Given some class of "geometric spaces", we can make a ring as follows.

1. (additive structure)  When U is an open subset of such a space X, [X] = [U] + [(X \ U)];
2. (multiplicative structure)  [X x Y] = [X] [Y].

In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be the case of "points on a line" --- polynomials in one variable.  (This talk is intended for a broad audience.)  This is joint work with Melanie Matchett
Wood.

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
(Joint work with K.Grove and G. Thorbergsson)

 
Thermoacoustic (TAT) and Photoacoustic Tomography (PAT) are examples of multiwave imaging methods allowing to combine the high imaging contrast of one wave (an electromagnetic or a photoacoustic one) with the high resolution of ultrasound. We present recent results obtained in collaboration with Gunther Uhlmann, Jianliang Qian and Hongkai Zhao on the mathematical theory behind TAT, PAT and other multiwave methods. We allows the acoustic speed to be variable, and consider the partial data case as well. We will also discuss the case of a discontinuous speed modeling brain imaging.  Numerical reconstructions will be shown as well.
Most of the progress is due to the use of microlocal methods. One of the goals of the talk is to show the usefulness of microlocal methods to solving real life 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).