There are very few known examples of compact Kahler manifolds that admit a holomorphic symplectic form. Two series of examples were discovered by Beauville in the 80's and then two sporadic (or exceptional) examples of dimension 6 and 10 were found by O'Grady. In this talk I will review the construction of these examples and describe some recent work describing the geometry of the two exceptional examples. One way to achieve this is to use Lagrangian fibrations. I will review what a Lagrangian fibration is as well as its basic properties, and then show how this structure allows one to study or construct examples of holomorphic symplectic manifolds.
Holomorphic symplectic manifolds and Lagrangian fibrations Sponsored by the Meyer Fund
Apr. 01, 2016 3pm (ECCR 265)
Kempner
John Hunter (University of California Davis)
X
Surface plasmons (SPs) are electromagnetic surface waves that propagate on the interface between an insulator and a conductor, such as air and gold. At short wavelengths, the frequency of SPs approaches a nonzero constant independent of their wavenumber, and we consider the effect of nonlinearity on SPs in this nondispersive limit. We show that the amplitude of weakly nonlinear SPs satisfies a cubically nonlinear, nonlocal asymptotic equation. We discuss the existence of smooth and weak solutions of this equation, and the effects of nonlinearity on the focusing and possible formation of singularities in SPs. This is joint work with Ryan Halabi.