In this series of two talks I will give an introduction to some of my recent research on the ineffable tree property. The ineffable tree property is a two cardinal combinatorial principle which can consistently hold at small cardinals. My recent work has been on generalizing results about the classical tree property to the setting of the ineffable tree property. The main theorem that I will work towards in these talks generalizes a theorem of Cummings and Foreman. From omega supercompact cardinals, Cummings and Foreman constructed a model where the tree property holds at all of the $\aleph_n$ with $1 < n < \omega$. I recently proved that in their model the $(\aleph_n,\lambda)$ ineffable tree property holds for all $n$ with $1 < n < \omega$ and $\lambda \geq \aleph_n$.

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).