Jonathan Hickman

I am lecturer in the analysis group at the University of Edinburgh. Previously, I held postdoctoral positions in the University of Chicago, MSRI and the University of St Andrews. A CV can be found here.

Research

My interests lie in Euclidean harmonic analysis, in particular questions pertaining to operators whose definition depends on some submanifold of $$\mathbb{R}^n$$ such as Fourier restriction/extension operators and generalised Radon transforms. I am also interested in discrete analogues of these objects.

Papers: research

• A non-archimedean variant of Littlewood--Paley theory for curves, with Jim Wright, submitted. Preprint.

• Sharp $$L^p$$ bounds for the helical maximal function, with David Beltran, Shaoming Guo and Andreas Seeger, accepted Amer. J. Math.. Preprint.

• A note on Fourier restriction and nested polynomial Wolff axioms, with Joshua Zahl, accepted J. d'Analyse Math.. Preprint.

• Improved bounds for the Kakeya maximal conjecture in higher dimensions, with Keith Rogers and Ruixiang Zhang, accepted Amer. J. Math.. Preprint.

• The circular maximal operator on Heisenberg radial functions, with David Beltran, Shaoming Guo and Andreas Seeger, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23, no. 2, 501--568, (2022). Preprint.

• Sharp $$L^p$$ estimates for oscillatory integral operators of arbitrary signature, with Marina Iliopoulou, Math. Z.. 301, no. 1, 1143-1189, (2022). Preprint.

• Sobolev improving for averages over curves in $$\mathbf{R^4}$$, with David Beltran, Shaoming Guo and Andreas Seeger, Adv. Math. 393, 108089, (2021). Preprint.

• Uniform $$L^p$$ resolvent estimates on the torus, Mathematics Research Reports, Volume 1, pp. 31-45, (2020). Preprint.

• Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, with David Beltran and Christopher D. Sogge, Anal. PDE 13, no. 2, pp. 403-433, (2020). Preprint.

• On a higher dimensional version of the Benjamin--Ono equation, with Felipe Linares, Oscar G. Riaño, Keith Rogers and Jim Wright, SIAM J. Math. Anal. 51, no. 6, pp. 4544-4569, (2019). Preprint.

• Sharp estimates for oscillatory integral operators via polynomial partitioning, with Larry Guth and Marina Iliopoulou, Acta Math., vol. 223, no. 2, pp. 251-376 (2019). Preprint.

• Counting factorisations of monomials over rings of integers modulo $$N$$, with Jim Wright, J. Théor. Nr. Bordx., vol. 31, no. 1, pp. 255-282, (2019). Preprint.

• Improved Fourier restriction estimates in higher dimensions, with Keith Rogers, Camb. J. Math., vol. 7, no. 3, pp. 219-282, (2019). Preprint.

• An abstract $$L^2$$ Fourier restriction theorem, with Jim Wright, Math. Res. Lett., vol. 26, no. 1, 75-100, (2019). Preprint.

• Some remarks on the Lipschitz regularity of Radon transforms, with Jonas Azzam and Sean Li, Proc. Amer. Math. Soc., vol. 146, no. 10, pp. 4331-4337, (2018). Preprint.

• The Fourier restriction and Kakeya problems over rings of integers modulo $$N$$, with Jim Wright, Discrete Analysis 2018:11, 54 pp, (2018). Preprint.

• Maximal operators and Hilbert transforms along variable non-flat homogeneous curves, with Shaoming Guo, Victor Lie and Joris Roos, Proc. Lond. Math. Soc. (3), vol. 115, pp. 177-219, (2017). Preprint.

• Uniform $$L^p_x - L^q_{x,r}$$ improving for dilated averages over polynomial curves, J. Funct. Anal., vol. 270, issue 2, pp. 560-608, (2016). Preprint.

• An affine Fourier restriction theorem for conical surfaces, Mathematika, vol. 60, issue 02, pp. 374-390, (2014). Preprint.

Papers: expository

• Sharp local smoothing estimates for Fourier integral operators, with David Beltran and Christopher D. Sogge, Geometric aspects of harmonic analysis, 29–105, Springer INdAM Ser., 45, Springer, Cham, (2021). Preprint.

Journals

I am an editor for Proceedings of the Royal Society of Edinburgh (Section A), specialising in harmonic analysis. This is a generalist journal considering high quality submissions across the whole of mathematics. If you wish to submit a paper or feature article, please follow the instructions on the journal webpage.

Fourier analysis @200

• Fourier Analysis @ 200
• Young Researchers Symposium

A one-week workshop, Fourier Analysis @200, was held 27 June - 1 July 2022.

The workshop was hosted by the ICMS in Edinburgh, Scotland, and marks the 200th anniversary of the publication of Fourier's treatise Théorie analytique de la chaleur. The aim was to bring together experts in harmonic analysis with interests in its many exciting (and expanding) interfaces with other areas of mathematics.

The workshop also took the opportunity to celebrate the 60th birthday of Jim Wright and his many contributions to modern Fourier analysis.

As a satellite event there was a 2-day Young Researchers Symposium at the ICMS in Edinburgh,   23rd June - 24th June 2022.

Seminar / colloquia / working groups / network meetings

I am or have been involved in the organisation of the following:

Teaching

During 2022/2023 I am teaching the following courses:
• SMSTC Measure and Integration (graduate), semester 1.
• Facets of Mathematics (undergraduate), semester 1, cotaught with Dr. James Maddison and Dr. Ozan Evkaya.
• Fourier analysis (undergraduate), semester 2.
During 2021/2022 I taught the following courses:
During 2020/2021 I taught the following courses:
During 2019/2020 I taught the following courses:

Contact

E-mail: jonathan.hickman [at] ed [dot] ac [dot] uk