Mathematics in progress
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Announcements, working observations, and expository notes.
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Earlier notes and expository material
Caveat emptor: the material below is likely riddled with typos.
- Higher-order transport commutators and cumulants of linear statistics — A brief note explaining how higher-order transport-commutator expressions arise naturally when computing cumulants of linear statistics of a Gibbs ensemble. It is stated for Coulomb/Riesz gas ensembles, but the argument is not specific to the interaction.
- Uniform LSI for the subcritical Kuramoto model — A short note explaining how work of Bauerschmidt and Bodineau on uniform logarithmic Sobolev inequalities for spin systems implies a uniform LSI for the Kuramoto model (the \(O(2)\), or classical XY, model) above the critical temperature.
- Uniform-in-time mean-field convergence for 2D Coulomb gradient flows — Exponentially fast relaxation to the uniform distribution and uniform-in-time mean-field convergence for any \(L^\infty\) probability density.
- Beyond Gronwall for gradient flows — A short note on noiseless mean-field convergence with a polynomial, rather than exponential, time factor in some Coulomb gradient flows.
- Pickl’s method for \(H^1\) data — A note using dispersion of the limiting equation to reduce the regularity assumptions in Pickl’s derivation of the three-dimensional Gross–Pitaevskii equation.
- Convolution inequalities for the Boltzmann collision operator — Notes based on work of Alonso, Carneiro, and Gamba.
- Critical conditional global well-posedness and scattering for cubic NLS in \(\mathbb{R}^3\) — Notes based on work of Kenig and Merle.
- Introduction to Fourier analysis on the torus — An introduction to Fourier analysis on \(\mathbb{T}^d\), including Kolmogorov’s construction of an \(L^1(\mathbb{T})\) function whose Dirichlet means diverge pointwise almost everywhere.
- Permutation-invariant functional inequalities for finite Ginibre I: The exact Poincaré rate and the centered LSI frontier — An unpublished technical research memorandum proving the exact permutation-invariant Poincaré rate and identifying several sharp LSI subproblems for finite Ginibre ensembles.
- A spectral gap for the unlabeled infinite Ginibre interacting Brownian motion — An unpublished note proving a particle-number-uniform finite symmetric-sector Poincaré inequality and a positive spectral-gap bound for the infinite Ginibre diffusion.