Research

Interests:

I work on questions that arise in Mathematical Physics, and more specifically the long-time behavior of solutions to partial differential equations. I pursue three main lines of research:

• Stability of solitons in nonlinear dispersive equations. The objective is to describe the flow near a traveling wave. The linearization of the equation near that solution, which is generically a quadratic dispersive equation with a potential, drives the dynamics in that situation. Some of my results give a description of the weakly nonlinear regime for such models. 

• Delocalization of Schrödinger eigenfunctions on hyperbolic manifolds. Given the chaotic nature of the underlying classical system, it is expected that these eigenfunctions should not concentrate. One way to quantify this is with their Lebesgue norms, which is the focus of my work in this area.

• Self-similar solutions to compressible Navier-Stokes equations. The goal is to understand singularity formation in compressible fluid dynamics. This is closely related to the existence of self-similar solutions. This question is addressed in the corresponding papers below for Fourier-Navier-Stokes models in the radial case.

Publications:

Here is a list of my publications and preprints:

• Spectral projectors on hyperbolic surfaces, with J.-P. Anker and P. Germain (2023), 46 pages, arXiv:2306.12827, submitted

• Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space, with P. Germain, J. Funct. Anal. 285 (2023), no.2, Paper No. 109918, 37 pp

• Internal mode-induced growth in nonlinear Klein-Gordon equations, with F. Pusateri, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 33 (2022), no.3, 695-727

• Internal modes and radiation damping for quadratic Klein-Gordon in 3D, with F. Pusateri (2021), 125 pages, arXiv:2112.13163, submitted

• Backward self-similar solutions for compressible Navier-Stokes equations, with P. Germain and T. Iwabuchi, Nonlinearity 34 (2021), no.2, 868-893

• On self-similar solutions to degenerate compressible Navier-Stokes equations, with P. Germain and T. Iwabuchi, Comm. Math. Phys. 381 (2021), no.3, 1001-1030

• Global existence and scattering for quadratic NLS with potential in 3D, Analysis & PDE 14 (2021) 1977-2046

• Scattering for a particle interacting with a Bose gas, Comm. Partial Differential Equations 45 (2020), no.10, 1381-1413

• 3D quadratic NLS with electromagnetic perturbations, Adv. Math. 375 (2020), 107407, 70 pp

• Quadratic NLS with potentials, Ph.D. Thesis, ProQuest LLC (2020), 239 pp

The corresponding preprints are available on my arXiv page.

Departmental activities

I am a co-organizer of the Analysis seminar at Princeton. It takes place on Mondays 3-4 in Fine Hall 314.

Previously I was a co-organizer of the Analysis of Fluids and Related Topics at Princeton.

Teaching

I have taught the following classes at NYU:

Instructor: 

• Calculus I (Summer 2017, Summer 2018, Summer 2019)

Recitations (teaching assistant): 

• TA for Analysis (Spring 2017, Fall 2017, Spring 2018, Spring 2019)

• TA for Honors Analysis (Fall 2017)

• TA for Honors Analysis II (Spring 2020)

• TA for ODEs (Fall 2018, Fall 2019)

• TA for PDEs (Spring 2017)

Grader:

• Grader for PDE I, Graduate course (Fall 2016) 

Outreach:

• I have also taught at CSplash. It is a non-profit event that aims at introducing university level Mathematics to local high-school students.