Conference

An Analysis and PDE Conference in Paris-Saclay, in honor of my PhD advisor Maciej Zworski, was held on May 27th to 31st, 2024.


Teaching

In Fall 2024, I am teaching MA 425/525 Elements of Complex Analysis.


PhD Student

I have advised one PhD student, Jacob Shapiro, who graduated in 2018.


Papers

• From resolvent expansions at zero to long time wave expansions, with Tanya Christiansen and Mengxuan Yang.

• Low energy resolvent asymptotics of the multipole Aharonov–Bohm Hamiltonian, with Tanya Christiansen and Mengxuan Yang.

• Low energy resolvent expansions in dimension two, with Tanya Christiansen.

• Explicit semiclassical resonances from many delta functions, by Ethan Brady. To appear in Mathematical Research Letters.

• Persistence and disappearance of negative eigenvalues in dimension two, with Tanya Christiansen and Colton Griffin. To appear in Journal of Spectral Theory.

• Propagation of singularities for the wave equation, with Dean Baskin. MATRIX Annals Book Series, Vol. 5. pp. 275–286, 2024.

• Newton polygons and resonances of multiple delta-potentials, with Jeremy L. Marzuola and Jared Wunsch. Transactions of the American Mathematical Society. Vol. 377, No. 3, pp. 2009–2025, 2024.

• Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation, with Jacob Shapiro. Mathematische Nachrichten. Vol. 296, No. 11, pp. 4978–4994, 2023.

• Low energy scattering asymptotics for planar obstacles, with Tanya Christiansen.
  Pure and Applied Analysis. Vol. 5, No. 3, pp. 767–794, 2023.

• Seven Useful Questions in Density Functional Theory, with Steven Crisostomo, Ryan Pederson, John Kozlowski, Bhupalee Kalita, Antonio C. Cancio, Adam Wasserman, Suhwan Song, and Kieron Burke. Letters in Mathematical Physics. Vol. 113, No. 7, Art. 42, 39 pp, 2023.

• Semiclassical resolvent bounds for compactly supported radial potentials, with Jeffrey Galkowski and Jacob Shapiro. Journal of Functional Analysis. Vol. 284, No. 7, Art. 109835, 28 pp., 2023.

• Wave asymptotics for waveguides and manifolds with infinite cylindrical ends, with Tanya Christiansen. International Mathematics Research Notices. Vol. 2022, No. 24, pp. 19431–19500, 2022.

• Semiclassical resonance asymptotics for the delta potential on the half line, with Nkhalo Malawo. Proceedings of the American Mathematical Society. Vol. 150, No. 11, pp. 4909–4921, 2022.

• Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides, with Tanya Christiansen. Mathematical Research Letters. Vol. 29, No. 1, pp. 101–130, 2022.

• Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues, with Tanya Christiansen. Bulletin of the London Mathematical Society. Vol. 53, No. 5, pp. 1351–1356, 2021.

• Resolvent estimates on asymptotically cylindrical manifolds and on the half line, with Tanya Christiansen. Annales Scientifiques de l'École Normale Supérieure. Vol. 54, No. 4, pp. 1051–1088, 2021.

• On the interaction of metric trapping and a boundary, with Jason Metcalfe, Jacob Shapiro, and Mihai Tohaneanu. Proceedings of the American Mathematical Society. Vol. 149, No. 9, pp. 3801–3812, 2021.

• Sharp polynomial decay rates for the damped wave equation with Hölder-like damping, with Perry Kleinhenz. Proceedings of the American Mathematical Society. Vol. 148, No. 8, pp. 3417–3425, 2020.

• Semiclassical estimates for scattering on the real line, with Jacob Shapiro. Communications in Mathematical Physics. Vol. 376, No. 3, pp. 2301–2308, 2020.

• Exponential lower resolvent bounds far away from trapped sets, with Long Jin. Journal of Spectral Theory. Vol. 10, No. 2, pp. 617–649, 2020.

• Resonance free regions for nontrapping manifolds with cusps. Analysis and PDE. Vol. 9, No. 4, pp. 907–953, 2016.

• Iterative reconstruction of the wavespeed for the wave equation with bounded frequency boundary data, with Maarten V. de Hoop. Inverse Problems. Vol. 32, No. 2, Art. 025008, 21 pp., 2016.

• Resonances and lower resolvent bounds, with Semyon Dyatlov and Maciej Zworski. Journal of Spectral Theory. Vol. 5, No. 3, pp. 599–615, 2015.

• Nontrapping surfaces of revolution with long living resonances, with Daniel Kang and Andre Kessler. Mathematical Research Letters. Vol. 22, No. 1, pp. 23–42, 2015.

• Quantitative limiting absorption principle in the semiclassical limit. Geometric and Functional Analysis. Vol. 24, No. 3, pp. 740–747, 2014.

• Sharp polynomial bounds on the number of Pollicott-Ruelle resonances, with Semyon Dyatlov and Maciej Zworski. Ergodic Theory and Dynamical Systems. Vol. 34, No. 4, pp. 1168–1183, 2014.

• Approximation and equidistribution of phase shifts: spherical symmetry, with Jesse Gell-Redman, Andrew Hassell, and Peter Humphries. Communications in Mathematical Physics. Vol. 326, No. 1, pp. 209–236, 2014.

• Fractal Weyl laws for asymptotically hyperbolic manifolds, with Semyon Dyatlov. Geometric and Functional Analysis. Vol. 23, No. 4, pp. 1145–1206, 2013.

• Extending cutoff resolvent estimates via propagation of singularities. Communications in Partial Differential Equations, Vol. 37, No. 8, pp. 1456–1461, 2012.

• Semiclassical resolvent estimates at trapped sets, with András Vasy. Annales de l'Institut Fourier, Vol. 62, No. 6, pp. 2379–2384, 2012.

• Propagation through trapped sets and semiclassical resolvent estimates, with András Vasy. Annales de l'Institut Fourier, Vol. 62, No. 6, pp. 2347–2377, 2012. See also a short report on the result.

• Gluing semiclassical resolvent estimates via propagation of singularities, with András Vasy. International Mathematics Research Notices, Vol. 2012, No. 23, pp. 5409–5443, 2012. See also my coauthor's short report on the result.

• Inverse problems in spectral geometry, with Hamid Hezari. Inverse Problems and Applications: Inside Out II. Mathematical Sciences Research Institute Publications, No. 60, pp. 455–486, 2012.

• Resonant uniqueness of radial semiclassical Schrödinger operators, with Hamid Hezari. Applied Mathematics Research Express, Vol. 2012, No. 1, pp. 105–113, 2012.

• Spectral uniqueness of radial semiclassical Schrödinger operators, with Hamid Hezari and Ivan Ventura. Mathematical Research Letters, Vol. 18, No. 3, pp. 521–529, 2011.

• Solitary waves for the Hartree equation with a slowly varying potential, with Ivan Ventura. Pacific Journal of Mathematics, Vol. 248, No. 1, pp. 63–90, 2010.

• Fast soliton scattering by attractive delta impurities, with Justin Holmer. Communications in Partial Differential Equations, Vol. 34, No. 9, pp. 1074–1113, 2009.

• Local smoothing for scattering manifolds with hyperbolic trapped sets. Communications in Mathematical Physics, Vol. 286, No. 3, pp. 837–850, 2009.

• Introduction to the method of complex scaling, 2006.

Notes

Introduction to density functional theory

The Laplacian in polar coordinates and spherical harmonics

How much detail should go into a proof?


Past Events

Midwestern Microlocal Meetings have been held at Northwestern in May 2023 and May 2015, and at Purdue in March 2019 and May 2017.

The meeting in March 2019 was a special one in honor of Plamen Stefanov.

I also co-organized a semester program in microlocal analysis at MSRI in Fall 2019.


Past Courses

MA 442 Honors Real Analysis II
MA 495/595 Partial Differential Equations
MA 530 Complex Analysis
MA 504 Real Analysis
MA 562 Introduction to Differential Geometry and Topology
MA 598 Wave Equations
MA 341 Foundations of Analysis
Scattering Theory, taught as part of this Microlocal Analysis Summer School.
MA 428 Introduction to Fourier Analysis
MA 510 Vector Calculus
MA 366 Ordinary Differential Equations
MA 527 Advanced Mathematics For Engineers And Physicists I
MA 362 Vector Calculus
MA 265 Linear Algebra
18.104 Seminar in Analysis: Distribution Theory
18.100B Real Analysis