Research

My interests lie in Euclidean harmonic analysis and analytic number theory with particular attention to their interactions.

A few of the topics I study include:

Maximal functions over hypersurfaces - continuous and discrete
Oscillatory integral and exponential sum estimates
discrete restriction theory, l^2 decoupling and efficient congruencing
Waring's problem
Variations of averages

Why these interests? Like many others, seemingly simple questions in number theory such as the distribution of primes seduced me. While an undergraduate I devoured as much analytic number theory as I could. My interest in harmonic analysis began in an incredible first year graduate analysis course with Bill Beckner.

Euclidean Harmonic analysis

I trained in the Calderon--Zygmund school of Euclidean harmonic analysis (my PhD advisor was Elias Stein). Thus I focus on maximal functions, singular integrals and oscillatory integrals. One class of objects I am interested in are maximal functions over hyper-surfaces such as Stein's spherical maximal function. For more general hyper-surfaces than the sphere, there is a nice interplay between the geometry of the hyper-surface and the boundedness of its maximal function. For instance Stein's spherical maximal function has the best range of boundedness since its curvature is everywhere bounded away from 0. To some extent, less curvature means weaker boundedness properties. An interesting consequence of bounds for these maximal functions is pointwise convergence of the averages under ergodic actions.

A particular interesting question in this direction: Is the lacunary circular maximal function weak-type (1,1)?

The results to date are

C. Calderon and Coifman--Wiess proved that the lacunary spherical maximal function is bounded on L^p(R^d) for all p > 1 and d > 1. Extrapolation arguments imply that the lacunary circular maximal function is bounded on L log L.
Christ showed that the lacunary spherical maximal function is bounded from H^1 to weak L^1
Seeger--Tao--Wright proved that the lacunary spherical maximal function is bounded from L^1 to L log log L.
Cladek--Krause improved the work of Seeger--Tao--Wright to a result closer to L^1.

Discrete restriction theory

I am also interested in Stein's restriction problem and its analogues. In 2014, I adapted Bourgain's arithmetic version of the Tomas--Stein argument to obtain sharp estimates in the super-critical regime for various families of hypersurfaces such as those arising in Waring's problem and Vinogradov's mean value conjecture. This work went unpublished because shortly after I completed it there was spectacular progress on Vinogradov restriction: Bourgain--Demeter resolved the l^2 decoupling conjecture, followed by work of Wooley and Bourgain--Demeter, culminating with Bourgain--Demeter--Guth's proof of Vinogradov's mean value conjectures. Digesting their decoupling and efficient congruencing techniques is a major focus at the moment.

In late 2015 I returned to this work after renewed interest in epsilon-removal lemmas in the work of Wooley and Henriot. This led to a fruitful collaboration with Kevin Henriot who independently studied similar problems motivated by applications in additive combinatorics. In our work, Kevin and I studied the restriction theory for k^th powers generalizing previous results of Bourgain for squares and for k-paraboloids generalizing previous results of Bourgain for paraboloids. Our emphasis is on trying to prove the sharp supercritical range of epsilon-removal lemmas; we obtain truncated restriction results in the supercritical regime along the way using Bourgain's circle method approach.

In Fall 2020, I returned again to these problems; Brian Cook, Eyvi Palsson and I extended my 2014 results to "Birch--Magyar hypersurfaces". There is much room for improvement in the ranges of our bounds as they are far from the expected truth.

A tantalizing open problem here, which Ciprian Demeter described as "one of Bourgain's favorite problems", is to obtain epsilon-removal lemmas in the subcritical regime.

Arithmetic Analogues in Harmonic Analysis

With Bourgain's work in the late '80s and early 90's the area of discrete analogues in harmonic analysis or as I prefer to call it arithmetic analogues in harmonic analysis has undergone a recrudescence. Most of my work to date combines ideas and techniques from harmonic analysis and analytic number theory.

Discrete spherical maximal functions and Waring's problem

I would like to understand the role of curvature in discrete averages and maximal functions much like Stein's program to understand maximal functions over hypersurfaces in the Euclidean setting. A natural family of hypersurfaces to study are the `k-spheres' arising in Waring's problem. In my Ph.D. thesis (preprint of the published paper linked here) I studied discrete maximal functions associated to the hyper-surfaces of Waring's problem. This built on work of Magyar and Magyar--Stein--Wainger (MSW). I attempted to find the sharpest possible ranges of l^p spaces. Improvements were technical, generalizing and tightening up arguments from MSW by incorporating better estimates for exponential sums and oscillatory integrals than used previously. A more recent development in this area is to consider a singular variant where the averages are not over integral points on spheres but instead over points whose coordinates are prime. This is a singular set since the density of such points tends to zero as the spheres increase in size. Therefore the boundedness of the maximal function over these averages does not follow from that of MSW. In work with Tess Anderson, Brian Cook and Angel Kumchev we studied the analogous problems from over the integers. We obtained the sharp range of p in the l^p(Z^d) spaces when the dimension was sufficiently large (d >= 7 dimensions/variables). However we believe that the range of dimension should match the integer case of MSW; that is, only 5+ dimensions/variables should be sufficient.

Conversations with Lillian Pierce encouraged me to study lacunary versions of discrete spherical maximal functions. This problem is surprisingly different from the Euclidean case. In the Euclidean case, C. Calderon and Coifman--Wiess proved that the lacunary spherical maximal function is bounded on L^p(R^d) for all p > 1 and d > 1. In fact this is simple to prove. By analogy, one naturally expects that the discrete spherical maximal function over a lacunary sequence of radii is bounded on l^p(Z^d) for all p>1 when d>4. Naively, I thought that this would be easy. In contrast, it is difficult to improve the range of l^p boundedness for the discrete spherical maximal function over a lacunary sequence of radii over the discrete spherical maximal function over the full set of radii. After spending months on this problem I was only able to make a sliver of progress in this paper. Not long after that paper was finished, Zienkiewicz showed me that the natural conjecture (which was folklore for some time) is false! Zienkiewicz showed that there are infinite sequences of radii that are arbitrarily thin, yet their associated maximal function is unbounded on l^p for 1 <p < p_d where p_d is a range that depends on dimension. One can generalize the problem to more general sets of radii and this sheds some light on the problem. Stein--Wainger, in unpublished work, investigated the Euclidean circular maximal function over a Cantor set of radii. Stein--Wainger's work motivated Seeger--Wainger--Wright, Duoandikoetxea--Vargas and more recently Seeger--Tao--Wright to study spherical maximal functions over other restricted sets of radii.Working to understand Zienkiwicz's counterexamples Jim, Jacek and I observed a deeper connection between this work and lacunary discrete spherical averages. This is work in progress; hopefully a preprint will be available in the fall of 2016.

Regularity of the Hardy--Littlewood maximal function

Also during my Ph.D. I worked with Jonathan Bober, Emanuel Carneiro and Lillian Pierce on the regularity of the discrete Hardy--Littlewood maximal function. The questions in this area begin with the observation that a nice averaging operator smooths functions. Then, heuristically, the maximal function for a nice sequence of averaging operators should not destroy any smoothness. With this in mind, Hajlasz--Onninen conjectured that the variation of the Hardy--Littlewood maximal function (that is, the HL maximal operator applied to a fixed function) is no worse than the variation of the original function. Several people studied this question with the L^p theory for p>1 becoming quickly well established. This left open the question about the L^1 theory because of the main difficulty that the Hardy--Littlewood maximal function is unbounded on L^1. Nevertheless, one can ask the question:

For a function f, is the total variation of its maximal function Mf bounded by the total variation of f?

Tanaka showed that this is true for the uncentered Hardy--Littlewood maximal function, but the centered maximal function remained open.

I was introduced to this problem in 2010 when my friend Emanuel Carneiro was a postdoc at the Institute for Advanced Study. This was my third year of grad school and after floundering for the first two years, trying to figure out what I wanted to work on, I settled on discrete analogues after attending Lillian Pierce's final public oral (aka defense) at the end of my second year. Naturally, Emanuel and I asked: what happens for these questions when one considers the discrete analogue; that is, maximal function on Z^n rather than R^n? We focused on the one dimensional case. Our first challenge was to determine what notion of variation we wanted to go with. The l^p theory followed immediately as in previous work of others, and we quickly focused on the l^1(Z) theory for the centered maximal function. This presented a major challenge to us during the fall of 2010 as there did not exist any arguments in other setting to try to adapt. In the spring Emanuel gave a seminar on the problem where Lillian Pierce and Jonathan Bober became interested. The four of us worked on this weekly until Emanuel moved back to Brazil. Fortunately shortly before that, Emanuel had a critical insight that showed that the variation of the maximal function is bounded by the l^1 norm of the function. This is weaker than bounding by the variation of the function, but was the first result supporting Hajlasz--Onninen's conjecture for the centered maximal function. Later Kurka proved that the desired result in R^1; however, the situation for higher dimensions remains open. As I finished my PhD studies, I visited Emanuel, and we extended our one dimensional work with Lillian and Bober to Hardy--Littlewood maximal functions over convex bodies in higher dimensions. After this, I turned my attention to other problems, but Emanuel and his students continued work on these problems, obtaining several interesting results. Many others have begun to consider this topic.

Last update: 7 March 2019