**
The Vinogradov Mean Value Theorem **

[after Wooley, and
Bourgain, Demeter, Guth]

(Bourbaki Seminar, June 2017)

Asterisque, to appear.

(abstract)
(arXiv)
(seminar)

This is the expository essay to accompany my June 2017 Bourbaki seminar
on the resolution of the Vinogradov Mean Value Theorem. 80 pages.

(hide abstract)

**
Endpoint Sobolev and BV Continuity for Maximal Operators **

* with co-authors E. Carneiro and J. Madrid *

J. Functional Analysis, 273 (2017) 3262-3294.

(submitted)

**
On \(\ell\)-torsion in class groups of number fields **

* with co-authors J. Ellenberg and M. Matchett Wood
*

Algebra and Number Theory, to appear.

(abstract)
(arXiv)

For each integer n, we prove an unconditional upper bound
on the size of the n-torsion subgroup of the class group,
which holds for all but a zero-density set of field
extensions of the rationals of degree d, for any fixed d=2,3,4,5
(with the additional restriction in the case d=4 that the
field be non-D4). For sufficiently large n (specified
explicitly), these results are as strong as a previously
known bound that is conditional on GRH. As part of our
argument, we develop a probabilistic "Chebyshev sieve,"
and give uniform, power-saving error terms for the
asymptotics of quartic (non-D4) and quintic fields with
chosen splitting types at a finite set of primes. 25 pages.

(hide abstract)

**
A polynomial Carleson operator along the paraboloid **

* with co-author Po-Lam Yung*

Revista Mat. Ibero., to appear.

(abstract)
(arXiv)

We consider a polynomial phase Carleson operator that is
of Radon type, namely it involves integration over a
paraboloid in Euclidean space of dimension at least 3. We
show that this operator is bounded on L^p for p strictly
between 1 and infinity, for certain classes of polynomial phase (in particular with
restrictions on the first and second order terms). 76 pages.

(hide abstract)

**
Polynomial Carleson operators along monomial curves in the plane **

* with co-authors Shaoming Guo, Joris Roos, Po-Lam Yung
*

Journal of Geometric Analysis, to appear.

(abstract)
(arXiv)

We prove L^p bounds for partial polynomial Carleson operators
along monomial curves in the plane with a phase
polynomial consisting of a single monomial. These
operators are "partial" in the sense that we consider
linearizing stopping-time functions that depend on only
one of the two ambient variables. A motivation for
studying these partial operators is the curious feature
that, despite their apparent limitations, for certain
combinations of curve and phase, L^2 bounds for partial
operators along curves imply the full strength of the L^2
bound for a one-dimensional Carleson operator, and for a
quadratic Carleson operator. 27 pages.

(hide abstract)

**
Averages and moments associated to class numbers of
imaginary quadratic fields **

* with co-author D.R. Heath-Brown *

Compositio. Math., to appear.

(abstract)
(arXiv)

For any odd prime q, consider the q-part of
the class number of the imaginary quadratic field
with square-free discriminant -d. Nontrivial pointwise upper
bounds are known only for q=3; nontrivial upper bounds
for averages of the q-part have previously been known
only for q=3,5 In this paper we prove nontrivial upper
bounds for the average of the q-part for all primes at
least 7, as well as nontrivial upper bounds for certain
higher moments for all primes at least 3. 25 pages.

(hide abstract)

**
Representations of integers by systems of three quadratic forms **

* with co-authors D. Schindler and M. Matchett Wood
*

Proceedings of the London Mathematical Society (2016) doi: 10.1112/plms/pdw027

(abstract)
(arXiv)
(journal)

We apply the circle method to count simultaneous
representations of "almost any"
tuple of three integers by a system of three quadratic
forms. In particular, we develop a three-dimensional
application of Kloosterman-style cancellation on the minor
arcs; inexplicit geometric criteria for smoothness play a
particularly important role. 66 pages.

(hide abstract)

**
Burgess bounds for multi-dimensional short mixed character sums **

Journal of Number Theory, 163 (2016) 172-210.

(abstract)
(arXiv)
(journal)

This paper applies multi-dimensional Vinogradov mean value
theorems in order to prove Burgess bounds for
multi-dimensional short mixed character sums. 33 pages.

(hide
abstract)

**
Lower bounds for the truncated Hilbert transform **

* with co-authors R. Alaifari and S. Steinerberger *

Revista Mat. Ibero., 32, no. 1 (2016) 23 - 56.

(abstract)
(arXiv)
(journal)

Given two intervals I and J on the real line,
we ask whether it is possible to reconstruct a
real-valued function supported on I from knowing its Hilbert transform
on the interval J.
When neither interval is fully contained in the other, this problem
has a unique answer (the nullspace is trivial) but is
severely ill-posed. We isolate the difficulty and show
that by restricting the problem to functions with controlled total variation, reconstruction becomes stable.

(hide abstract)

**
Burgess bounds for short mixed character sums **

* with co-author D.R. Heath-Brown *

Journal of the London Math. Soc., 91, no. 3 (2015) 693-708.

(abstract)
(arXiv)
(journal)

In this paper we introduce Vinogradov mean value theorems in
order to prove Burgess bounds for short mixed character sums.

(hide abstract)

**
Simultaneous integer values of pairs of quadratic forms **

* with co-author D.R. Heath-Brown *

J. Reine Angew. Math., in press

(abstract)
(arXiv)
(journal)

We prove that a pair of integral quadratic forms in 5 or more
variables will simultaneously represent almost all pairs of integers
that satisfy the necessary local conditions, provided that the forms
satisfy a suitable nonsingularity condition. In particular such
forms simultaneously attain prime values if the obvious local
conditions hold. The proof uses the circle method, and
in particular pioneers a two-dimensional version of a Kloosterman
refinement.

(hide abstract)

**
Counting rational points on smooth cyclic covers**

* with co-author D.R. Heath-Brown *

Journal of Number Theory, 132 (2012), pp. 1741-1757.

(abstract)
(pdf)
(arXiv)
(journal)

A conjecture of Serre concerns the number of rational
points of bounded height on a finite cover of (n-1)-dimensional projective
space. In this paper, we achieve Serre's conjecture in the
special case of smooth cyclic covers of any degree when n
is at least 10 and surpass it for covers of degree at
least 3 when n is at least 11. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method.

(hide abstract)

**
On a discrete version of Tanaka's theorem for maximal functions**

*
with co-authors Jonathan Bober, Emanuel Carneiro, and Kevin Hughes
*

Proceedings of the Amer. Math. Soc. 140 (2012) 1669-1680.

(abstract)
(journal)
revised version on (arXiv)

In this paper we prove a discrete version of Tanaka's Theorem for the Hardy-Littlewood maximal operator in one dimension, both in the non-centered and centered cases. Up to date
version available on arXiv.

(hide abstract)

**
A note on discrete fractional integral operators on the Heisenberg group **

Internat. Math. Res. Not. Vol. 2012, No. 1, (2012) 17-33.

(abstract)
(pdf)
(journal link)

We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.

(hide abstract)

**
Discrete fractional Radon transforms and quadratic forms**

Duke Math. Journal 161, No. 1 (2012)
69-106. Correction 162 (2013) 1203-1204.

(abstract)
(pdf)

We consider discrete analogues of fractional Radon
transforms involving integration over paraboloids defined by
positive definite quadratic forms. We prove certain
conditions are sufficient for them to extend to bounded
operators from l^p to l^q for certain p,q. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.

(hide abstract)

**
On discrete fractional integral operators and mean values of Weyl sums**

Bull. London Math. Soc., 43 (2011) 597-612.

(abstract)
(pdf)
(journal)

In this paper we prove new operator norms for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator.
From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number of representations of a positive integer as a sum of s positive k-th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.

(hide abstract)

**
A note on discrete twisted singular Radon transforms**

Mathematical Research Letters 17, no. 4 (2010) 701-720.

(abstract)
(pdf)
(journal)

In this paper we consider three types of discrete
operators stemming from singular Radon transforms. We
first extend an l^p result for translation
invariant discrete singular Radon transforms to a class of
twisted operators including an additional oscillatory
component, via a simple method of descent
argument. Second, we note an l^2 bound for
quasi-translation invariant discrete twisted Radon
transforms. Finally, we extend an existing l^2 bound for a
closely related non-translation invariant discrete
oscillatory integral operator with singular kernel to an
l^p bound. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function.

(hide abstract)

**
A bound for the 3-part of class numbers of quadratic fields
by means of the square sieve**

Forum Math. 18 (2006) no. 4, 677-698.

(abstract)
(pdf)
(journal)

This paper proves a nontrivial bound for the 3-part of the class number of a
real or imaginary quadratic field, by developing a variant of the square sieve and
the q-analogue of van der Corput's method.

(hide abstract)