I will be discussing a problem I am working on involving the spectral properties of Schrdinger operators with Sturmian potentials. This problem originated with a Nobel Prize-winning observation by a chemist in 1984. After being studied by math physicists, it was reduced to the purely dynamical behavior of a polynomial map on a manifold. The word "interdisciplinary" in the title of this talk refers both to a mathematical problem arising in another discipline and to the collaboration within mathematics across different research areas.

 
This talk will discuss the problem of finding effective laws that the govern the overall evolution of free boundaries propagating in a heterogeneous media.  This is motivated by a number of phenomena in mechanics and materials physics including phase boundaries, peeling of adhesive tape, dislocations, fracture and wetting fronts.   While there is a rigorous mathematical theory of homogenization in the context of properties that are characterized by a variational principle, much remains unknown about equations that describe evolutionary processes.   The talk will discuss the mathematical issues, difficulties and results, and illustrate the implication on materials through selected examples.  The talk will conclude with current work on free discontinuity problems.

 
The Cauchy-Riemann operator for domains in a complex manifold is well understood for domains in complex spaces. However, much less is known for the solvability and regularity for the Cauchy-Riemann operator in a complex manifold which is not complex spaces  or Stein. Recently, some progress has been made for the L2 theory of the Cauchy-Riemann equations on product domains in complex manifolds. An analogous formula of the classical Kunneth formula for the harmonic forms are also obtained. We have also discuss an L2 version of the Serre duality for domains on complex manifolds. Furthermore, duality between the harmonic spaces and the Bergman space in complex manifolds will also be presented (Joint work with Debraj Chakrabarti).

Cancer stem cells (CSCs) have been identified in primary breast cancer tissues and cell lines. The size of CSC population varies a lot among cancer tissues and cell lines but is associated with aggressiveness of breast cancer. In this study, we develop a mathematical model to explore the key factors which control the size of CSC during tumor cell growth both in vitro and in vivo. Our mathematical model and experimental data suggest that there is a negative feedback mechanism to control the balance between CSC and non-stem cancer cells. We further calculate how feedback sensitivities and robustness can be regulated by different intrinsic and extrinsic factors.

We show how to efficiently count exactly the number of solutions of a system of n polynomials in n variables over certain local fields L, for a new class of polynomials systems. The fields we handle include the reals and the p-adic rationals. The polynomial systems amenable to our methods are made up of certain A-discriminant chambers, and our algorithms are the first to attain polynomial-time in this setting. We also discuss connections to Baker's refinement of the abc-Conjecture, Smale's 17th Problem, and tropical geometry. The results presented are, in various combinations, joint with Martin Avendano, Philippe Pebay, Korben Rusek, and David C. Thompson.