Mini Courses (FMSP Lectures)Lecture Hall, Graduate School of Mathematical SciencesMatthew Gursky (Univ. Nortre Dame) March 17, 18 and 19, 9:00--9:50 at Lecture Hall Critical metrics for quadratic Riemannian functionals in dimension four Abstract: In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky. These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP^{2} and the product metric on S^{2 }X S^{2}. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.I will also explain some recent work which attempts to understand the moduli space of critical metrics. Gábor Székelyhidi (Univ. Nortre Dame) March 17, 18 and 19, 10:10--11:00 at Lecture Hall. Hessian type equations on compact Kähler manifoldsAbstract: I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations. |