Research

Interests:

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

• Stability of solitons in nonlinear dispersive equations. Solitons are ubiquitous objects in nonlinear analysis. They are used in many scientific fields as a modeling tool (e.g. ocean and plasma waves, signals in fiber optics, deformations in plastic materials, DNA molecules...). Yet questions related to their formation and evolution are still unanswered. This project aims to understand the stability of these special structures. 

Mathematically, the objective is to describe the flow of a nonlinear dispersive PDE near a traveling wave. The linearization of the equation near that solution, which is generically a quadratic dispersive equation with a potential,  is expected to drive the dynamics. See papers [6], [7], [10], [11], [12] and my PhD thesis [13] below for my results in this direction. 

• Delocalization of Schrödinger eigenfunctions on hyperbolic manifolds. Most physical systems are chaotic, thus making long-time deterministic predictions of the evolution extremely challenging. As a result a more probabilistic approach is often taken. In the specific problem of statistical energy level distribution of complex nuclei (paramount in nuclear fission), great agreement was experimentally observed with the simpler model of particle motion on a negatively curved space. This connection is now believed to go well beyond the aforementioned example, and be common to many chaotic systems. Yet this remains largely unexplained theoretically. 

To make progress on the question, this project seeks to refine the understanding of motion on negatively curved manifolds through the study of eigenfunctions of the Laplace-Beltrami operator on hyperbolic manifolds. This can then be compared with analogous results in large random matrix models. More specifically, I am interested in measuring the concentration of these eigenfunctions by providing sharp estimates of their Lebesgue norms. Such results can be found in the papers [4], [5] below in the case of real hyperbolic spaces and hyperbolic surfaces of infinite area respectively.

• Dynamics of kinetic equations. Kinetic theory provides equations that model the evolution of systems that are too complex to be simulated numerically. The most celebrated example is the Boltzmann equation for gas particles. This general program was also successfully implemented in kinetic wave theory for systems of waves in a variety of contexts (water waves for navigation, superfluid helium, Bose-Einstein condensates, waves in plasmas of fusion devices...)

However the mathematical study of these equations is still in its infancy. Even basic questions of local well-posedness or small data behavior are largely open. Some results in this direction can be found in the papers [2] and [3] below for kinetic wave equations, and [1] for the classical and quantum Boltzmann equations. 

Publications:

Here is a list of my publications and preprints:

1. Convolution estimates for the Boltzmann gain operator with hard spheres, with I. Ampatzoglou (2025), 38 pages, arXiv:2505.09554

2. On the ill-posedness of kinetic wave equations, with I. Ampatzoglou (2024), 24 pages, arXiv:2411.12868, submitted

3. Scattering theory for the Inhomogeneous Kinetic Wave Equation, with I. Ampatzoglou (2024), 41 pages, arXiv:2408.05818, submitted

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

5. Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space, with P. Germain, Journal of Functional Analysis 285 (2023), no.2, Paper No. 109918, 37 pp

6. 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

7. Internal modes and radiation damping for quadratic Klein-Gordon in 3D, with F. Pusateri (2021), 126 pages, Accepted in Memoirs of the Amer. Math. Soc.

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

9. 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

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

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

12. 3D quadratic NLS with electromagnetic perturbations, Advances in Mathematics 375 (2020), 107407, 70 pp

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

The corresponding preprints are available on my arXiv page.

Departmental activities

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

Teaching

At Yale:

• Math 246, Ordinary Differential Equations (Fall 2025)

• Math 256, Analysis - intensive (Spring 2025)

At Princeton:

• Mat201, Multivariable calculus (Fall 2024)

At NYU:

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

Outreach:

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