Professor Fred Wan will describe some highlight of his 50 years long journey in applied mathematics research at four institutions. He will touch on scientific topics such as mechanics, economics, and the biology of Morphogens that guide tissue development.

In this talk, we consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. We establish a uniform boundary Harnack principle with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b = \Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets.

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.