Dominique Maldague
University of California-Los Angeles, CA·Mathematics & Statistics
The Fourier transform breaks a signal or function into its component frequencies. It underlies medical imaging such as MRI, engineering, and physics, including the study of waves and of the Schrödinger equation from quantum mechanics.
- Fields
Mathematics & Statistics
- Topics
harmonic analysis
Fourier transform
Fourier restriction
- Grants
- 1 known grant, through Jun 2029
What they work on
- Study special cases of the Mizohata-Takeuchi conjecture in the plane. This conjecture bounds how waves can concentrate along tubes, and the work will use high-low decoupling together with new geometric insights about weights and wave packets.
- Begin a systematic classification of the settings in which Mizohata-Takeuchi estimates can be proved, drawing on recent breakthroughs on Kakeya-type questions about tubes pointing in many directions.
- Prove moment estimates for cubic Weyl sums, which are exponential sums central to number theory, in one and two dimensions, using refined Fourier restriction and decoupling tools.
- Determine whether the main obstacles to cubic Weyl sum estimates are geometric or arithmetic in nature, building on multiscale geometry of balls and tubes.
More about the research
This project studies how the Fourier transform behaves in special geometric settings. It then uses that understanding to make progress on hard problems in number theory and fractal geometry.
Why it matters
This is basic mathematical research. It aims to sharpen what is known about the Fourier transform near curved surfaces and tube-like structures. The results could open new routes to long-standing exponential sum problems in number theory that classical estimates cannot reach. They could also strengthen links between harmonic analysis, fractal geometry and number theory. Better understanding of this kind may eventually inform the analysis of wave and Schrödinger equations used in physics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Ciprian Demeter
Indiana University, IN·Mathematics & Statistics
Develop methodology to analyze the Schrödinger maximal function in the periodic setting, incorporating Fourier analysis and finer number theory into the existing combinatorial framework for this measure of worst-case solution size.
Frank Thorne
University of South Carolina-Columbia, SC·Mathematics & Statistics
Carry out finite Fourier transform computations for families of arithmetic objects, extending approaches the PI and collaborators have previously developed in arithmetic statistics.
Hang Xue
University of Arizona, AZ·Mathematics & Statistics
Study the local Gan-Gross-Prasad conjecture for real orthogonal groups (symmetry groups that preserve lengths) using theta correspondences together with a new relative version of Shahidi's local coefficients.
The school
University of California-Los Angeles: 84 labs on PI Pages, $1.9B a year on research, 803 research doctorates a year.
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