I will report on recent joint work with L. Lanzani on three basic projection operators, each associated to an appropriate domain in C^n. These are: variants of Cauchy-Fantappie integrals; the Cauchy-Szego projection: and the Bergman projection. The results obtained give L^p boundedness of these oprators under minimal smoothness assumptions of the boundary of the domain.
Our present framework for physics is difficult to modify without destroying its marvelous, successful properties. This provides a strong check on theoretical speculations and helps guide us to a small set of candidates for new laws. In this talk, Nima Arkani-Hamed, Professor in the School of Natural Sciences, illustrates these ideas in action by explaining why theoretical physicists knew the Higgs boson had to exist long before it was discovered at the Large Hadron Collider in July 2012. While the discovery of the Higgs is a triumph for both experimental and theoretical physics, its existence opens up a set of profound conceptual paradoxes, whose resolution is likely to involve radical new ideas. The talk concludes with a description of possible avenues of attack on these mysteries, and what we might learn from the LHC in this decade.