Associate Professor
Department of Mathematics
Purdue University
150 North University Street
West Lafayette, IN 479072067
Office: 602 Mathematical Sciences Building
Teaching
In Spring 2021 I am teaching two courses:
MA 428 Introduction to Fourier Analysis
MA 442 Honors Real Analysis II
PhD Student
I have advised one PhD student,
Jacob Shapiro, who graduated in 2018.
Papers

On the interaction of metric trapping and a boundary,
with Jason Metcalfe, Jacob Shapiro, and Mihai Tohaneanu.

Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues ,
with Tanya Christiansen.

Wave asymptotics for waveguides and manifolds with infinite cylindrical ends,
with Tanya Christiansen.

Resolvent estimates, wave decay, and resonancefree regions for starshaped waveguides,
with Tanya Christiansen. To appear in Mathematical Research Letters.

Resolvent estimates on asymptotically cylindrical manifolds and on the half line,
with Tanya Christiansen. To appear in Annales Scientifiques de l'École Normale Supérieure.

Sharp polynomial decay rates for the damped wave equation with Hölderlike 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, 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 PollicottRuelle 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 GellRedman, 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.
Events
I coorganized a
semester program in microlocal analysis at MSRI in Fall 2019.
I coorganized a
conference in honor of Plamen Stefanov at Purdue from March 22nd to 24th, 2019.
The above conference was also the third in a series of Midwestern Microlocal Meetings. The
first was at Northwestern in May 2015 and the
second at Purdue in May 2017.
Previous Courses
MA 510 Vector Calculus.
MA 366 Ordinary Differential Equations.
MA 341 Foundations of Analysis.
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.