New York University

Given two probability distributions in R^d, a transport map is a function which maps samples from one distribution into samples from the other. For absolutely continuous measures, Brenier proved a remarkable theorem identifying a unique canonical transport map, which is "monotone" in a suitable sense. We study the question of whether this map can be efficiently estimated from samples. The minimax rates for this problem were recently established by Hutter and Rigollet (2021), but the estimator they propose is computationally infeasible in dimensions greater than three. We propose two new estimators---one minimax optimal, one not---which are significantly more practical to compute and implement. The analysis of these estimators is based on new stability results for the optimal transport problem and its regularized variants. Based on joint work with Manole, Balakrishnan, and Wasserman and with Pooladian.

I will discuss new work on practically popular algorithms, including the kernel polynomial method (KPM) and moment matching method, for approximating the spectral density (eigenvalue distribution) of an n x n symmetric matrix A. We will see that natural variants of these algorithms achieve strong worst-case approximation guarantees: they can approximate any spectral density to epsilon accuracy in the Wasserstein-1 distance with roughly O(1/epsilon) matrix-vector multiplications with A. Moreover, we will show that the methods are robust to *in accuracy* in these matrix-vector multiplications, which allows them to be combined with any approximation multiplication algorithm. As an application, we develop a randomized sublinear time algorithm for approximating the spectral density of a normalized graph adjacency or Laplacian matrices. The talk will cover the main tools used in our work, which include random importance sampling methods and stability results for computing orthogonal polynomials via three-term recurrence relations.

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I will discuss modified Newton methods for solving nonlinear systems of PDEs. These methods introduce additional variables before deriving the Newton linearization. These variables can then often be eliminated analytically before solving the Newton system, such that existing solvers can be adapted easily and the computational cost does not increase compared to a standard Newton method. The resulting algorithms yield favorable convergence properties. After illustrating the ideas on a simple example, I will show its application for the solution of incompressible Stokes flow problems with viscoplastic constitutive relation, where the additionally introduced variable is the stress tensor. These models are commonly used in earth science models. This is joint work with Johann Rudi (Argonne) and Melody Shih (NYU).

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In this talk I will review the work that has been done by me, N. Gromov, V. Kazakov, G. Korchemsky and G. Sizov on the analysis of fishnet Feynman graphs in a particular scaling limit of $\mathcal N=4$ SYM, a theory dubbed $\chi$FT$_4$. After introducing said theory, in which the Feynman graphs take a very simple fishnet form — in the planar limit — I will review how to exploit integrable techniques to compute these graphs and, consequently, extract the anomalous dimensions of a simple class of operators.

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

Dynamics in nature often proceed in the form of reactive events, aka activated processes. The system under study spends very long periods of time in various metastable states; only very rarely does it transition from one such state to another. Understanding the dynamics of such events requires us to study the ensemble of transition paths between the different metastable states. Transition path theory (TPT) is a general mathematical framework developed for this purpose. It is also the foundation for developing modern numerical algorithms such as the string method for finding the transition pathways or milestoning to calculate the reaction rate, and it can also be used in the context of Markov State Models (MSMs). In this talk, I will review the basic ingredients of the transition path theory and discuss connections with transition state theory (TST) as well as approaches to metastability based on potential theory and large deviation theory. I will also discuss how the string method arises in order to find approximate solutions in the framework of the transition path theory, the connections between milestoning and TPT, and the way the theory help building MSMs. The concepts and methods will be illustrated using examples from molecular dynamics, material science and atmosphere/ocean sciences.

Notice that the time is 12:30, not 12:00!

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The following is joint work with Sylvia Serfaty and Cyrill Muratov.

We study the asymptotic behavior of the screened sharp interface

Ohta-Kawasaki energy in dimension 2 using the framework of Γ-convergence.

In that model, two phases appear, and they interact via a nonlocal Coulomb

type energy. We focus on the regime where one of the phases has very small

volume fraction, thus creating ``droplets" of that phase in a sea of the

other phase. We consider perturbations to the critical volume fraction

where droplets first appear, show the number of droplets increases

monotonically with respect to the perturbation factor, and describe their

arrangement in all regimes, whether their number is bounded or unbounded.

When their number is unbounded, the most interesting case we compute the

Γ limit of the `zeroth' order energy and yield averaged information for

almost minimizers, namely that the density of droplets should be uniform.

We then go to the next order, and derive a next order Γ-limit energy,

which is exactly the ``Coulombian renormalized energy W" introduced in the

work of Sandier/Serfaty, and obtained there as a limiting interaction

energy for vortices in Ginzburg-Landau. The derivation is based on their

abstract scheme, that serves to obtain lower bounds for 2-scale energies

and express them through some probabilities on patterns via the

multiparameter ergodic theorem. Without thus appealing at all to the

Euler-Lagrange equation, we establish here for all configurations which

have ``almost minimal energy," the asymptotic roundness and radius of the

droplets as done by Muratov, and the fact that they asymptotically shrink

to points whose arrangement should minimize the renormalized energy W, in

some averaged sense. This leads to expecting to see hexagonal lattices of

droplets.

The computation of flows of compressible fluids will be

discussed, exploiting the symmetric form of the equations describing

compressible flow.