Ciprian Demeter
Indiana University, IN·Mathematics & Statistics
Waves traveling in different directions can reinforce or cancel one another, and how much they interact turns out to control problems far beyond physics. The PI has developed a tool called decoupling that quantifies this interaction.
- Fields
Mathematics & Statistics
- Topics
decoupling inequalities
harmonic analysis
Fourier analysis
- Grants
- 1 known grant, through May 2027
What they work on
- 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.
- Obtain sharp estimates for Fourier transforms of fractal measures supported on curved manifolds, combining decoupling with sharp bounds on incidences between balls and tubes.
- Investigate the newly introduced tight decoupling phenomenon and its connections to number theory and to Lambda(p) estimates, which describe when sets of frequencies behave independently.
- Extend decoupling inequalities toward further problems in harmonic analysis, geometric measure theory, and the counting of solutions to Diophantine systems, while making the tools accessible to a wider mathematical community.
More about the research
Its surprising reach comes from a translation: whole numbers can be treated as frequencies, and therefore as waves, so counting the number of whole-number solutions to a system of equations becomes a question about how waves interfere. Diophantine equations, systems of equations in whole numbers, are one example; they also appear in generating the scrambling and diffusion keys used to encrypt data. This translation underlay the PI's resolution of the Main Conjecture in Vinogradov's Mean Value Theorem, a long-standing problem about counting solutions to such systems.
Why it matters
This is foundational mathematics: it sharpens a general method for measuring wave interference and transfers it to counting solutions of whole-number equation systems, the kind that arise in constructing data-encryption keys, and to controlling solutions of the Schrödinger equation. The payoff is deeper understanding and more broadly usable tools across harmonic analysis, geometric measure theory, and number theory, plus training for graduate students in these areas.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Dominique Maldague
University of California-Los Angeles, CA·Mathematics & Statistics
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.
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.
Simon Bortz
University of Alabama, AL·Mathematics & Statistics
Characterize parabolic uniform rectifiability of time-varying boundaries through properties of caloric measure, the distribution of heat flowing in from the boundary, and through estimates for bounded solutions of the heat equation.
The school
Indiana University: 21 labs on PI Pages, $897M a year on research, 438 research doctorates a year.
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