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.

PhD Opportunities

We encourage any interested students to apply to our programme, in particular women and members of underrepresented groups. There is no application fee and successful applicants are fully funded: see here for further details.

Research

My interests lie in Euclidean harmonic analysis, in particular questions about 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

On a planar Pierce--Yung operator, with David Beltran and Shaoming Guo. Preprint.
Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates, with Mingfeng Chen, Shengwen Gan, Shaoming Guo, Marina Iliopoulou and Jim Wright. Preprint.
Localised variants of multilinear restriction, with David Beltran and Jennifer Duncan, 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, J. Anal. Math. 152, no. 1, 19–52, (2024). Preprint.
Off-diagonal estimates for the helical maximal function, with David Beltran and Jennifer Duncan. Proc. Lond. Math. Soc. (3) 128, no.4, (2024). Preprint.
Lacunary maximal functions on homogeneous groups, with Aswin Govindan Sheri and Jim Wright, J. Funct. Anal 286, no. 3, (2024). Preprint.
A non-archimedean variant of Littlewood--Paley theory for curves, with Jim Wright, J. Geom. Anal. 33, no. 3, Paper No. 104, (2023). Preprint.
Improved bounds for the Kakeya maximal conjecture in higher dimensions, with Keith Rogers and Ruixiang Zhang, Amer. J. Math. 144, no. 6, 1511–1560, (2022). 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

Pointwise convergence for the Schrödinger equation [After Xiumin Du and Ruixiang Zhang], Séminaire Bourbaki, 75^e année, 2022–2023, n^o 1205, Avril 2023. Preprint.
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.

PhD supervision

Hrit Roy, 2021 -- .
Aswin Govindan Sheri, MIGSAA, 2018 -- 2023.

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 2023/2024 I am on sabbatical in semester 1 and am teaching the following course in semester 2:

Fourier analysis (undergraduate), semester 2.

During 2022/2023 I taught the following courses:

SMSTC Measure and Integration (graduate), semester 1.
Fourier and Harmonic Analysis (graduate reading course), semester 1.
Facets of Mathematics (undergraduate), semester 1, cotaught with Dr. James Maddison and Dr. Ozan Evkaya.
Fourier analysis (undergraduate), semester 2.

Previously, I have organised the following courses (accompanying notes may be of wider interest):

Mini lecture series on Bourgain's circular maximal theorem (graduate, 2021/22).
Incidence geometry: continuous and discrete (graduate, 2020/21).
Modern Developments in Fourier Analysis (graduate, 2019/20).

Media

Contact

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

Address:
Room 5322,
James Clerk Maxwell Building,
The King's Buildings,
Peter Guthrie Tait Road,
Edinburgh,
EH9 3FD.