Overview
Links to papers, lecture notes and slides from various talks in the past few years are organized by area. Some articles appear in more than one list. This website provides an overview of my work. My recent math preprints are all on the arxiv, especially papers related to the Nash embedding theorems. I hope the brief blurbs here will provide some insight into the motivation and scope of several projects.
Please email me directly for preprints/materials related to talks if you can't find them below. The notes on the embedding problem below are technical reports that cover several parts of my program and include many original results on model development, Gibbs measures for SDP, analogies with RMT, simplified PDE. These notes were written when the ideas were being developed from 2018-2022. For completed results, please see preprints on the arxiv on isometric embedding and the dynamics of optimization. I expect to organize these reports into lecture notes and preprints in 2024.
Related lecture notes that include the interplay of dynamics and algorithms, and a new take on pattern theory are included below. The pattern theory class includes several Gibbs measures for applications in machine learning, as well as a discussion of fast algorithms. Two chapters on thermodynamics of learning are not included in the current version. These notes provide a quick start on information theory and Gibbs measures (should be useful if you're an analyst trying to figure out what the fuss is all about).
Publications organized by area
Dynamical systems and integrable systems.
Fluid mechanics.
Kinetic theory, phase transitions, materials.
Models of turbulence and embedding.
Random matrix theory and numerical linear algebra.
Self-assembly.
Recent papers not on the arxiv
Information theory and the embedding problem for Riemannian manifolds , GSI 2021.
The second law: information theory and self-assembly , Biophysical Journal 2021.
These papers develop the conceptual basis and applications for the "stochastic Nash evolution" framework that I have been developing since 2016. However, it took another two years to find the right evolution equations beginning with Nash's proofs. We now know that it is the Riemannian Langevin equation (RLE) and the simplest argument for the probabilistic basis for isometric embedding is provided in our recent paper with Dominik Inauen. The RLE framework provides geometric stochastic flows for Nash embedding. It also yields Gibbs sampling schemes that we term Riemannian Langevin Monte Carlo (RLMC). The use of Riemannian geometry provides a common framework for the dynamics of gradient descent in classical optimization (conic programs) as well as deep learning. The specific nature of the Riemannian geometry in deep learning is very subtle. However, we can gain several insights using phenomenological models such as the deep linear network.
Slides and texts of some talks
Building polyhedra by self-assembly, (AMS Fall 2017).
Informal description in the Notices of the AMS.
Stochastic Loewner evolution with branching and the Dyson superprocess, (PCMI July 2017). Video of the talk may be found here .
The related preprint may be found on the arxiv.
Harish-Chandra's integral, the Calogero-Moser system and the complex Burgers equation. (CMSA, Harvard, Jan 2017). This talk is on the `standard' U(N) integral and contains a new Lax pair. My student Colin McSwiggen has developed a complete representation theoretic description of the asymptotics of HC integrals over compact, semisimple Lie groups.
Kinetic theory of grain boundary networks , last updated Oct. 2016.
Complete integrability of shock clustering , last updated Oct. 2015.
Conservation laws with random data and the equation free method , Dec. 2015.
How long does it take to compute the eigenvalues of a random matrix, last updated April 2016.
Lecture notes and technical reports
Dynamical systems , last updated Jan 2024.
Pattern theory, last updated Jan 2021.
Gaussian processes, the isometric embedding problem and turbulence , last updated June 2020.
The isometric embedding problem and random matrix theory last updated, Aug. 2020.
Gibbs measures for semidefinite programming , Sept. 2020.
Mathematics and Materials. Edited with Mark Bowick, David Kinderlehrer and Charles Radin. Vol. 23, IAS/Park City Mathematics Series, AMS/SIAM (2017).
A quick introduction to kinetic theory. Lectures at Hebrew University, University of Chicago and TIFR, 2016-2017.
Statistical theories of turbulence. Graduate course Spring 2016, last updated April 2016).
Random matrix theory and Numerical Algorithms. Book in progress with Tom Trogdom. Last updates July 2023. This is missing a couple of chapters on dynamics of algorithms, but is usable in its current form.
Partial differential equations graduate sequence in 2005-2006.
2005 notes.
Conservation laws.
Translation of Hopf's paper on Navier-Stokes