization

can be applied to compute the effect of constraining the motion of a

quantum particle on the ambient manifold to the submanifold.

Dirac-Born-Infeld Kinematics