Overview
Postdoctoral Research:
My current research interests are in dyadic harmonic analysis, weighted inequalities,
multiparameter harmonic analysis, and operator theory.
For more details, see the most recent publications below.
Graduate Research:
My graduate research interests were infinitedimensional analysis, probability, functional
analysis and machine learning.
Together with my adviser, I worked on developing the Gaussian Radon
transform for Banach spaces, an infinitedimensional
generalization of the classical Radon transform,
and applications of this transform to machine learning.
For more details,
see my
Graduate Research Statement,
or my thesis.
Undergraduate Research:
As an undergraduate at LSU, I worked as a research student at
CCT.
In particular, I worked with Horst Beyer on the Kerr metric of rotating black holes.
Publications

Commutators with Fractional Integral Operators
 joint with Robert Rahm and Scott Spencer (Studia Mathematica
233, no.3, 2016).
 In this paper we investigate weighted norm inequalities
for the commutator of a fractional integral operator and multiplication by a function.
In particular, we show that, for $\mu, \lambda \in A_{p,q}$ and
$\alpha/n + 1/q = 1/p$, the norm
$\ [b, I_{\alpha}] : L^p(\mu^p) \to L^q(\lambda^q) \$
is equivalent to the norm of $b$ in the weighted BMO space $BMO(\nu)$,
where $\nu = \mu\lambda^{1}$. This work extends some of the results
on this topic existing in the literature, and continues a line of investigation
which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

Two Weight Inequalities for Iterated Commutators with CalderónZygmund Operators
 joint with Brett D. Wick (Submitted).
 Given a CalderónZygmund operator $T$, a classic result
of CoifmanRochbergWeiss relates the norm of the commutator $[b, T]$ with the BMO norm
of $b$. We focus on a weighted version of this result, obtained by
Bloom and later generalized by Lacey and the authors, which relates
$\ [b, T]: L^p(\mathbb{R}^n; \mu) \rightarrow L^p(\mathbb{R}^n; \lambda) \$
to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights
$\mu$ and $\lambda$. We extend this result to higher iterates
of the commutator and recover a oneweight result of ChungPereyraPerez in the process.

Commutators in the TwoWeight Setting
 joint with Michael T. Lacey and Brett D. Wick (Mathematische Annalen, 2016).
 Generalizes the previous paper, by proving the boundedness of the commutator
$[b, T] : L^p(\mu) \rightarrow L^p(\lambda)$ for any CalderónZygmund operator $T$,
where $\mu, \lambda$ are Muckenhoupt $A_p$ weights on $\mathbb{R}^n$, provided that $b$ is
in a special weighted $BMO$ class. Conversely, specializing to Riesz transforms, we characterize this
$BMO$ class in terms of the boundedness of the commutators.

Bloom's Inequality: Commutators in a TwoWeight Setting
 joint with Michael T. Lacey and Brett D. Wick (Archiv der Mathematik 106, no. 1, 2016).
 Gives a modern proof of a result by Bloom which characterizes the boundedness of the
commutator $[b, H] : L^2(\mu) \rightarrow L^2(\lambda)$, where $H$ is the Hilbert transform
and $\mu, \lambda$ are Muckenhoupt $A_2$ weights on $\mathbb{R}$, in terms of a novel weighted $BMO$ condition.

The Gaussian
Radon Transform in Classical Wiener Space
 joint with Ambar Sengupta
(Communications on Stochastic Analysis 8, no. 2, 2014).
 Provides concrete examples and computations of the Gaussian Radon transform in the setting of
the classical Wiener space of Brownian motion, and establishes a Fock space decomposition for Gaussian
measure conditioned to closed affine subspaces in Hilbert spaces.

The Gaussian Radon Transform and Machine Learning
 joint with
Ambar Sengupta
(Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18, no. 3, 2015).
 This work is motivated by an increasing number of recent
publications exploring
probabilistic interpretations of kernelbased prediction methods.
While there is a clear Bayesian
interpretation to some of these methods when working in finite dimensions,
the absence of Lebesgue measure
in infinitedimensional Hilbert spaces poses an obstacle. We show that the
Gaussian Radon transform
can be used to provide a stochastic interpretation to the ridge regression
problem
in infinite dimensions.

An Inversion Formula for the Gaussian Radon Transform for Banach Spaces
(Infinite Dimensional Analysis, Quantum Probability and Related Topics, 16, no. 4, 2013).
 Proves a disintegration formula leading
to the expression
of the Gaussian Radon transform as a conditional expectation, and an inversion
procedure involving the SegalBargmann transform.

A Gaussian Radon
Transform for Banach Spaces

joint with Ambar Sengupta
(Journal of Functional Analysis 263, no. 11, 2012).
 Establishes the existence
and uniqueness of the
Gaussian Radon transform and proves an analogue of the Helgason support theorem.

On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole
 joint with Horst R. Beyer
(Classical and Quantum Gravity, 25, no. 13, 2008)
 Derives the constant of motion of a scalar field
in the gravitational field of a Kerr black hole
which is associated to a Killing tensor of that spacetime.

On a Problem in the Stability Discussion of Rotating Black Holes
(Proceedings of The National Conference On Undergraduate Research (NCUR) 2006)
 This is an undergraduate paper which I presented at the 2006
National
Conference for Undergraduate Research in Asheville, North Carolina.