Adding a column of numbers produces `carries' along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? (Many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae.) The examples give a gentle introduction to the emerging fields of one-dependent and determinantal point processes. This work is joint with Alexei Borodin and Persi Diaconis.

Major outstanding questions regarding vertebrate limb development
concern how the numbers of skeletal
elements along the proximodistal (P-D) and anteroposterior (A-P) axes
are determined and how the shape of
a growing limb affects skeletal element formation. Recently [Alber etal., The morphostatic limit for a
model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified
two-equation reaction-diffusion system
was developed to describe the interaction of two of the key morphogens: the activator and an activator-dependent
inhibitor of precartilage condensation formation. In this talk, I will present a discontinuous Galerkin (DG) finite element method to solve this nonlinear system on complex domains
to study the effects of domain geometry on the pattern generated. Moreover, recently we have extended these previous results and developed a DG finite element model in a moving and deforming domain for skeletal
pattern formation in the vertebrate limb. Simulations reflect the actual dynamics of limb development and
indicate the important role played
by the geometry of the undifferentiated apical zone. This computational model can also be applied to
simulate various fossil limbs.

Three different singularly perturbed eigenvalue problems in perforated
domains, or in domains with perforated boundaries, with direct
biological applications, are studied asymptotically. In the context
of cellular signal transduction, a common scenario is that a diffusing
surface-bound molecule must arrive at a localized signalling region,
or trap, on the cell membrane before a signalling cascade can be
initiated. In order to determine the time-scale for this process,
asymptotic results are given for the mean first passage time (MFPT) of
a diffusing particle confined to the surface of a sphere that has
absorbing traps of small radii. In addition, asymptotic results are
given for the related narrow escape problem of calculating the MFPT
for a diffusing particle inside a sphere that has small traps on an
otherwise reflecting boundary. The MFPT for this narrow escape problem
is shown to be minimized for particular trap configurations that
minimize a certain discrete variational problem (DVP). This DVP is
closely related to the classic Fekete point problem of determining the
minimum energy configuration for repelling Coulomb charges on the unit
sphere. Finally, in the context of spatial ecology, a long-standing
problem is to determine the persistence threshold for extinction of a
species in a heterogeneous spatial landscape consisting of either
favorable or unfavorable local habitats. For a 2-D spatial landscape
consisting of such localized patches, the persistence threshold is
calculated asymptotically and the effects of both habitat
fragmentation and habitat location on the persistence threshold is
examined. From a mathematical viewpoint, the persistence threshold
represents the principal eigenvalue of an indefinite weight singularly
perturbed eigenvalue problem, resulting from a linearization of the
diffusive logistic model.

The analysis of these three PDE eigenvalue problems is based on the
development of a common singular perturbation methodology to treat
localized patches or traps in combination with some detailed
analytical properties of the Neumann Green's function for the
Laplacian. With this asymptotic framework, the problem of optimizing
the principal eigenvalue for the each of these three problems is
reduced to the simpler task of determining optimal configurations for
certain discrete variational problems.

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Speaker: Michael Ward (UBC). Joint work with Dan Coombs (UBC), Alexei
Chekhov (U. Sask), Alan Lindsay (UBC), Anthony Peirce (UBC), Samara
Pillay (JP Morgan), Ronny Straube (Max Planck, Magdeburg).

Spectral properties of discrete Schrodinger operators with potentials generated by substitutions can be studied using so called trace maps and their dynamical properties. The aim of the talk is to describe the recent results (joint with D.Damanik) obtained in this direction for Fibonacci Hamiltonian, and to list some related problems that could potentially turn into research projects for interested graduate students.