# NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Fall 2020

Wednesday, August 26, 15:00-16:00Zoom meeting:LinkSpeaker: Michael Shearer, North Carolina State University, USATitle:Riemann Problems for the BBM EquationAbstract:The BBM equation $u_t+uu_x=u_{xxx}$ is a nonlinear dispersive scalar PDE related to the KdV equation. However, it has a non-convex dispersion relation that introduces a variety of novel wave structures. These waves are highlighted by considering numerical solutions of Riemann problems, in which a smoothed step function initial condition $u(x,0)$ exhibits long-time behavior that is a challenge for asymptotic analysis and the theory of dispersive hydrodynamics. In this talk, I describe the new waves and the analysis explaining their structure. This is joint work with Thibault Congy, Gennady El and Mark Hoefer.

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Wednesday, September 02, 15:00-16:00

Zoom meeting:LinkSpeaker: Luis Briceno,Universidad Técnica Federico Santa María, ChileTitle:Splitting algorithms for non-smooth convex optimization: Review, projections, and applicationsAbstract:In this talk we review some classical algorithms for solving structured convex optimization problems, passing from gradient descent to proximal iterations and going further to modern proximal primal-dual splitting algorithms in the case of more complicated objective functions. We put special attention to constrained convex optimization, in which we accelerate the performance of the algorithms by including additional projections onto a selection of the constraints. Applications to mean-field games and image processing are included.

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Wednesday, September 09, 15:00-16:00

Zoom meeting:LinkSpeaker: Boris Muha, University of Zagreb, CroatiaTitle:Analysis of Moving Boundary Fluid-Structure Interaction Problems Arising in HemodynamicsAbstract:Fluid-structure interaction (FSI) problems describe the dynamics of multi-physics systems that involve fluid and solid components. These are everyday phenomena in nature, and arise in various applications ranging from biomedicine to engineering. Mathematically, FSI problems are typically non-linear systems of partial differential equations (PDEs) of mixed hyperbolic-parabolic type, defined on time-changing domains.

In this lecture we will study an FSI problem describing blood flow through a compliant vessel. First we will explain main modeling assumptions, illustrate the main challenges arising in the PDE analysis of such FSI problems, and prove the existence of a weak solution of Leray-Hopf type. The proof will use a partitioned numerical scheme called ”kinematically coupled scheme” for the construction of the approximate solutions. The special emphasis will be on the interplay of numerics and the existence proof. Finally, some extension of these ideas to different FSI problems will be discussed._____________________________________________________________________________________________

Wednesday, September 16, 15:00-16:00

Zoom meeting:LinkSpeaker: Konstantin Tikhomirov, Georgia Institute of Technology, USATitle:Littlewood-Offord inequalities, and random matricesAbstract:I will discuss some aspects of two recent results on singularity of Bernoulli matrices. I will emphasize the use of new Littlewood-Offord-type inequalities in the proofs of the results. Partially based on a joint work with A.Litvak.

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Wednesday, September 23, 15:00-16:00Zoom meeting:LinkSpeaker: Rupert L. Frank, California Institute of TechnologyTitle:A ‘liquid-solid’ phase transition in a simple model for swarmingAbstract:We consider a non-local optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof is a strict rearrangement inequality with a quantitative error estimate. We formulate several open problems which might be amenable to PDE techniques.

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Wednesday, September 30, 15:00-16:00

Zoom meeting:LinkSpeaker: Petronela Radu, University of Nebraska-Lincoln, USATitle:Nonlocal models: theoretical and applied aspects

Abstract:The emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present some recent results on systems of integral equations with weakly singular kernels, flux-type boundary conditions, as well as some recent results on nonlocal Helmholtz-Hodge type decompositions with far-reaching applications at both, theoretical, and applied levels.

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Wednesday, October 07, 15:00-16:00

Zoom meeting:LinkSpeaker: Teemu Saksala, North Carolina State UniversityTitle:Generic uniqueness and stability for the mixed ray transform

Abstract:We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 and 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of qS-polarized elastic waves provide partial information about the mixed ray transform of 2+2 tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the Dirichlet-to-Neumann map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

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Wednesday, October 14, 15:00-16:00

Zoom meeting:LinkSpeaker: Oliver Tse, Eindhoven University of TechnologyTitle:Jump processes as generalized gradient flows

Abstract:The study of evolution equations in spaces of measures has seen tremendous growth in the last decades, of which resulted in general metric space theories for analyzing variational evolutions—evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of generalized gradient flows—gradient flows with non-homogeneous dissipation potentials—which are not covered in metric space theories. In this talk, we introduce dynamical-variational transport costs—a large class of large-deviation inspired functionals that provide a variational generalization of existing transport distances—to remedy this deficiency. The role in which these objects play in the theory of generalized gradient flows will be illustrated with an example on Markov jump processes. Finally, open questions and challenges will be mentioned.

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Wednesday, October 21, 15:00-16:00

Zoom meeting:LinkSpeaker: Francisco J. Silva, Université de LimogesTitle:Analytical and numerical aspects of variational mean field games

Abstract:Mean Field Games (MFGs) have been introduced independently by Lasry-Lions and Huang, Malhamé and Caines in 2006. The main purpose of this theory is to simplify the analysis of stochastic differential games with a large number of small and indistinguishable players. Applications of MFGs include models in Economics, Mathematical Finance, Social Sciences and Engineering. In its simplest form, an equilibrium in MFG theory is characterized by a system of PDEs consisting of a Hamilton-Jacobi-Bellman equation and a Fokker-Planck-Kolmogorov equation. In some particular cases, this PDE system corresponds, at least formally, to the first order optimality condition of an associated variational problem. This class of MFGs is called “variational MFGs”.In this talk we will review some theoretical and numerical techniques for variational MFGs that allow us to characterize and compute the associated Nash equilibria.

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Wednesday, October 28, 15:00-16:00

Zoom meeting:LinkSpeaker: Braxton Osting, University of Utah, USATitle:Consistency of archetypal analysis

Abstract:Archetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. For fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared distance between the data and the polytope is minimal. In this talk, I’ll give a consistency result that shows if the data is independently sampled from a probability measure with bounded support, then the archetype points converge to a solution of the continuum version of the problem, of which we identify and establish several properties. We also obtain the convergence rate of the optimal objective values under appropriate assumptions on the distribution. If the data is independently sampled from a distribution with unbounded support, we also prove a consistency result for a modified method that penalizes the dispersion of the archetype points. Our analysis is supported by detailed computational experiments of the archetype points for data sampled from the uniform distribution in a disk, the normal distribution, an annular distribution, and a Gaussian mixture model. This is joint work with Dong Wang, Yiming Xu, and Dominique Zosso.

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Wednesday, November 04, 15:00-16:00

Zoom meeting:LinkSpeaker: Nick Barron, University of Loyola, USATitle:Applications of Quasiconvex functions to HJ Equations and Optimal Control

Abstract:Quasiconvex functions, a major generalization of convex functions, naturally arise in calculus of variations, optimal control and differential games in L-infinity. This connection with HJ equations, representation formulas, obstacle problems, and reach-avoid problems will be discussed.

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Wednesday, November 11, 15:00-16:00

Zoom meeting:LinkSpeaker: Geng Chen, University of KansasTitle:Poiseuille flow of nematic liquid crystals via Ericksen-Leslie model

Abstract:In this talk, we will discuss a recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. In this recent progress, we established a new method to study the full model. A singularity formation result will also be discussed, together with the global existence result showing that the solution will in general live in the Holder continuous space. The earlier related result on the stability of variational wave equation using the optimal transport method, and the future work on other wave equations will also be discussed. The talk is on the joint work with Tao Huang & Weishi Liu.

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Wednesday, November 18, 15:00-16:00

Zoom meeting:LinkSpeaker: Paata Ivanisvili, North Carolina State University, USATitle:Enflo’s problemAbstract:A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with Ramon van Handel and Sasha Volberg where we prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier’s inequality on the Hamming cube.