# People A-Z

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## JohnGreen

### Research Interests

Harmonic analysis, Real Analysis

### Biographical Statement

I am a PhD student interested in real and harmonic analysis working under the supervision of Jim Wright. I am particularly interested in oscillatory integral estimates with uniformity within a class of functions, and the relation between such estimates and the corresponding sublevel set estimates. Recently, I have been thinking about ways of proving oscillatory integral estimates from the corresponding sublevel set estimates in the presence of certain qualitative, geometric structural assumptions.

Publications and preprints:

1. "Lower bounds on L^p quasi-norms and the Uniform Sublevel Set Problem", Mathematika 67 (2021) 296–323. Published version (open access): https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12076 Preprint: https://arxiv.org/abs/2008.01708

2. "Oscillating spectral multipliers on groups of Heisenberg type", Rev. Mat. Iberoam. (to appear in print), joint work with Roberto Bramati, Paolo Ciatti and James Wright. Online version: https://www.ems-ph.org/journals/of_article.php?jrn=rmi&doi=1302 Preprint: https://arxiv.org/abs/2011.13987

3. "Algebraic Stability of Oscillatory Integral Estimates: A Calculus for Uniform Estimates", preprint. https://arxiv.org/abs/2104.12238

4. "Uniform Oscillatory Integral estimates for Convex Phases via Sublevel Set estimates", preprint. https://arxiv.org/abs/2111.05395

Talks:

1. "Connections between scalar Oscillatory Integrals and Sublevel Set estimates", LMS Harmonic Analysis and PDE Network Meeting, April 2022.

2. "Estimates for Oscillatory Integrals via Sublevel Sets", Madison Analysis seminar, December 2021.

3. "The stationary set estimate: O-minimal structures and oscillatory integrals" (with Reuben Wheeler), University of Edinburgh, November 2021.

4. "Discrete analogues of the Kakeya set conjecture: The cases of Finite Fields and Z/NZ for square-free N", University of Edinburgh, January 2021.

5. "Wigderson & Wigderson's Method: Uncertainty Principles for some Infinite-dimensional Vector Spaces" (with Constantin Bilz), University of Edinburgh, July 2020

6. "Dimension estimates for Besicovitch sets using Additive Combinatorics" (with Michael Tang), University of Edinburgh, December 2019

### Education Background

MMath University of Warwick (2018)

Jim Wright