Abstract: This talk will provide an overview of recent developments in Fourier restriction theory, which is the study of exponential sums over restricted frequency sets with geometric structure, typically arising in pde or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight recent work which uses the latest tools developed in decoupling theory to prove much more delicate sharp square function estimates for frequencies lying in the cone in R^3 (Guth-Wang-Zhang) and moment curves (t,t^2,...,t^n) in all dimensions (Guth-Maldague).
Abstract: This talk showcases the speaker’s recent results in the field of Optimal Recovery, viewed as a trustworthy Learning Theory focusing on the worst case. At the core of several results presented here is a scenario, resolved in the global and the local settings, where the model set is the intersection of two hyperellipsoids. This has implications in optimal recovery from deterministically inaccurate data and in optimal recovery under a multifidelity-inspired model. In both situations, the theory becomes richer when considering the optimal estimation of linear functionals. This particular case also comes with additional results in the presence of randomly inaccurate data.
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