In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss new convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.

Over a series of four papers (three with John Loftin, Rutgers-Newark) we've built up an understanding of how to define and study a moduli space for equivariant minimal surfaces in noncompact symmetric spaces using Higgs bundles. I'll talk about what we learned from studying the case of maps into the complex hyperbolic plane, whose group of holomorphic isometries is PU(2,1).

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