interests

Operator theory, weighted composition operators, Bloch functions, functional analysis, pattern formation in PDEs, mathematical biology, Lie algebras, operator algebras.

current projects

Operators on Spaces of Holomorphic Functions
I investigate weighted composition operators on spaces of holomorphic functions defined by $W_{\psi,\varphi} f = \psi(f\circ\varphi)$. I study these operators on spaces such as the Bloch space, Hardy spaces $H^p$, Bergman and weighted Bergman spaces, and Dirichlet and weighted Dirichlet spaces. I study these spaces in the unit disk $\mathbb{D}$ and the unit ball and polydisk $\mathbb{B}_n$ and $\mathbb{D}^n$. I also study equivalent spaces on discrete structures such as trees. (with Flavia Colonna, Katherine Heller, and Matthew Pons)

Embeddings of Discrete Structures in the Hyperbolic Ball
We are defining discrete structures such as trees to embed into the unit ball in $\mathbb{R}^n$ under the hyperbolic geometry. We are attempting to use this embedding to solve an interpolation problem: if we define a function on the vertices of the structure, can we find a function on the unit ball which agrees on the vertices? (with Tushar Das and Eddie Kim)

Mathematical Biology and Sociology
We are looking at how SIR models can be modified with nonlinear mass-action and stochasticity to describe various situations in biology and sociology. These topics include the spread of disease and rumors through populations. (with Eric Eager)

journal articles

  1. Composition operators on the Lipschitz space of a Tree (with Flavia Colonna and Glenn R. Easley), Mediterranean Journal of Mathematics, to appear.
  2. Multiplication operators on the weighted Lipschitz space on a tree (with Flavia Colonna and Glenn R. Easley), Journal of Operator Theory 69 (2013), 209-231.
  3. Multiplication operators on the iterated logarithmic Lipschitz spaces of a tree (with Flavia Colonna and Glenn R. Easley), Mediterranean Journal of Mathematics 9 (2012), 575-600.
  4. Multiplication operators between Lipschitz-type spaces of a tree (with Flavia Colonna and Glenn R. Easley), International Journal of Mathematics and Mathematical Sciences, 2011, Art. ID 47495, 33pp.
  5. Weighted composition operators between \(H^\infty\) and the Bloch space of a bounded homogeneous domain (with Flavia Colonna), Integral Equations and Operator Theory 66 (2010), 21-40.
  6. Weighted composition operators on the Bloch space of a bounded homogeneous domain (with Flavia Colonna), Topics in Operator Theory Volume 1: Operators, Matrices and Analytic Functions, ISBN:978-3-0346-0157-3, 2010, 11-38.
  7. Multiplication operators on the Bloch space of a bounded homogeneous domain (with Flavia Colonna), Computational Methods and Function Theory 9 (2009), 679-693.
  8. On the isometric composition operators on the Bloch space in \(\mathbb{C}^n\) (with Flavia Colonna), Journal of Mathematical Analysis and Applications 355 (2009), 675-688.
  9. Isometries and spectra of multiplication operators on the Bloch space (with Flavia Colonna), Bulletin of the Australian Mathematical Society 79 (2009), 147-160.

manuscripts under review

  1. Isometric composition operators on the analytic Besov spaces (with Katherine Heller and Matthew A. Pons), submitted to Journal of Mathematical Analysis and Applications.
  2. Multiplication operators on $S^2(\mathbb{D})$ (with Katherine Heller and Matthew A. Pons), submitted to Computational Methods and Function Theory.
  3. Compact differences of composition operators on weighted Dirichlet spaces (with Katherine Heller and Matthew A. Pons), submitted to Central European Journal of Mathematics.
  4. Weighted composition operators from the Bloch space to \(H_\mu^\infty\) of a bounded homogeneous domain, submitted to Complex Variables and Operator Theory.

manuscripts in progress

  1. Weighted composition operators between $S^p$ spaces (with Katherine Heller and Matthew A. Pons).
  2. Surviving an outbreak of zombiism (with Cassandra Jens and Theodore Wendt).

dissertations and theses

  1. A class of operators with symbol on the Bloch space of a bounded homogeneous domain, Ph.D. dissertation, George Mason University.
  2. Turing instabilities with spatial pattern formation in one-dimension, Honors theses, George Mason University.
  3. The C-- programming language, Undergraduate thesis, University of Virginia.

talks

Conferences
  1. "Surviving an Outbreak of Zombiism", MAA-DC/MD/VA Sectional Meeting, Salisbury MD, April 2013
  2. "Mass-Action in Action: From Zombiism to Undergraduate Research", MAA-WI Sectional Meeting, Baraboo, WI, April 2013.
  3. "Zombies, Ecology, and Epidemiology...Oh My!", MAA MathFest 2012, Sesson on Undergraduate Research in Mathematical and Computational Biology, MAA MathFest, Madison, WI, August 2012.
  4. "Surviving an Outbreak of Zombiism", MAA-WI Sectional Meeting, Milwaukee, WI, April 2012.
  5. "Using WeBWorK in Mathematics and Statistics Courses", Best Practices in Science, Math, and Engineering Teaching Conference, Baraboo, WI, August 2011.
  6. "Mathematical Modeling Meets Pop Culture", MAA MathFest 2011, Session on Mathematical Modeling Projects that Matter, MAA MathFest, Lexington, KY, August 2011.
  7. "Functional Analysis and Operator Theory for Undergraduates", MAA-WI Sectional Meeting, Menomonie, WI, April 2011.
  8. "Using WeBWorK in Mathematics and Statistics Courses", MAA-WI Sectional Meeting, Menomonie, WI, April 2011.
  9. "Multiplication Operators Between Lipschitz-Type Spaces of a Tree", Joint AMS/MAA Mathematics Meeting, Session on Operator Theory, New Orleans, LA, January 2011.
  10. "Motivating Class Participation and Cooperation Through Pop Culture", Best Practices in Science, Math, and Engineering Teaching Conference, Baraboo, WI, August 2010.
  11. "Isometries on the Bloch Space"', MAA-WI Sectional Meeting, Oshkosh, WI, April 2010.
  12. "Weighted Composition Operators from \(H^\infty\) to the Bloch Space of a Bounded Homogeneous Domain", Joint AMS/MAA Mathematics Meeting, Session on Functional Analysis and Operator Theory, San Francisco, CA, January 2010.
  13. "Multiplication Operators on the Bloch Space of a Bounded Homogeneous Domain", Joint AMS/MAA Mathematics Meeting, Special Session on Function Theoretic Operator Theory, Washington, D.C., January 2009.
  14. "Weighted Composition Operators on the Bloch Space on a Bounded Homogeneous Domain", International Workshop on Operator Theory and its Applications XIX, Special Session on Composition Operators, College of William & Mary, July 2008.
  15. "On the Spectrum of an Isometric Composition Operator on the Bloch Space of the Polydisk", South Eastern Analysis Meeting XXIII, University of Richmond, March 2007.
Seminars and Colloquia
  1. "Surviving an Outbreak of Zombiism", MSCP Lecture Series, Roanoke College, March 2012.
  2. "Composition Operators: Something for Everyone", Mathematics Colloquium, George Mason University, March 2012.
  3. "Surviving an Outbreak of Zombiism", Pi Mu Epsilon Prime Time Mathematics Colloquium, North Central College, October 2011.
  4. "Multipliers on Lipschitz-Type Spaces of a Tree", Mathematics Department Colloquium, University of Wisconsin-La Crosse, September 2010.
  5. "Turing Instabilities in Reaction-Diffusion Equations: A Model for the Formation of Mammalian Coat Patterns", Mathematics Research Seminar, University of Wisconsin-La Crosse, March 2010.
  6. "Isometries on the Bloch Space", Mathematics Research Seminar, University of Wisconsin-La Crosse, February 2010.
  7. "Analysis on Discrete Structures", Mathematics Research Seminar, University of Wisconsin-La Crosse, October 2009.
  8. "Classical and Modern Operator Theory on the Bloch Space", Mathematics Research Seminar, University of Wisconsin-La Crosse, September 2009.
  9. "Why I Should Have Paid More Attention in Linear Algebra", Math Department Colloquium, University of Wisconsin-La Crosse, February 2009.
  10. "An Introduction to Lie Algebras and their Cohomology", Combinatorics, Algebra and Geometry Seminar, George Mason University, November 2008.
  11. "Turing Instabilities in Reaction-Diffusion Equations: A Model for the Formation of Mammalian Coat Patterns", Applied & Computational Math Seminar, George Mason University, September 2008.
  12. "Isometries on the Bloch Space", Graduate Student Seminar, George Washington University, April 2008.
  13. "Isometric Composition Operators on the Bloch Space of the Unit Disk", Complex Analysis & Potential Theory Seminar, George Mason University, November 2007.
  14. "Embeddings of Trees in the Hyperbolic Disk", Graduate Seminar, University of Virginia, November 2004.
Panel Discussions
  1. "The Academic Job Talk", George Mason University, Fairfax VA, March 2012.
  2. "Effective Use of Online Homework to Enhance Student Learning and Retention", Project NeXT Panel, Lexington KY, August 2011.