Research Interests:

My current research interests lie at the intersection of mathematical physics, nonlinear partial differential equations, and probability. More specifically, I have been recently interested in the mathematics of nonlinear dispersive, fluid, and kinetic equations and their derivation through scaling limits from underlying physical problems, such as large classical & quantum systems of particles or waves.  More details on my research can be found here.

My research is funded in part by the Simons Foundation through the Simons Collaboration on Wave Turbulence and by the NSF through grant DMS-2206085.

The Simons Collaboration on Wave Turbulence is sponsoring a summer school July 24-28, 2023 at MIT. Details and application may be found here.

Papers and Preprints:

1. w/ S. Aryan, G. Staffilani. The trend to equilibrium for flows with random diffusion, in preparation.

2. w/ A. Hannani, G. Staffilani, M.B. Tran. On a transport-diffusion equation originating in wave turbulence theory, in preparation.

3. w/ A. Hannani,  G. Staffilani, M.B. Tran. On the wave turbulence theory for a stochastic KdV type equation - generalization for the inhomogeneous kinetic limit, https://arxiv.org/abs/2210.17445, 109 pgs., preprint (2022).

4. w/ A. Chodron de Courcel, S. Serfaty. The attractive log gas: phase transitions, (non)uniqueness and nonlinear (in)stability of equilibria, and uniform-in-time mean-field convergence, in preparation.

5. w/ A. Chodron de Courcel, S. Serfaty. Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows, https://arxiv.org/abs/2304.05315, 63 pgs, preprint (2023).

6. Scaling-critical mean-field convergence for systems with Riesz interactions, in preparation.

7. w/ J.K. Miller, A.R. Nahmod, N. Pavlović, G. Staffilani. A Rigorous Derivation of the Hamiltonian Structure for the Vlasov Equation, https://arxiv.org/abs/2206.07589, 54 pgs., preprint (2022).

8. w/ S. Serfaty. Sharp estimates for variations of Coulomb and Riesz modulated energies, applications to supercritical mean-field limits, in preparation.

9. A Rigorous Justification of the Stochastic Point Vortex Model with Multiplicative Noise, in preparation.

10. From Quantum Many-Body Systems to Ideal Fluids,  https://arxiv.org/abs/2110.04195, 29 pgs., preprint (2021).

11. w/ G. Staffilani. Global Solutions of Aggregation Equations and Other Flows with Random Diffusion. Prob. Theory Related Fields 185 (2023), no. 3-4, 1219-1262.

12. w/ S. Serfaty, Global-in-time mean-field convergence for singular Riesz-type diffusive flows. Ann. Appl. Prob. 33 (2023), no. 2, 754-798.

13. On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law. Lett. Math. Phys. 113 (2023), no. 1, paper. no 13. 

14. w/ G. Staffilani, Uniqueness of solutions to the spectral hierarchy in kinetic wave turbulence theory. Phys. D 433 (2022), Paper No. 133148, 16 pp.

15. The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise, https://arxiv.org/abs/2011.12180, 34 pgs., (2020), accepted by Comm. Pure Appl. Math.

16. The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical $L^\infty$ space. Nonlinearity 35 (2022), no. 6, 2722-2766.

17. Mean-field convergence of point vortices to the incompressible Euler equation with vorticity in $L^\infty$. Arch. Ration. Mech. Anal. 243 (2022), no. 3, 1361-1431.

18. Old and New Perspectives on Effective Equations: A Study of Quantum Many-Body Systems, dissertation, 398 pgs., (2020).

19. Global Well-Posedness and Scattering for the Hartree Equation at $L^{2}$-Critical Regularity,  draft, 137 pgs., (2019).

20. The Mean-Field Limit of the Lieb-Liniger Model,  Discrete Contin. Dyn. Syst. 42 (2022), no. 6, 3005-3037.

21.  w/ D. Mendelson A.R. Nahmod, N. Pavlović, G. Staffilani. Poisson commuting energies for a system of infinitely many bosons. Adv. Math. 406 (2022), Paper No. 108525, 148 pp.

22. w/ D. Mendelson, A.R. Nahmod, N. Pavlović, G. Staffilani. A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrodinger Equation. Adv. Math. 365 (2020), 107054, 115 pp.

23. Justification of the Point Vortex Approximation for Modified Surface Quasi-Geostrophic Equations. SIAM J. Math. Anal. 52, No. 2 (2020), 1690-1728.

24. Global Well-Posedness and Scattering for the Elliptic-Elliptic Davey-Stewartson System at $L^{2}$-Critical Regularity,  https://arxiv.org/abs/1808.01955, 129 pgs., preprint (2018).

Notes and Expository Material:

Caveat Emptor: the material below is likely riddled with typos.

1. Uniform-in-time mean-field convergence for 2D Coulomb gradient flows - A short note showing that for the gradient flow of the 2D Coulomb gas on the torus, one can has an exponentially fast relaxation of solutions to the uniform distribution as $t\rightarrow\infty$ and one can prove uniform-in-time mean-field convergence for any $L^\infty$ probability density. This seems to be the first such result for a model with a singular pair potential.

2. Beyond Gronwall for gradient flows - A short note showing that in some cases (e.g. Coulomb gradient flow on $\mathbb{R}^d$), one can prove mean-field convergence for noiseless systems of particles with a convergence rate having time factor $O(t^{\delta})$, for some $\delta>0$. While well-known that noise improves the growth in time of the rate (possibly making the rate uniform), it doesn't seem to have been previously observed you can go beyond the Gronwall $e^{Ct}$ growth without noise.

3. Pickl's method for $H^1$ data - A note showing how to use dispersion of the limiting equation to reduce the regularity assumptions in Pickl's derivation of the 3D Gross-Pitaevskii equation.

4. Convolution inequalities for Boltzmann collision operator - These notes are based on the work by Alonso, Carneiro, and Gamba.

5. Critical conditional global well-posedness and scattering for cubic NLS in $\mathbb{R}^{3}$ - These notes are based on the work by Kenig and Merle.

6. Introduction to Fourier analysis on the torus -- These notes give an introduction to Fourier analysis on the torus $\mathbb{T}^{d}$. The notes include a treatment of Kolmogorov's construction of an $L^{1}(\mathbb{T})$ function whose Dirichlet means diverge pointwise almost everywhere.