The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality.
We will present advances on the boundedness of geometric maximal operators, focusing on a recent result from joint work with Paul Hagelstein and Alex Stokolos, which employs probabilistic techniques in the construction of Kakeya-type sets. The material presented extends ideas of M. Bateman and N. Katz.
The `middle third integer Cantor set' consists of those integers which do not have a 2 in their base 3 representation. We will review and extend some results about such sets. For a general integer Cantor set K, with 0 as an allowed digit, it is known that K is intersective, a result of Furstenberg-Katznelson.
We first investigate reproducing pairs in Hilbert spaces, with a focus on the discrete case. Reproducing pairs generalize frames and consist of two sequences $\Psi$ and $\Phi$, along with a bounded invertible operator $S_{\Psi,\Phi}$. The work examines sequences that are overcomplete by one element—that is, they become exact upon removal of a single element. A central result shows that if such a sequence admits a reproducing partner, the resulting exact subsequence must form a Schauder basis.
In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.
An important class of problems at the intersection of harmonic analysis and geometric measure theory asks how large the Hausdorff dimension of a set must be to ensure that it contains certain types of geometric point configurations. We apply these tools to study configurations associated to the problem of bounding the VC-dimension of a naturally arising class of indicator functions on fractal sets.
For \( c\in(1,2) \) we consider the following operators
\[
\mathcal{C}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n} \bigg|\text{,}
\]
\[
\mathcal{C}^{\mathsf{sgn}}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign}(n) \lfloor |n|^{c} \rfloor}}{n} \bigg| \text{,}
\]
and prove that both extend boundedly on \( \ell^p(\mathbb{Z}) \), \( p\in(1,\infty) \).
In 1992, Olson and Zalik conjectured that no system of translates can be a Schauder basis for L^2(R). This conjecture remains open as of the time of writing. Although some partial results regarding Olson-Zalik conjecture have been proved to be true, a characterization of subspaces of L^2(R) that do not admit a Schauder basis, or an unconditional basis is still unknown.
We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths; this is carried out within the context of separable Banach spaces. Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces.
It is well known that $H^2(\mathbb{D}^2)$ is a RKHS with the reproducing kernel $K( \lambda, z) = \frac{1}{(1-\overline{\lambda_1}z_1)(1 - \overline{\lambda_2}z_2)}$ and that for any submodule $M \subseteq H^2(\mathbb{D}^2)$ its reproducing kernel is $K^M( \lambda, z) = P_M K( \lambda, z)$ where $P_M$ is the orthogonal projection onto $M$. Associated with any submodule $M$ are the core function $G^M( \lambda, z) = \frac{K^M( \lambda, z)}{K( \lambda, z)}$ and the core operator $C_M$, an integral transform on $H^2(\mathbb{D}^2)$ with kernel function $G^M$.
