Applied Analysis: from the calculus of variations to materials science, finance and data science – a celebration of the science of Bob Kohn

John Ball
• Title: Image comparison via nonlinear elasticity
• Abstract: The talk will describe how models based on nonlinear elasticity can be used to compare images and parts of images. The requirement that the corresponding minimization algorithm delivers a linear map between linearly related images leads to a new condition of quasiconvexity. This is joint work with Chris Horner.
 
Kaushik Bhattacharya
• Title: Learning effective properties of heterogeneous media
• Abstract: Various phenomena in the physical sciences and engineering leads to systems of parametrized partial differential equations with rapidly varying parameters. Homogenization theory provides a framework to approximate the solutions these systems (microscale problem) with those of a new system with slowly varying parameters (macroscale problem). However, the macroscale problem and the associated parameters (effective properties) are rarely defined explicitly, but commonly implicitly through the solution of the microscale problem. Bob Kohn and his collaborators have made seminal contributions in characterizing effective properties through bounds and estimates in linear and nonlinear phenomena. This talk revisits this issue using emerging machine learning methods and examines whether one can use data generated by the repeated solution of small-scale problems to characterize the effective properties and gather insights into the underlying physics. This naturally leads to neural operators and its ability to approximate the solution of parametrized partial differential equations. We provide examples where. the approach shows promise, and comment how an understanding of the underlying mathematics allows the construction of good approximations.
 
Lia Bronsard
• Title: Core shells and double bubbles in a weighted nonlocal isoperimetric problem
• Abstract: We study the Nakazawa-Ohta ternary inhibitory system, which describes domain morphologies in a triblock copolymer as a nonlocal isoperimetric problem for three
interacting phase domains. The free energy consists of two parts: the local interface energy measures the total perimeter of the phase boundaries, while a longer-range Coulomb interaction energy reflects the connectivity of the polymer chains and promotes splitting into micro-domains. We consider global minimizers on the two-dimensional torus, in a limit in which two of the species have vanishingly small mass but the interaction strength is correspondingly large. In this limit there is splitting of the masses, and each vanishing component rescales to a minimizer of an isoperimetric problem for clusters in 2D. Depending on the relative strengths of the coefficients of the interaction terms we may see different structures for the global minimizers, ranging from a lattice of isolated simple droplets of each minority species to double bubbles or core-shells. This represents work with S. Alama, X. Lu, and C. Wang.
 
Sergio Conti
• Title: Variational models of dislocation microstructures in crystals
• Abstract: Dislocations are topological singularities of the strain field, which play a crucial role in the plastic deformation of crystals. Their models offer an interesting playground in the calculus of variations. I shall give an overview of recent progress in the variational analysis of dislocation models in linear and finite elasticity, and in the derivation of continuum strain-gradient plasticity from dislocation model. The talk is based on joint work with Adriana Garroni, Stefan M¨uller and Michael Ortiz.
 
Camillo De Lellis
• Title: Area-minimizing integral currents: singularities and structure
• Abstract: Almgren’s famous Big Regularity Paper proves that the interior singular set of any mdimensional area-minimizing integral current T in any smooth Riemannian manifold M has (Hausdorff) dimension at most m − 2. Except for the case m = 2, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover a recent theorem by Liu proves that we cannot expect it to be a C 1 m − 2 dimensional submanifold (unless the ambient M is real-analytic) as in fact it can be a fractal set of any Hausdorff dimension α ≤ m − 2. On the other hand it seems likely that it is an (m − 2)-rectifiable set, i.e. that it can be covered by countably many C 1 submanifolds. In this talk I will explain why the problem is very challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.
 
Guido De Philippis
• Title: Min-Max construction of anisotropic minimal surfaces
• Abstract: Since the seminal work of Birkhoff min max techniques have been a powerful tool in showing existence of stationary point of geometric variational problem. Almgren-Pitts developed in the 80`s extended the min-max framework to the area functional, to show existence of minimal surfaces. In this talk I will discuss the extension of the Almgren-Pitts theory to anisotropic energies. The lack of local area bounds requires to devise a new strategy to the regularity theory. This is a joint work with A. De Rosa.
 
Antonio De Simone
• Title: Some recent problems in biological and bio-inspired locomotion
• Abstract: In recent years we have studied a number of problems related to locomotion and shape control in several biological systems. This has required the use of tools ranging from control theory and theoretical/computational mechanics, to experiment and observations at the microscope, to manufacturing of prototypes. We will review some of the most instructive lessons that have emerged from this research line.
 
Qiang Du
• Title: Trace theorems for nonlocal function spaces with or without localization
• Abstract: We consider function spaces associated with nonlocal boundary value problems on a bounded domain involving a finite range of nonlocal interactions. We provide recent characterizations given in [1] of the trace spaces of such function spaces as generalizations of the classical trace theorems for local Sobolev spaces, with the latter being the local limits. We also present analogous results for nonlocal problems involving interactions heterogeneously localized at the boundary [2,3]. 1. Du Q, Tian X, Wright C, Yu Y. Nonlocal trace spaces and extension results for nonlocal calculus. Journal of Functiona Analysis. 2022; 282(12):109453. 2. Tian X, Du Q. Trace theorems for some nonlocal function spaces with heterogeneous localization. SIAM Journal on Mathematical Analysis. 2017;49(2):1621-44. 3. Du Q, Mengesha T, Tian X. Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization. Analysis and Applications. 2022;20(03):579-614.
 
Irene Fonseca
• Title: Phase Separation in Heterogeneous Media
• Abstract: Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. In the subcritical case with moving wells where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. The supercritical case for fixed wells is also addressed, and a partial characterization of the limit energy is given. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).
 
Gilles Francfort
• Title: Hyperbolicity as a possible path to the uniqueness of minimizers for energies with linear growth
• Abstract:In this joint work with JF Babadjian we investigate a typical functional of the gradient which exhibits linear growth at infinity. The relaxed functional has BV minimizers. In 2d, their uniqueness is intimately tied to the properties of a spatial continuity equation for which the theory of regular Lagrangian flows does unfortunately not apply. However, techniques related to the work of Jabin-Otto Perthame on 2d Ginzburg-Landau models permit a better understanding of the associated characteristic flow.
 
Adrianna Garroni
• Title: Surface tension for grain boundaries in polycrystals.
• Abstract: We consider a sharp interface model for grain boundaries in 2 dimensions and its variational derivation starting from a semi discrete model accounting for elastic deformation and incompatibilities at the boundaries. The model was proposed by Lauteri and Luckhaus in order to theoretically validate the Read and Shockley formula for small angle grain boundaries. It turns out that the technics developed by Lauteri and Luckhaus are very powerful and a careful localization and adaptation of their results allows to completely derive as a Gamma limit the sharp interface limit. The limiting energy density is defined through an asymptotic cell problem formula and exhibits a logarithmic behaviour for small angles in agreement with Read Shockley formula. The talk is based on a paper in collaboration with Emanuele Spadaro.
 
Leslie Greengard
• Title: A new integral representation for unsteady, incompressible flow
• Abstract: We describe a new representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each of these equations is well conditioned, and we show that using them in alternating step fashion leads to an effective numerical method. The fundamental unknowns in this ”mixed potential representation” are densities supported on the boundary of the domain. In the three dimensional case, one density is a tangential vector field which controls the vorticity, while the other is a scalar function which controls the pressure. (This is joint work with Shidong Jiang.)
 
David Jerison
• Title: How curved are level sets of harmonic functions and eigenfunctions?
• Abstract: This talk will present a new geometric approach to the Polya-Szego isoperimetric capacity conjecture and other conjectures concerning level sets of eigenfunctions and harmonic functions.
 
Jianfeng Lu
• Title: Analysis of score-based generative models
• Abstract: Diffusion models, such as score-based generative models, have achieved impressive performance in generation of samples from complex data distributions such as that of images and audio. It is based on evolving a stochastic differential equation (SDE) that transforms white noise into a sample from the learned distribution, using estimates of the score function, or gradient log-pdf. In this talk, we will discuss some
recent progress in analysis of such approaches, in particular, assuming L2-accurate score estimates, we obtain polynomial complexity bound for approximating the target distribution without structural assumptions such as log-Sobolev inequalities. (Joint work with Hongrui Chen, Holden Lee, and Yixin Tan)
 
Maria-Giovanna Mora
• Title: Explicit minimizers for anisotropic Coulomb energies
• Abstract: Nonlocal interaction energies play a pivotal role in describing the behavior of large systems of particles, in a variety of applications. Traditionally, the focus of the mathematical literature on nonlocal energies has been on radially symmetric potentials, which model interactions depending on the mutual distance between particles. The mathematical study of anisotropic potentials, despite their natural occurrence in modeling interactions where a preferred direction of interaction is present, has on the other hand been very limited until recently. In this talk we will consider a general class of anisotropic energies of Coulomb type in three dimensions and give a complete characterization of their minimizers, under the sole assumption of non-negativity for the Fourier transform of the interaction kernel.
 
Peter Sternberg
• Title: Entire solutions to vector Allen-Cahn with triple junction structure for asymmetric potentials
• Abstract: I will describe a new result for vector Allen-Cahn assuming a triple-well potential. We find an entire, locally minimizing solution u from R2 to R2 whose ‘blowdowns’ from infinity exhibit triple junction structure. This is joint work with Etienne Sandier (Paris XII).
 
Vladimir Sverak
• Title: On the Caffarelli-Kohn-Nirenberg Theorem
• Abstract: The 1982 paper ”Partial regularity of the Suitable Weak Solution of the Navier-Stokes Equation” by Luis Caffarelli, Bob Kohn, and Louis Nirenberg is one of the classics of PDE analysis. The ideas in the paper have been very influential and have led to many advances. I will discuss examples showing how the results in the paper are crucial for some more recent results.
 
Ian Tobasco
• Title: Pattern forming and shape changing thin elastic sheets
• Abstract: What do wrinkled shells and shape-morphing kirigami have in common? Two answers: Bob worked on both; also, they are models for understanding the rich interplay between energy minimization and geometrical constraints. This talk will present two recent developments on (i) a hidden duality between the wrinkles of positively and negatively curved confined shells, and (ii) the aggregate deformations of kirigami sheets made by removing a lattice of holes. In both cases a homogenization-style argument leads to a coarse-grained description of the pattern which is easier to study, and is even explicitly solvable sometimes. Wrinkle patterns become the subdifferentials of pairs of convex functions; kirigami is shown to deform along a family of non Euclidean isometries set by the holes. These results are obtained via matching upper and lower bounds on the minimum energy.
 
Felix Otto
• Title: Universality of the magnetization ripple: A singular SPDE-perspective
• Abstract: The ripple in a thin-film ferromagnet is the magnetization’s anisotropic response to poly crystallinity; it is experimentally well-documented. The randomly oriented grains come with a favorite axis for the magnetization that acts as a random field, a source of quenched noise. On a mesoscopic level, the ripple is modeled by a 2-d anisotropic and non-local variational problem with a random field term converging to white noise. White noise itself is too rough for the Burgers’ type nonlinearity to have a classical sense, so that a renormalization is called for. We follow the strategy of rough path/regularity structures in the sense that we enhance the noise description. The main result is that of universality: The mesoscopic ensemble of the ripple is independent on how the limit of white noise is approached, it is thus independent of the statistics of the grain arrangement. In view of the non convexity of the variational problem, this statement is formulated in terms of a suitable Γ-topology. On the technical side, we appeal to the spectral gap inequality to avoid the assumption of Gaussianity of the approximation. We also leverage a fluid-type identity for the Burgers equation to obtain coercivity of the renormalized variational problem. This is joint work with Radu Ignat, Tobias Ried, and Pavlos Tsatsoulis.
 
Gunther Uhlmann
• Title: The Calderon Problem: 40 Years Later
• Abstract: Motivated by a talk by Bob Kohn at MSRI in early 1983, I started working on Calderon’s Inverse Problem: can one determine the conductivity of a medium by making voltage and current measurements at the boundary? I will report on some of the progress that has been made on this problem since Bob’s inspiring lecture.
 
Eric Vanden-Eijnden
• Title: Learning to sample better
• Abstract: Sampling high-dimensional probability distributions is a common task in computational chemistry, Bayesian inference, etc. Markov Chain Monte Carlo (MCMC) is the method of choice to perform these calculations, but it is often plagued by slow convergence properties. I will discuss how methods from deep learning (DL) can help enhance the performance of MCMC via a feedback loop in which we simultaneously use DL to learn better samplers based e.g. on generative models such as normalizing flows, and MCMC to obtain the data for the training of these models. I will draw connection between these methods and score-based diffusion models that have proven successful for image generation. I will also illustrate these techniques via several examples, including the calculation of free energies and Bayes factors.
 
Michael Vogelius
• Title: Non-scattering phenomena
• Abstract: In this talk I shall discuss some recent results relating the regularity, the geometry and the contents of an inhomogeneity to its potential for being non-scattering in the context of the Helmholtz equation. This is joint work with F. Cakoni and J. Xiao.
 
Michael Weinstein
• Title: Edge states in continuum and discrete systems
• Abstract: Effective equations are discrete (tight-binding) or continuum (homogenized PDE) approximations, which elucidate—in a specified energy range—the wave properties of crystalline media. In this talk I’ll focus on the tight binding approximation of the Schroedinger equation (non-magnetic and strongly magnetic), which is of interest for the study of 2D quantum materials and their artificial analogues [1]. I’ll then present recent work on the tight binding model of graphene, sharply terminated along a “rational edge”, a line I parallel to a direction of translational symmetry of the underlying period lattice. We give a complete classification of the zero-energy / flat-band edge states, and provide strong evidence for the existence of (non-flat) dispersive edge state curves for most l [2].
 
References:
1) J. Shapiro and M.I. Weinstein,
Tight binding reduction and topological equivalence in strong magnetic fields,
Advances in Mathematics, vol. 403 108343 (2022).
2) C.L. Fefferman, S. Fliss and M.I. Weinstein, Discrete honeycombs, rational edges and edge states,
to appear in Communications on Pure and Applied Mathematics.

Kaushik Bhattacharya, Caltech
Lia Bronsard, McMaster University
Rustum Choksi, McGill University
Leslie Greengard, Courant Institute and Flatiron Institute
Sylvia Serfaty, Courant Institute
Raghav Venkatraman, Courant Institute

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