Jonathan Hickman

Me

I am currently a Research Fellow in the analysis group at the University of St Andrews. Previously, I was an L. E. Dickson Instructor in the University of Chicago. I completed my PhD in the University of Edinburgh under the supervision of Prof. Jim Wright. A brief 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

Improved Fourier restriction estimates in higher dimensions, with Keith Rogers. Preprint.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, with David Beltran and Christopher D. Sogge, submitted. Preprint.

Counting factorisations of monomials over rings of integers modulo \(N\), with Jim Wright, submitted. Preprint.

Sharp estimates for oscillatory integral operators via polynomial partitioning, with Larry Guth and Marina Iliopoulou, submitted. Preprint.

An abstract \(L^2\) Fourier restriction theorem, with Jim Wright, to appear Math. Res. Let. Preprint.

Some remarks on the Lipschitz regularity of Radon transforms, with Jonas Azzam and Sean Li, Proc. Amer. Math. Soc., v. 146, no. 10, pp. 43314337, (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), volume 115, pp. 177-219, (2017). Preprint.

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

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

Teaching

Whilst in Chicago I taught a number of courses.

During 2017-2018 I lectured Math 162 Honours Calculus and Math 163 Honours Calculus.

During 2016-2017 I lectured Math 153 Calculus and Math 273 Basic Theory of Ordinary Differential Equations.

During 2015-2016 I lectured the Math 160 Honours Calculus series and Math 256 Algebra.

Expository notes

Together with Marco Vitturi and Odysseas Bakas I presented a series of lectures describing Bourgain and Demeter's recent work on Wolff-type decoupling inequalities: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5.

I also have some (somewhat rougher) notes on generalised Radon transforms, which were written to accompany a course I gave in the Analysis Working Group in Edinburgh: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Picture 1, Picture 2.

I once wrote an essay on Furstenberg's proof of Szemeredi's theorem.

Here are slides from expository talks on Szemeredi's theorem, Roth's theorem and Kakeya sets.

I'd be grateful to hear about any mistakes / typos!

Contact

E-mail: jeh25 [at] st-andrews [dot] ac [dot] uk

Address:
Mathematical Institute,
North Haugh,
St Andrews,
Fife,
KY16 9SS.