Research Interests:

I am interested in functional analysis and operator theory of unbounded operators, specifically spectral analysis, expansion theorems of said operators, and lifting unbounded derivations. Currently I am researching the lifting problem with Slawomir Klimek from IUPUI. Given an unbounded derivation defined densely on some C*-algebra, B, the goal is to lift that derivation to be densely defined on a C*-algebra, A, where B is the quotient algebra of A and some ideal in A. From there the goal is to construct meaningful spectral triples by implementing these unbounded derivations. I am also researching the APS index problem for Dirac type operators in non-commutative spaces. The Atiyah-Patodi-Singer (APS) index problem, studies Dirac operators on closed manifolds with boundary. Since there is no known generic formula for Dirac operators in this set-up, we are studying d-bar and Dirac-type operators to different kinds of global boundary conditions, that are similar to APS boundary conditions, as the APS boundary condition does not always work for these operators on certain domains. I am also researching compact linear relations and their spectral properties with Keshav Archaya from Embry-Riddle.
For more details you can view my research statement here: Research Statement.
You can also view my CV as well: CV
Below is a list of my publications with a corresponding link to each paper. There are also links to the slides of some talks I have given in the past.

Publications:

Here is a list of my publications:
  1. D-bar operators in Quantum Domains (with S. Klimek), Math. Phys. Anal. Geom., 13, 357 - 390, 2010. ( pdf)
  2. A note on Dirac Operators on the Quantum Punctured Disk (with S. Klimek), SIGMA, 6, 56 - 68, 2010. ( pdf)
  3. Classical limit of the d-bar operator on quantum domains (with S. Klimek), Jour. Math. Phys., 52, 1 - 16, 2011. ( pdf)
  4. Global boundary conditions for a Dirac operator on the solid torus (with S. Klimek), Jour. Math. Phys., 52, 1 - 14, 2011. ( pdf)
  5. A Note on Gluing Dirac Type Operators on a Quantum Two-Sphere (with S. Klimek), SIGMA, 36, 1 - 15, 2014. ( pdf)
  6. A P-adic Spectral Triple (with S. Klimek and S. Rathnayake), Jour. Math. Phys., 55, 1 - 17, 2014. ( pdf)
  7. The Marchenko Representation of Reflectionless Jacobi and Schrodinger Operators (with I. Hur and C. Remling), Trans. Amer. Math. Soc., 9947, 1 - 20, 2015. ( pdf)
  8. The Quantum Pair of Pants (with S. Klimek, S. Rathnayake, and K. Sakai), SIGMA, 012, 1 - 22, 2015. ( pdf)
  9. Estimates in Generalized Morrey Spaces for Linear Parabolic Systems, Jour. Math. Anal. Appl., 1, 1 - 12, 2017. ( pdf)
  10. Derivations and Spectral Triples on Quantum Domains I: Quantum Disk (with S. Klimek, S. Rathnayake, K. Sakai, and H. Wang), SIGMA, 013, 1 - 26, 2017. ( pdf)
  11. Derivations and Spectral Triples on Quantum Domains II: Quantum Annulus (with S. Klimek and S. Rathnayake), Sci. Chi. Math., doi:10.1007/s11425-018-9317-3, 2018. ( pdf)
  12. Action of Complex Symplectic Matrices on the Siegel Half Space (with K. Acharya), Lin. Alg. Appl., 563, 47 - 62, 2019. ( pdf)
  13. Unbounded Derivations in Bunce-Deddens-Toeplitz Algebras (with S. Klimek, S. Rathnayake, K. Sakai, and H. Wang), Jour. Math. Anal. Appl., 15, 988 - 1020, 2019. ( pdf)
  14. Derivations and Reflection Positivity in the Quantum Annulus (with S. Klimek), Sci. Chi. Math., doi:10.1007/s11425-019-9533-3, 2019. ( pdf)
  15. A Note on Spectral Triples on the Quantum Disk (with S. Klimek and J. Peoples), SIGMA, 015, 1 - 8, 2019. ( pdf)
  16. A Value Region Problem for Continued Fractions and Discrete Dirac Equations (with S. Klimek, S. Rathnayake, and K. Sakai), to appear in Hokk. Math. Jour. ( pdf)
  17. Dirac type Operators on the Quantum Solid Torus (with S. Klimek), to appear in Jour. Math. Anal. Appl. ( pdf)
  18. Unbounded Derivations on Algebras Associated with Monothetic Groups (with S. Klimek), to appear in Jour. Aust. Math. Soc. ( pdf)
  19. Compact Linear Relations and their Spectral Properties (with K. Acharya), submitted, arXiv: 1909.05124. ( pdf)
  20. Spectral Properties of Generalized Cesaro Operators on Hardy and Bergman Spaces of the Unit Ball (with S. Ballamoole, T.L. Miller, and V. Miller), preprint
  21. The m-Legged Pair of Quantum Pants, preprint

Talks:

Here are slides from some talks I have given:

University of North Texas PDE Conference 2014

Oklahoma State University PDE Workshop 2013

AMS Sectional Meeting at Washington University in St. Louis 2013

Math Fest 2013

Wabash Conference 2010

Wabash Conference 2009