Research Interests

My primary research interests lie in complex analysis and operator theory, where I focus on the structure and boundary behavior of multivariate functions and related operators on polydisks. I am currently studying singular rational functions, models for Schur-Agler functions, and multivariate generalizations of the shift. I have also studied problems in harmonic analysis, and recently, have become interested in Crouzeix’s conjecture.

Many of these research questions involve polynomials and/or rational functions and structured matrices. Thus, they have aspects that are accessible to undergraduate researchers who have taken Bucknell's calculus sequence (Math 201-211), linear algebra (Math 245), and introduction to proofs (Math 280) courses. Many phenomena can even be illustrated via pictures like the ones below:

Unimodular level sets of a two-variable rational inner function with two singularities on the 2-torus.

Circles used to to build the (non-circular) numerical range of a two-variable compression of the shift.

Much of my research has been supported by the National Science Foundation grants:

NSF-DMS # 2000088, RUI: Rational Schur Functions and their Applications, June 2020–May 2023
NSF–DMS #1448846, Function Theory on Polydiscs, June 2014– July 2018

All of my research papers are available on the arXiv preprint server here. However, for the final version of the paper, please see the journal where it is published.

Accepted or In Press Papers

Crouzeix’s Conjecture and Related Problems (with P. Gorkin, A. Greenbaum, T. Ransford, F.. Schwenninger, and E. Wegert), 2020. To appear in. Comput. Methods Funct. Theory. Preprint available at https://arxiv.org/pdf/2006.04901.pdf.
Singularities of Rational Inner Functions in Higher Dimensions (with J.E. Pascoe and A. Sola). 2019. To appear in Amer. J. Math. Preprint available at https://arxiv.org/pdf/1906.10913.pdf.
Two-weight Tb Theorems for Well-localized Operators (with T. Korhonen and B.D. Wick). 2019. To appear in Math. Nachr. Preprint available at https://arxiv.org/pdf/1906.03372.pdf.
Level Curve Portraits of Rational Inner Functions (with J.E. Pascoe and A. Sola). 2018. To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. Preprint available at https://arxiv.org/pdf/1803.04666.pdf.

Published Papers

Numerical Range and Compressions of the Shift (with P. Gorkin). Numerical range and compressions of the shift. Complex analysis and spectral theory, 241–261, Contemp. Math., 743, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, 2020.
Applications of Envelopes (with P. Gorkin and T. Trung). Complex Anal. Synerg. 6 (2020), no. 1, 14 pp.
Two Weight Estimates with Matrix Measures for Well Localized Operators (with A. Culiuc, S. Treil, and B. Wick). Trans. Amer. Math. Soc. 371 (2019), no. 9, 6213–6240.
A Multiplier Algebra Functional Calculus (with M. Hartz and J.E. McCarthy). Trans. Amer. Math. Soc. 370 (2018), no. 12, 8467–8482.
Derivatives of Rational Inner Functions: Geometry of Singularities and Integrability at the Boundary (with J.E. Pascoe and A. Sola). Proc. Lond. Math. Soc. (3). 116 (2018), no. 2, 281–329.
Compressions of the Shift on the Bidisk and their Numerical Ranges (with P. Gorkin). J. Operator Theory. 79 (2018), no. 1, 225–265.
Characterizations of A₂ Matrix Power Weights (with K. Lunceford and N. Mukhtar). J. Math. Anal. Appl. 453 (2017), no. 2, 985–999.
Properties of Beurling-Type Submodules via Agler Decompositions (with C. Liaw). J. Funct. Anal. 272 (2017), no. 1, 83–111.
Properties of Vector-Valued Submodules on the Bidisk (with C. Liaw). Recent progress on operator theory and approximation in spaces of analytic functions, 1–11, Contemp. Math., 679, Amer. Math. Soc., Providence, RI, 2016.
Bounds for the Hilbert Transform with Matrix A₂ Weights (with S. Petermichl and B. Wick). J. Funct. Anal. 270 (2016), no. 5, 1719–1743.
A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators (with B. Wick). J. Math. Anal. Appl. 435 (2016), no. 1, 229–243.
Well-Localized Operators on Matrix Weighted L² Spaces (with B. Wick). Houston J. Math. 42 (2016), no. 1, 249–283.
Canonical Agler Decompositions and Transfer Function Realizations (with G. Knese). Trans. Amer. Math. Soc. 368 (2016), no. 9, 6293–6324.
The Linear Bound for Haar Multiplier Paraproducts (with E. Sawyer and B. Wick). Invariant Subspaces of the Shift Operator. (Centre de Recherches Mathématiques, Montreal, 2013), 267-286, Contemp. Math., 638, Amer. Math. Soc., Providence, RI, 2015.
Inner functions on the Bidisk and Associated Hilbert Spaces (with G. Knese). J. Funct. Anal., 265 (2013) no. 11, 2753–2790.
Differentiating Matrix Functions. Operators and Matrices, 7 (2013), no. 1, 71-90.
Fundamental Agler Decompositions. Integral Equations and Operator Theory, 74 (2012) no. 2, 233-257.