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 equation.

Publications:

1. (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]
2. "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].
3. "Abel's theorem simplifies reduction of order." College Mathematics Journal, Vol. 42, No. 5, November 2011 , pp. 399-402(4). [preprint], [Journal Version].
4. "Using Abel's theorem to explain repeated roots of the characteristic equation." CODEE Journal, July 2011. URL http://www.codee.org/ref/CJ11-0679.
5. (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].
6. (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.
7. (with M.B. Erdogan) "A weighted dispersive estimate for Schrödinger operators in dimension two." Comm. Math. Phys. 319 (2013), 791--811. [preprint]. [Journal Version].
8. (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].
9. (with Dirk Hundertmark) "Exponential decay for dispersion managed solitons for general dispersion profiles," [preprint]. Submitted 2014, 24 pp.
10. "Time decay estimates for the wave equation with potential in dimension two," [preprint], Journal of Differential Equations 257 (2014) pp. 868-919. Journal Version
11. (with M.B. Erdogan and M.J. Goldberg) "Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy." [preprint], accepted by Comm. PDE. 32 pp.