BCAM Course

Overview of (wave) turbulence and statistical mechanics. Formal derivation of Wave Kinetic Equations from semi-linear, dispersive PDEs. Random initial data RP and RPA. Resonance vs quasi-resonances, counting lemmas and scaling laws.

Introduction to Gaussian random variables and Gaussian Hilbert Spaces. Isserlis’/Wick’s theorem, Feynmann diagrams and combinatorics. Wick products, Wiener chaos decomposition and Gaussian hypercontractivity estimates.

The cubic NLS equation with random initial data. Picard iteration scheme. Closed formulas for Picard iterates in terms of trees. Control of the remainder terms and high-order TT* argument.

Probabilistic bounds on Picard iterates, two and three-vector counting and general counting algorithms.

Derivation of the Wave Kinetic equation from Picard iteration: Poisson summation, stationary phase argument, Gauss sums and Hua’s lemma.

I am an Assistant Professor (RTD-A) at SISSA.

Previously, I was a Postdoctoral Researcher at École Normale Supérieure working with Isabelle Gallagher and Laure Saint-Raymond. During 2020-21, I was a Postdoctoral Assistant Professor at the University of Michigan – Ann Arbor working with Zaher Hani. I obtained my PhD from MIT in 2020, under the supervision of Gigliola Staffilani.

More about me can be found in my CV.

Address: Office A-727, Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy
Email: rgrandei [at] sissa [dot] it

I am interested in Hydrodynamic and Wave Turbulence, Dispersive and Kinetic PDEs, Probability and Harmonic Analysis.
Here are some of my publications:

1. Resonant large deviations principle for the beating NLS equation, preprint (2024)
2. Rigorous derivation of damped-driven wave turbulence theory (with Z. Hani), preprint (2024)
3. On the convergence rates of discrete solutions to the Wave Kinetic Equation (with M. Dolce), Math. Eng. 6 (4), 536-558 (2024)
4. Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields (with G. Beck, C.-E. Bréhier, L. Chevillard and W. Ruffenach), Phys. Rev. Research 6, 033048 (2024)
5. A Linear Stochastic Model of Turbulent Cascades and Fractional Fields (with G.B. Apolinario, G. Beck, L. Chevillard and I. Gallagher), to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)
6. Large deviations principle for the cubic NLS equation (with M.A. Garrido, K. M. Kurianski and G. Staffilani), Comm. Pure Appl. Math. (2023)
7. Continuum limit for discrete NLS with memory effect, to appear in Journal of Nonlinear Modeling and Analysis (2024)
8. On the nonlinear Dysthe equation (with K. M. Kurianski and G. Staffilani), Nonlinear Analysis 207, 112292 (2021)
9. Space-time fractional nonlinear Schrödinger equation, SIAM J. Math. Anal (2019), 51(5), 4172-4212
10. Equisizable partial sum families (with I. Kovács, K. Kutnar, A. Malnič, L. Martínez and D. Marušič), J Algebr Comb 51, 273–296 (2020)
11. On embeddings of circulant graphs (with Marston Conder), Electronic Journal of Combinatorics 22 (2015), P2.28