Abstract: There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been recent work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does old partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about an old argument that shows VMVT "asymptotically" due to Karatsuba and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung. 

Abstract: We discuss families of limit functionals and weak type (quasi)- norms which represent the standard Sobolev norms, extending and unifying work by Nguyen and by Brezis, Van Schaftingen and Yung. We also consider versions with fractional smoothness and applications, including a characterization of approximation spaces for nonlinear wavelet approximation. 

Joint works with H. Brezis, J. Van Schaftingen and P. Yung, and with Ó. Domínguez, B. Street, J. Van Schaftingen and P. Yung. 

Abstract: Unique continuation property is a fundamental property of harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes to infinite order at a point, it must be zero everywhere. In the same spirit, we can use the local growth rate of harmonic functions to deduce global information, such as estimating the size of the singular set for elliptic PDEs. This is joint work with Carlos Kenig. 

Abstract: We will discuss bi(sub)linear and tri(sub)linear embeddings for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form. Bilinear embeddings can be thought of as sharpenings and generalizations of estimates for second-order singular integrals. In the context of complex elliptic operators such $L^p$ bounds were shown by Carbonaro and Dragičević, who emphasized and crucially used certain generalized convexity properties of powers. We remove this obstruction and generalize their approach to the level of Orlicz-space norms that only “behave like powers”. Next, what we call a trilinear embedding is a paraproduct-type estimate. It incorporates bounds for the conical square function and finds an application to fractional Leibniz-type rules. In the proofs we use two carefully constructed auxiliary functions that generalize a classic Bellman function constructed by Nazarov and Treil in two different ways. The talk is based on joint work with Andrea Carbonaro, Oliver Dragičević, and Kristina Škreb. 

Abstract: Since Szemerédi's seminal work in the 70s, regularity lemmas have proven to be of fundamental importance in many areas of discrete mathematics. This talk will survey recent work on regularity decompositions of subsets of finite groups under additional assumptions such as stability or bounded VC-dimension, which turn out to have particularly desirable properties. In the second half of the talk, we will describe very recent joint work with Caroline Terry (Ohio State University) which extends these ideas to the realm of higher-order Fourier analysis.

Abstract: We discuss a range of Lp bounds for singular Brascamp-Lieb forms with cubical structure. This extends an earlier result which only allowed for a single tuple of the Lebesgue exponents.  We pass through local and sparse bounds. This is a joint work with L. Slavíková and C. Thiele.