Corona problem was initiated from function algebra, then it became one of the most important problems in harmonic analysis and complex analysis. For the case of one complex variable, the problem was solved by L. Carleson in 1960s. In the several complex variables, the problem remains open. In this talk, I will introduce the development of the Corona Problems of one and several complex variables. I will also introduce the Hormander’s weighted L2-estimate for ∂, and demonstrate how to use the weighted L2 estimates to prove some old and new results on Corona Problem.

We provide a general introduction to the field of inverse boundary problems for elliptic PDE, with the celebrated Calderon problem serving as a prototypical example. The emphasis of the talk is on the techniques based on Carleman estimates and their role in the construction of complex geometric optics solutions. We also survey some of the more recent developments, including partial data problems, inverse boundary problems on manifolds, as well as inverse boundary problems for non-linear equations.

A typical result in graph/hypergraph theory has the following structure: Every G satisfying certain conditions must have some target property P. For example, a classical theorem by Dirac asserts that every graph on n vertices and with minimum degree at least n/2 must contain a hamiltonian cycle (that is, a cycle that passes through every vertex).

After establishing such a theorem, it is natural to ask how ``robust'' is G with respect to this property P. In this talk we discuss some possible measures of ``robustness'' and illustrate them with many examples.