Abstract: In a relatively general setting, it is known that decay estimates for scalar oscillatory integrals imply volume estimates for the sublevel sets of the phase function, and this implication is stable in the sense that whenever oscillatory integrals with different phases all satisfy the same estimates, the resulting estimates for sublevel sets are uniform over that set of phase functions. Reversing this implication is much more difficult, and understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall consider some situations in which the reverse implication holds in the presence of additional, qualitative assumptions on the structure of the phase function. We first give some results in one dimension, then address the setting of convex functions in higher dimensions.

Abstract: I am going to present very recent progress on the classical subellipticity problem for the d-bar-equation on a smooth pseudoconvex domain in C^n. Despite deep contributions by Kohn, Greiner, Stein, Catlin and D'Angelo (among others), the question of determining the sharp order of subellipticity in terms of the boundary geometry of the domain is wide open.

I am not going to assume familiarity with the theory of the d-bar-equation in several complex variables. This is joint work with S. Mongodi (Università Milano-Bicocca).

Abstract: This talk concerns a circle of questions on L^p majorant properties of trigonometric polynomials going back to work of Hardy and Littlewood in the 1930s.

After a brief historical review, we will focus on the strict majorant property in d dimensions and

give a necessary and sufficient condition for frequency sets that determines when this property must fail for some p>0.

This is joint work with Phil Gressman, Shaoming Guo, Lillian Pierce and Po-Lam Yung.

Abstract: This talk will focus on small-data global existence results for coupled wave and Klein-Gordon equations in two and three spatial dimensions. Surprisingly, these problems are less systematically understood, and several open systems remain across the possible spectrum of wave--wave, wave--KG and KG--KG interactions. I will particularly focus on the Dirac equation which, depending on the Dirac field mass, can encode both wave or Klein-Gordon behaviour. I will present a recent proof of small data global-existence and sharp asymptotics for a Dirac--Klein-Gordon system in two spatial dimensions with Shijie Dong (SUSTech).