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Harmonic Analysis witnessed a number of breakthroughs in recent years. The most famous one is perhaps the Kakeya conjecture in three dimensions, recently solved by Wang and Zahl. Very roughly speaking, the major conjectures in the field are either of oscillatory or non-oscillatory nature. A major modern challenge is to build bridges between these universes by using tools of the latter kind to solve problems of the former.
The quest of blowup solutions to the NSE remains a challenging topic. We will discuss a recent construction that illustrates a blowup phenomenon unexplored previously for the NSE. In particular, the constructed solutions start from smooth initial data and exhibit an instantaneous blowup saturating Type-I blowup rate at a finite time. Moreover, there are infinitely many such blowup solutions; and the non-uniqueness occurs in borderline spaces of known criteria which ensure uniqueness. This is joint work with Alexey Cheskidov and Stan Palasek.
We completely characterize the range of $L^p$-boundedness of certain multilinear Radon-like transforms involving vertical projections in the Heisenberg group. This result is now available on arXiv:2603.17147.
I will present on recent work - joint with John Green, Terence Harris, Kevin Ren, and Yumeng Ou - towards proving lower bounds for the dimensions of Furstenberg sets of circles and sine curves in the plane. A circular $(u,v)$-Furstenberg set is a set that contains a $u$-dimensional subset of each circle from a $v$-dimensional family of circles.
The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved.
The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem extends the classical restriction theorem for measures on smooth manifolds to fractal measures. We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions. The proof uses number fields to construct fractal measures in R^d. This work is joint with Robert Fraser and Kyle Hambrook.
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