Analysis Seminar:

About Me:

I received my undergraduate degree from the University of Oklahoma in 2002, where I wrote an undergraduate thesis on Numerical Semigroups with Professor Andy Miller. Then, I went to UC-Berkeley where I finished my Ph.D. working with Professor Daniel Tataru in 2007. My thesis topic was the asymptotic behavior of solutions to the Nonlinear Schrödinger Equation (NLSE).

From 2007-2008 and 2009-2010, I was at Columbia University as a National Science Foundation Postdoctoral Research Fellow working with Professor Michael Weinstein. However, I spent the academic year 2008-2009 as a Hausdorff Center Postdoc with Professor Herbert Koch at the University of Bonn.

Presently, I am an Assistant Professor in the Analysis Group at the University of North Carolina, Chapel Hill, where I am lucky enough to study/discuss PDE using tools from microlocal analysis almost every day with my colleagues Hans Christianson, Jason Metcalfe, Michael Taylor and Mark Williams. In addition, I get to talk about models, applications and numerics with Chris Jones as well as members of our Applied Fluids Group such as Roberto Camassa and Rich McLaughlin.

My research interests lie mainly in phenomenological descriptions of solutions to linear/nonlinear PDE's related to quantum mechanics and other physical models that give rise to Hamiltonian structure.

Research Topics:

Soliton existence/stability, well-posedness, scattering theory, spectral theory, Strichartz/dispersive estimates, microlocal analysis, properties of eigenfunctions on billiards, classical dynamics descriptions of PDE solutions, numerical simulations, asymptotic analysis.

Papers, Preprints and Presentations:

• 1. "Eigenfunctions For Partially Rectangular Billiards" (written with the help of Maciej Zworski). Published in Communications in Partial Differential Equations, Vol. 31 (2006), 775-790.

• 2. "Fast Soliton Scattering by Delta Impurities" (with Maciej Zworski and Justin Holmer). Pubished in Communications in Mathematical Physics, Vol. 274, No. 1 (2007), 187-216.

• 3. "Soliton Splitting by External Delta Potentials" (with Maciej Zworski and Justin Holmer). This is a numerical companion to the result above. If you are interested in the numerical methods, feel free to e-mail me. Published in Journal of Nonlinear Science, Vol. 17, No. 4 (2007), 349-367.

• 4. "Wave packet parametrices for evolutions governed by PDO's with rough symbols" (with Jason Metcalfe and Daniel Tataru). Published in Proceedings of the American Mathematical Society, Vol. 136, No. 2 (2008), 597-604.

• 5. "Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations" (with Jason Metcalfe and Daniel Tataru). Published in Journal of Functional Analysis, Vol. 255, Issue 6 (2008), 1497-1553.

• 6. My UC Berkeley Ph.D. Thesis supervised by Daniel Tataru. Also, here is a copy of my talk regarding the research. There will be some further details added for the resulting publications, but the main ideas are all here. Filed (2007).

• 7. "Strichartz estimates on Schwarzschild black hole backgrounds" (with Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu). Published in Communications in Mathematical Physics, Vol. 293, No. 1 (2010), 37-83.

• 8. "Counting numerical sets with no small atoms" (with Andy Miller). This article reports findings in Table 1 that follow directly from running code that computes the cardinality of numerical sets with no small atom. Similarly, in Table 2 data is recorded from running code that computes the cardinality of symmetric numerical sets with no small atom. Both codes are written in Fortran 90. Published in Journal of Combinatorial Theory A, Vol. 117, Issue 6 (2010), 650-667.

• 9. "An eigenfunction concentration result for polygonal billiards" (with Andrew Hassell and Luc Hillairet). Published in Communications in Partial Differential Equations, Vol. 34, Issue 5 (2009), 475-485.

• 10. "Dispersive estimates using scattering theory for matrix Hamiltonian equations". Published in Discrete and Continuous Dynamical Systems - Series A, Vol. 30, No. 4 (2011), 995-1036.

• 11. "A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations". Published in SIAM Journal of Mathematical Analysis, Vol. 42, Issue 3 (2010), 1382-1403.

• 12. "Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space" (with Hans Christianson). Published in Nonlinearity, Vol. 23 (2010), 89-106.

• 13. "A system of ODE's for a perturbation of a minimal mass soliton" (with Sarah Raynor and Gideon Simpson). Published in Journal of Nonlinear Science, Vol. 20, No. 4 (2010), 425-461.

• 14. "Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations" (with Michael Weinstein). Published in Discrete and Continuous Dynamical Systems A, Vol. 28, No. 4 (2010), 1505-1554.

• 15. "Small data scattering and soliton stability in H^{-1/6} for the quartic KdV equation." (with Herbert Koch). To appear in Analysis & PDE (2012).

• 16. "Strichartz estimates for the Schrödinger equation on polygonal domains" (with Matt Blair, G. Austin Ford and Sebastian Herr). Published in Journal of Geometric Analysis, Vol. 22, No. 2 (2012), 339-351.

• 17. "Spectral analysis for matrix Hamiltonian operators" (with Gideon Simpson). Published in Nonlinearity, Vol. 24 (2011), 389-429.

• 18."Wave operator bounds for singular perturbations of the Laplacian in 1d" (with Vincent Duchêne and Michael Weinstein). Published in Journal of Mathematical Physics, Vol. 52, Issue 1 (2011), 013505.

• 19. "Nonconcentration for partially rectangular billiards" (with Luc Hillairet). To appear in Analysis & PDE (2012).

• 20. "Bose-Einstein condensation transition studies for atoms confined in Laguerre-Gaussian laser modes" (with T.G. Akin, S. Kennedy, B. Dribus, L. Johnson, J. Alexander and E.R.I. Abraham). This is a physics paper written by Akin and Abraham regarding work I collaborated on as an undergraduate at the University of Oklahoma. Published in Optics Communications, Vol. 285 (2012), 84-89.

• 21. "On instability for the quintic nonlinear Schrödinger equation of some approximate periodic solutions" (with Scipio Cuccagna). Submitted to Indiana University Mathematics Journal (2011).

• 22. "Nonlinear quasimodes near elliptic periodic geodesics" (with Pierre Albin, Hans Christianson and Laurent Thomann). Published in Physica D: Nonlinear Phenomena, Vol. 241 (2012), 409-425.

• 23. "Strichartz estimates for the wave equation on flat cones" (with Matt Blair and G. Austin Ford). To appear in International Mathematics Research Notices (2012).

• 24. "Quasilinear Schrödinger equations I: Small data and quadratic interactions" (with Jason Metcalfe and Daniel Tataru). Submitted (2011).

• 25. "Ill-posedness of the gravity-capillary equations" (with Ming Chen, Dan Spirn and J. Doug Wright). Submitted (2011).

• 26. "Nonlinear bound states on weakly homogeneous spaces" (with Hans Christianson, Jason Metcalfe and Michael Taylor). Submitted (2012).

• Note that all finished papers above are presented in PDF format and can be found on the ArXiv's.

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