The objective of the tutorial Computer-Assisted Proofs in Nonlinear Analysis is to introduce participants to fundamental concepts of a posteriori validation techniques. This mini-course will cover topics ranging from finite-dimensional problems, such as finding periodic orbits of maps, to infinite-dimensional problems, including solving the Cauchy problem, proving the existence of periodic orbits, and computing invariant manifolds of equilibria. Each session will last 2 hours: 1 hour of theory followed by 1 hour of hands-on practical applications. The practical exercises will focus on implementing on implementing computer-assisted proofs using the Julia programming language.

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Please bring your laptop to the tutorial and find the attached syllabus for more information. (to download the syllabus) https://saarc.kaist.ac.kr/boards/view/seminars/32

The objective of the tutorial Computer-Assisted Proofs in Nonlinear Analysis is to introduce participants to fundamental concepts of a posteriori validation techniques. This mini-course will cover topics ranging from finite-dimensional problems, such as finding periodic orbits of maps, to infinite-dimensional problems, including solving the Cauchy problem, proving the existence of periodic orbits, and computing invariant manifolds of equilibria. Each session will last 2 hours: 1 hour of theory followed by 1 hour of hands-on practical applications. The practical exercises will focus on implementing on implementing computer-assisted proofs using the Julia programming language.

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Please bring your laptop to the tutorial and find the attached syllabus for more information. (to download the syllabus) https://saarc.kaist.ac.kr/boards/view/seminars/32

Wave turbulence refers to the statistical theory of weakly nonlinear dispersive waves. In the weakly turbulent regime of a system of dispersive waves, its statistics can be described via a coarse-grained dynamics, governed by the kinetic wave equation. Remarkably, kinetic wave equations admit exact power-law solutions, called Kolmogorov-Zakharov spectra, which resemble Kolmogorov spectrum of hydrodynamic turbulence, and is often interpreted as a transient equilibrium between excitation and dissipation. In this talk, we will outline a local well-posedness result for kinetic wave equation for a toy model for wave turbulence. The result includes well-posedness near K-Z spectra, and demonstrates a surprising smoothing effect of the kinetic wave equation. The talk is based on the joint work with Pierre Germain (ICL) and Katherine Zhiyuan Zhang (Northeastern).

Modern machine learning methods such as multi-layer neural networks often have millions of parameters achieving near-zero training errors. Nevertheless, they maintain strong generalization capabilities, challenging traditional statistical theories based on the uniform law of large numbers. Motivated by this phenomenon, we consider high-dimensional binary classification with linearly separable data. For Gaussian covariates, we characterize linear classification problems for which the minimum norm interpolating prediction rule, namely the max-margin classification, has near-optimal generalization error. In the second part of the talk, we consider max-margin classification with non-Gaussian covariates. In particular, we leverage universality arguments to characterize the generalization error of non-linear random features model, a two-layer neural network with random first layer weights. In the wide-network limit, where the number of neurons tends to infinity, we show how non-linear max-margin classification with random features collapse to a linear classifier with a soft-margin objective.

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schrodinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. Such margin-constrained random matrix turns out to be sharply concentrated around a certain deterministic matrix, which we call the "typical table". Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two dual variables, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We also propose a Sinkhorn-type alternating minimization algorithm for computing typical tables, which speicalizes to the classical Sinkhorn algorithm for the Poisson base measure. We derive several new results for random contingency tables from our general framework. This talk is based on a Joint work with Sumit Mukherjee (Columbia).

In this talk, we investigate some regularity results for non-uniformly elliptic problems. We first present uniformly elliptic problems and the definition of non-uniform ellipticity. We then introduce a double phase problem which is characterized by the fact that its ellipticity rate and growth radically change with the position. We show gradient Hölder continuity and Calderón-Zygmund type estimates for distributional solutions to double phase problems. We also consider double phase problems with two modulating coefficients.