Interests

My main mathematical interests are in applications of harmonic analysis to partial differential equations and mathematical physics. I am particularly interested in oscillatory and singular integrals and integral operators that arise in the analysis of dispersive PDEs such as the Schrödinger, wave and Dirac equations.
Since September 2017 my work has been partially supported by the Simons Foundation under grant #511825.

Links

Google Scholar Profile
MathSciNet (Subscription Required)
arXiv

Publications

  1. (with C. Lane*, B. Lyons*, S. Ravishankar*, and A. Shaw*) "The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions," [preprint], 40 pages, submitted.
  2. (with M.B. Erdogan and M.J. Goldberg) "Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials," [preprint], 25 pages submitted.
  3. (with M.B. Erdogan) "A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators."
         J. Differential Equations, 355, (2023), 144-161. [preprint], [Journal Version].
  4. (with M.B. Erdogan and M.J. Goldberg) "Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators,"
         J. Funct. Anal. 285, (2023) no. 5, 110008. 16 pages. [preprint], [Journal Version].
  5. (with K. Ott) "Analysis and differential equations at primarily undergraduate institution,"
        Notices of the American Mathematical Society, 70 (2023), no. 4, 603-605. [Journal Version].
  6. (with K. Ott) "Organizing Virtual Panels,"
        MAA Focus, 42, (2022), no. 1, 22-24. [Journal Version]
  7. (with M.B. Erdogan) "The $L^p$-continuity of wave operators for higher order Schrödinger operators."
         Advances in Mathematics Volume 404, Part B, 6 August 2022, 108450. [preprint], [Journal Version]. The preprint linked here corrects an error in Lemma 4.2 of the published version.
  8. (with M.B. Erdogan and E. Toprak) "What is... the Dirac equation?"
         Notices of the American Mathematical Society 68 (2021), no. 10, 1782-1785. [Journal Version].
  9. (with M.B. Erdogan) "On the one dimensional Dirac equation with potential."
         Journal de Mathématiques Pures et Appliquées Volume 151, July 2021, Pages 132-170. [preprint], [Journal Version].
  10. (with M.J. Goldberg) "On the $L^p$ boundedness of Wave Operators for fourth order Schrödinger operators,"
         Trans. Amer. Math. Soc., 374 (2021), 4075-4092. [preprint], [Journal Version]
  11. (with M.J. Goldberg) "Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions,"
         Bulletin of the London Mathematical Society, Volume 54, no. 2, 428-448. [preprint], [Journal Version].
  12. (with J. McSweeney) "A Speaker Series of Graduate Students,"
        MAA Focus, 40, (2020), no. 3, 24-26. [Journal Version]
  13. (with M.B. Erdogan and M.J. Goldberg) "Strichartz Estimates for the Schrödinger Equation with a Measure-Valued Potential,"
         Proc. Amer. Math. Soc. Ser. B 8 (2021), 336-348. [preprint], [Journal Version]
  14. (with M.B. Erdogan and E. Toprak) "On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances,"
         J. Differential Equations, 271, (2021), 152-185. [preprint], [Journal Version].
  15. "The natural frequency: More natural and more frequent than expected,"
        College Mathematics Journal, 51:5, 372-374. [Preprint], [Journal Version]
  16. (with E. Toprak) "On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances,"
        J. Differential Equations, 267 , (2019), no. 3, 1899-1954. [preprint], [Journal Version].
  17. (with S. Carlisle) "Solving systems of differential equations in the case of a defective coefficient matrix."
        College Mathematics Journal, 50:5, 372-374. [preprint], [Journal Version]
  18. (with M.B. Erdogan and M.J. Goldberg) "The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates,"
         Journal of Spectral Theory, Vol. 11, no. 3, 935-979. [preprint], [Journal Version].
  19. (with M.B. Erdogan and E. Toprak) "Dispersive estimates for massive Dirac operators in dimension two",
        J. Differential Equations, 264, (2018), no. 9, 5802-5837. [preprint], [Journal Version].
  20. (with M.B. Erdogan and M.J. Goldberg) "Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions,"
        Comm. Math. Phys. 367 (2019), no. 1, 241-263. [preprint], [Journal Version].
  21. (with M.B. Erdogan and M.J. Goldberg) "On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger Operators with threshold obstructions,"
        J. Funct. Anal. 274, (2018) no. 7, 2139-2161. [preprint], [Journal Version]
  22. (with M.B. Erdogan and E. Toprak) "Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies,"
        Amer. J. Math., 141, no. 5, 1217-1258. [preprint], [Journal Version]
  23. (with M.J. Goldberg) "On the $L^p$ boundedness for four-dimensional wave operators for Schrödinger Operators with a threshold eigenvalue,"
        Ann. Henri Poincaré 18 (2017), no.4, 1269-1288. [preprint], [Journal Version] , [Free Journal Version].
  24. (with M.B. Erdogan) "The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions."
        Comm. Math. Phys. 352 (2017), no. 2, 719-757. [preprint], [Journal Version]
  25. (with E. Toprak) "Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy."
        Differential and Integral Equations, Volume 30, Number 5/6 (2017), 329-386. [preprint], [Journal Version].
  26. (with M.J. Goldberg) "The $L^p$ boundedness of wave operators for higher dimensional Schrödinger Operators with threshold singularities,"
        Advances in Mathematics, 303 (2016), 360-389. [preprint], [Journal Version]
  27. (with M.J. Goldberg) "Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues II: The even dimensional case,"
        J. Spectr. Theory 7 (2017), 33-86. [preprint], [Journal Version].
  28. (with M.J. Goldberg) "Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd dimensional case,"
        J. Funct. Anal. 269, (2015) no. 3, 633-682. [preprint], [Journal Version]
  29. (with M.B. Erdogan and M.J. Goldberg) "Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy."
        Comm. PDE 39 (2014), no. 10, 1936-1964. [preprint], [Journal Version]
  30. "Time decay estimates for the wave equation with potential in dimension two,"
        Journal of Differential Equations 257 (2014) pp. 868-919. [preprint], [Journal Version]
  31. (with Dirk Hundertmark) "Exponential decay for dispersion managed solitons for general dispersion profiles,"
        Letters in Mathematical Physics, Volume 106, Issue 2, 221-249. [preprint]. [Journal Version].
  32. (with M.B. Erdogan) "Dispersive estimates for matrix Schrödinger operators in dimension two."
        DCDS-A. Pages: 4473 - 4495, Volume 33, Issue 10, October 2013. [preprint]. [Journal Version].
  33. (with M.B. Erdogan) "A weighted dispersive estimate for Schrödinger operators in dimension two."
        Comm. Math. Phys. 319 (2013), 791--811. [preprint]. [Journal Version].
  34. (with H.I. Dwyer) "Integrating factors and repeated roots of the characteristic equation."
        CODEE Journal, January 2012. URL http://www.codee.org/ref/CJ12-0930.
  35. (with M.B. Erdogan) "Dispersive estimates for two-dimensional Schrödinger operators with obstructions at zero energy."
        Trans. Amer. Math. Soc. 365 (2013), 6403-6440. [preprint], [Journal Version].
  36. "Using Abel's theorem to explain repeated roots of the characteristic equation."
        CODEE Journal, July 2011. URL http://www.codee.org/ref/CJ11-0679.
  37. "Abel's theorem simplifies reduction of order."
        College Mathematics Journal, Vol. 42, No. 5, November 2011 , pp. 399-402(4). [preprint], [Journal Version].
  38. "Dispersive estimates for the matrix and scalar Schrödinger equation in dimension five,"
        Illinois J. Math. Volume 56, Number 2 (2012), 307-341. [preprint], [Journal Version].
  39. (with M.B. Erdogan) "Dispersive estimates for the Schrödinger equation for C$^{\frac{n-3}{2}}$ potentials in odd dimensions,"
        International Mathematics Research Notices 2010:13, 2532-2565. [preprint], [Journal Version]

* Denotes undergraduate student

Undergraduate Research Supervised

  1. Casey Garner (2018-19). Senior thesis on the Unified Transform Method applied to the Dirac equation. SIAM Undergraduate Research Online Vol. 12, 2019. [preprint], [Journal Version].
  2. Connor Lane, Benjamin Lyons, Shyam Ravishankar, and Aden Shaw (2023-24). Independent study on Analysis of a three-dimensional massless Dirac equation. "The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions," [preprint], 40 pages, submitted.



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