# Research

I take an interdisciplinary approach to harmonic analysis with applications in image processing and data analysis. The modern world is powered by a deluge of data, and much of human progress hinges on developing better methods to process this data. Many powerful data processing algorithms rely on a well-designed representation system, and we can leverage beautiful mathematics to design such systems. Much of my research concerns the design of representation systems involving rich symmetries and combinatorial structures that allow one to extract critical features from various classes of data.Representation systems play a crucial role in both low complexity models (where they are known as dictionaries) and redundant transforms (where they are known as frames). By and large, my research is guided by two fundamental questions along these lines:

- How can low complexity models be used to develop tools and analyze success in data analysis and machine learning?
- How can algebraic, geometric, and combinatorial methods be leveraged to solve open problems in frame theory and quantum information theory?

Frames are redundant systems in Hilbert spaces which have reconstruction properties akin to orthonormal bases. In finite dimensions, one often seeks frames which represent line configurations which are optimally geometrically spread apart and are in some cases also equivalent to special measurement systems in quantum information theory. In recent years, more has been discovered about the deep connections between such frames and mathematical fields like algebraic combinatorics, real algebraic geometry, and algebraic number theory. One goal of mine is to attack open problems in finite frame theory using these tools [JKM19, Kin19, Kin16, Kin15, FJKM18, KT18, BK18]. I am also interested in characterizing frames in \(L^2(G)\), where \(G\) is a locally compact abelian group [KS10, Kin13, KS18], and using representation theory to generate new transforms in \(L^2(\mathbb{R}^d)\) [CK12, CK14].