Title: Whitney-type Extension and Trace Problems and Convexity in Analysis
Speaker: Majorie Drake
Abstract: We will introduce analysis research in Whitney-type extension and trace problems. While theoretical in nature, this field is motivated by practical questions that arise in the interpolation of data. We will discuss recent innovation and opportunities for more research in two areas: understanding constrained interpolation when the constraint is nonlinear (i.e., convexity or nonnegativity) and understanding extension in the Sobolev space L^{m,p}(R^n) for the range of exponents n/m<p<n (where there is no embedding in Holder spaces). We will also discuss open conjectures about the smooth approximation, interpolation, and extension of convex functions.( Because of the role convexity plays in optimization, calculus of variations, and partial differential equations, the smoothness, or regularity, of convex functions is often required.)