We consider three constitutive relations to describe tumor growth: Darcys law, Stokes law, and the combined Darcy-Stokes law. Darcys law is used to describe fluid flow in a porous medium. Stokes law describes the flow of a viscous fluid. In this talk, we will discuss using linear theories to study tumor shape stability (the ability of the tumor to return to being spherical or exhibit protrusions) described by the three physical relations and to evaluate the consistency between theoretical model predictions and experimental data. The motivation behind this work is that shape instabilities (growing protrusions) are associated with local invasiveness, which is often a precursor to tumor metastasis (infiltration of the distant organs). We will discuss the results and further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modeling framework that can be extended to the multiscale study of cancer. Numerical methods are used to simulate the nonlinear effects of stress on solid tumor growth and invasiveness.

In this talk we consider linear periodic differential-algebraic equations (DAEs) that depend analytically on a spectral parameter. In particular, we extend the results of M. G. Kre ̆ın and G. Ja. Ljubarski ̆ı [Amer. Math. Soc. Transl. (2) Vol. 89 (1970), pp. 128] to linear periodic DAEs of definite type and study the analytic properties of Bloch waves and their Floquet multipliers as functions of the spectral parameter.
Our main result is the connection between a non-diagonalizable Jordan normal form of the monodromy matrix for the reduced differential system associated with the DAEs and the occurrence of slow Bloch waves for the periodic DAEs, i.e., Bloch solutions of the periodic DAEs which propagate with near zero group velocity.
We show that our results can be applied to the study of slow light in photonic crystals [A. Figotin and I. Vitebskiy, Slow Light in Photonic Crystals, Waves Random Complex Media, 16 (2006), pp. 293382].

The limit shape of Young diagrams under the Plancherel
measure was found by Vershik \& Kerov (1977) and Logan \& Shepp
(1977). We obtain a central limit theorem for fluctuations of Young
diagrams in the bulk of the partition '`spectrum''. More
specifically, under a suitable (logarithmic) normalization, the
corresponding random process converges (in the FDD sense) to a
Gaussian process with independent values. We also discuss a link
with an earlier result by Kerov (1993) on the convergence to a
generalized Gaussian process. The proof is based on poissonization
of the Plancherel measure and an application of a general central
limit theorem for determinantal point processes. (Joint work with
Zhonggen Su.)