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Time & Location*:
Wednesdays from 11am – 12pm in Hill 705
(*Some talks may be scheduled for different times or locations. Such details will be provided additionally.)
Organizers:
Narek Hovsepyan
narek.hovsepyan@rutgers.edu
Ewerton Rocha Vieira
ewerton.vieira@dimacs.rutgers.edu
(For inquiries, e.g. to be added to the mailing list, please contact either one of the organizers.)

 

17 January
Narek Hovsepyan, Rutgers University
“On the lack of external response of a nonlinear medium in the second-harmonic generation process.”
Abstract

Second Harmonic Generation is a process in which the input wave (e.g. laser beam) interacts with a nonlinear medium and generates a new wave, called the second harmonic, at double the frequency of the original input wave. We investigate whether there are situations in which the generated second harmonic wave does not scatter and is localized inside the medium, i.e., the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated inside the medium with non-standard boundary conditions.

 

31 January
Manas Rachh, Flatiron Institute
“Static currents in type-1 superconductors.”
Abstract

In this talk, we describe the classical magneto-static approach to the theory of type-I superconductors. The magnetic field and the current in type-I superconductors are related by the London equations and tend to decay exponentially inside the superconducting material with support of the fields contained primarily in O(λ_L) neighborhood of the superconductor. We present a Debye source based integral representation for the numerical solution of the London equations, and demonstrate the efficacy of our approach for moderate values of λ_L on complex three dimensional geometries. However, for typical materials λ_L ∼ O(10−7), which makes the PDE and integral equation increasingly difficult to solve in the limit λ_L → 0 due to the presence of two different length scales in the problem. We derive a limiting PDE and a corresponding integral equation, and show that the solutions of this limiting PDE and integral equations are O(λ_L) accurate as compared to the corresponding solutions of the London equations and the Debye source integral equations respectively. We demonstrate the effectiveness of this asymptotic approach both in terms of speed and accuracy through several numerical examples.

 

7 February
Aida Maraj, Harvard University
“Understanding Multivariate Gaussian Models via Toric Geometry.”
Abstract

Lately, algebraic geometry has been offering a novel approach to advancing problems on multivariate Gaussian models. This is done by identifying Gaussian distributions with symmetric matrices and analyzing the set of polynomials that vanish on these matrices, referred to as ideals. The talk will focus on Brownian motion tree (BMT) models – Gaussian models in phylogenetics. These models have a hidden toric geometry, which we use to advance questions on their maximum likelihood estimate. Lastly, motivated by the need to classify toric statistical models, we introduce the symmetry Lie group of an ideal and an associated algorithm as a means to detect non-toric structures in a general setting. No prior knowledge on toric ideals or BMT models is assumed.

 

14 February
Rui Tuo, Georgia Institute of Technology
“Some recent advances in statistical asymptotic theory via RKHS approximations.”
Abstract
Reproducing Kernel Hilbert Spaces (RKHS) techniques are a cornerstone in various statistical
methods and machine learning algorithms. They provide a robust mathematical foundation
for developing new algorithms and understanding existing ones, particularly in terms of their
approximation capabilities, generalization properties, and computational efficiency. My
presentation will highlight two recent advances in statistical asymptotic theory, leveraging
RKHS approximation properties. The first part addresses the approximation efficacy of
Gaussian process regression, a Bayesian nonparametric method for reconstructing functions
from scattered data. Our research establishes its statistical asymptotic theory. We utilize
maximum inequalities for Gaussian processes to transform stochastic errors into a
deterministic framework, and latter is shown to be closely related to an approximation
problem in the RKHS. The second segment focuses on kernel ridge regression, a popular
nonparametric regression method. While its global convergence is well-documented, the
local properties of these estimators remain less understood. We analyze these estimators’
linear functionals by separating bias and variance. Remarkably, both can be reinterpreted
through RKHS approximation indicators. This leads to exact convergence rates and a central
limit theorem for various local estimators. This talk aims to provide insights into these
complex statistical tools, enhancing their accessibility and comprehension in the broader
fields of statistics and machine learning.

 

*16 February (Friday), from 2 – 3 pm in Hill 425

Joint with Nonlinear Analysis and Hyperbolic & Dispersive PDE Seminars

Marta Lewicka, University of Pittsburgh
“The Monge-Ampere system and the isometric immersion system: convex integration in arbitrary dimension and codimension.”
Abstract

The Monge-Ampere equation \det\nabla^2 v =f posed on a d=2 dimensional domain \omega and in which we are seeking a scalar (i.e. dimension k=1) field v, has a natural weak formulation that appears as the constraint condition in the \Gamma-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: \frac{1}{2}\nabla v\otimes \nabla v) + \sym\nabla w= – (\curl \curl)^{-1}f  and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in C^0(\omega), at the regularity C^{1,\alpha} for any \alpha<1/3, no matter the sign of the right hand side function f.

Does a similar result hold in higher dimensions d>2 and codimensions k>1? Indeed it does, but one has to replace the Monge-Ampere equation by the Monge-Ampere system, by altering \curl \curl to the corresponding operator that arises from the prescribed Riemann curvature problem, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d.

Our main result is a proof of flexibility of the Monge-Ampere system at C^{1,\alpha} for  \alpha<1/(1+d(d+1)/k). This finding extends our previous result where d=2, k=1, and stays in agreement with the known flexibility thresholds for the isometric immersion problem: the Conti-Delellis-Szekelyhidi result \alpha<1/(1+d(d+1)) when k=1, as well as the Kallen result where \alpha\to 1 as k\to\infty. For d=2, the flexibility exponent may be even improved to \alpha<1/(1+2/k), using the conformal invariance of 2d metrics to the flat metric.

We will also discuss other possible improvements and parallel results and techniques valid for the isometric immersion system.

 

28 February
Raghav Venkatraman, Courant Institute, NYU
“The geometry of data through the lens of homogenization.”
Abstract

The “manifold hypothesis” posits that many high dimensional data sets that occur in the real world actually are actually scattered on or around a much lower dimensional manifold embedded in the high dimensional space. Estimating attributes of this “ground truth” manifold from finitely many samples (point cloud) is a problem of statistical inference. Given such a point cloud that is modelled as an independent and identically distributed (i.i.d) sample from a (nice) density on a closed manifold, over the past decade there is a body of literature which considers the question: forming a random geometric (weighted) graph on the point cloud (by, for example, joining points that are within a threshold distance by a weighted edge) how well can one estimate the spectrum (eigenvalues, eigenfunctions) of the (weighted) Laplace-Beltrami operator on the ground truth manifold, from that of the graph laplacian associated with the random geometric graph?

After introducing the problem, we will show how this question is one of “stochastic homogenization”, a traditionally well-studied theme in partial differential equations originating in the theory of composite materials. Warming up with results that are new even for the classical “periodic” homogenization problem, we will describe how one can obtain optimal convergence rates for the spectrum of the graph laplacian using tools from the recent theory of quantitative stochastic homogenization. Briefly: borrowing tools from percolation theory, the argument proceeds by “coarse-graining” the random geometry in the problem to large scales, where the environment “appears Euclidean”. Then, one adapts  arguments from the more recent quantitative theory of homogenization.

This talk represents joint work with Scott Armstrong (Courant).

 

6 March
Jonathan Jaquette, New Jersey Institute of Technology
“Reliability and robustness of oscillations in some slow-fast chaotic systems.”
Abstract
A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator, we propose a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales.
We show that systems displaying relaxation cycles going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales, thus consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may generate erratic activity also at slow timescales. We analyze in detail these phenomena in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics, and validate this numerically in other models.

 

*20 March, from 11am – 12 pm in Hill 005
Yury Grabovsky, Temple University
“On feasibility of extrapolation of completely monotone functions.”
Abstract

In many areas of science and engineering people measure positive linear combinations of decaying exponentials and want to recover their parameters. The goal of this talk is to discuss feasibility of such a recovery. Positive linear combinations form a well-known class of completely monotone functions (CMFs). To quantify our question, we look for pairs of CMFs that have relative discrepancy $\epsilon$ on a given interval (a,b) as measured by the $L^{2}$-norm. Our goal is to estimate the largest relative discrepancy between them at a given point $x_{0}\not\in(a,b)$. We show that this discrepancy can be made as large as one wishes, when $x_{0}\le a$, and scales as $\epsilon^{\gamma(x_{0})}$, when $x_{0}>b$. An explicit formula for $\gamma(x_{0})\in(0,1)$ will be presented. This is a joint work with my graduate student Henry Brown and builds on a prior joint work with Narek Hovsepyan.

 

*22 March (Friday), from 2 – 3 pm in Hill 525
Benedetto Piccoli, Rutgers University
“Mathematical approaches for the smoothing of traffic via autonomous vehicles.”
Abstract
The problem of control of large multi-agent systems, such as vehicular traffic, poses many challenges both for the development of mathematical models and their analysis and the application to real systems. First, we discuss how conservation laws can be used for macroscopic description of traffic, then present some results for mean-field limit of controlled systems. Finally, we describe on a recent experiment involving 100 autonomous vehicles to dampen stop-and-go waves on an open highway.

 

27 March
Michael Weinstein, Columbia University / IAS / Princeton
“Pseudo-magnetism and Landau Levels in Deformed Photonic Honeycomb Structures.”
Abstract

A non-uniform deformation of a honeycomb medium induces effective-magnetic and effective electric fields.
One may choose a deformation which gives rise to a constant perpendicular effective magnetic field with Landau-level spectrum (flat bands / high density of states), and hence a mechanism to produce strong light-matter interactions. In contrast to the setting of graphene, in condensed matter physics, for photonic crystals the tight binding (discrete) model is generally not applicable. I will present a continuum (homogenized / PDE) theory (joint work with J. Guglielmon and M. Rechtsman – Phys. Rev. A 103 2021), and then discuss very recent experimental confirmation of this theory (Barsukova et al. Nature Photonics, to appear). I will incorporate rigorous and formal asymptotic mathematical results throughout.

 

*5 April (Friday), from 1 – 2 pm in Hill 425
Malena Espanol, Arizona State University
“Regularization Methods for Inverse Problems in Imaging.”
Abstract

Discrete linear and nonlinear inverse problems arise from many different imaging systems, exhibiting inherent ill-posedness wherein solution sensitivity to data perturbations prevails. This sensitivity is exacerbated by errors arising from imaging system components (e.g., cameras, sensors, etc.), necessitating the development of robust regularization methods to attain meaningful solutions. This talk starts with a review of distinct imaging systems and their mathematical formalism and subsequently introduces regularization techniques tailored for linear inverse problems. Then, we will look into the variable projection method, a powerful tool to address separable nonlinear least squares problems.

 

*8 April (Monday), from 11 am – 12 pm in Hill 705
Giovanni Fantuzzi, Friedrich-Alexander-Universität Erlangen Nürnberg
“Data-driven system analysis: Polynomial optimization meets Koopman.”
Abstract

Understanding the stability and long-term behavior of dynamical systems is vital in numerous applications. These properties can be studied through Lyapunov frameworks that, when the dynamics are governed by known polynomial models, can be implemented computationally using polynomial optimization. But what if the model is not polynomial or, worse, not even known? In this talk, I will show that polynomial optimization can be combined with extended dynamic mode decomposition to perform system analysis directly from measured data. This is possible thanks to a connection between Lyapunov frameworks and the Koopman operator. After introducing the basic theory behind this connection, I will show how data-driven system analysis can guarantee stability and unravel chaotic dynamics on a range of examples.

 

10 April
Kui Ren, Columbia University
“Inverse problems to mean field game systems: analysis and computation.”
Abstract

Mean field game models have been developed in different application areas. We will provide an overview of recent developments in inverse problems to mean field game models where we are interested in reconstructing missing information from observed data. We present a few different scenarios where differential data allows for the unique identification of model parameters in various forms, as well as numerical methods for computing the inverse solutions.

 

17 April
Hunter King, Rutgers University
“Two tales of slenderness and confinement: mechanics of wrinkled sheets and jammed filaments.”
Abstract
The first part of my talk will consider a thin, circular, elastic sheet adhered to the surface of a liquid droplet of increasing curvature. Geometric frustration causes it to buckle, first forming a pattern of smooth wrinkles, then sharp crumpled features at higher curvature. These steps can be seen as two distinct symmetry breaking instabilities: that of shape and stress field, respectively, driven by one non-dimensional confinement parameter, according to coupled experiments and theoretical analysis.
In the second part, I will present a preliminary study of the emergent mechanics of packings of unbonded, semiflexible, athermal fibers, inspired by the highly non-intuitive nest-building strategy of birds.  Through complementary physical and computational measurements of mechanical resonse under oedometric compression, we reveal micromechanical origins of some basic features of this class of disordered metamaterial, relating non-linear stiffness and quasistatic hysteresis to the reversible formation and motion of internal contacts.
In either case, we find that generic confinement of slender bodies gives rise to phenomenology that is surprisingly rich, but can be reduced to understandable models and motifs.

 

*19 April (Friday), from 1 pm – 2 pm in Hill 425
Jean-Philippe Lessard, McGill University
“Computer-assisted proofs for differential equations with non-polynomial nonlinearities via the FFT.”
Abstract

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) to establish the existence of solutions for nonlinear differential equations with non-polynomial analytic nonlinearities. Our approach integrates the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to construct CAPs effectively. Notably, to rigorously manage the Fourier coefficients of the nonlinear term Fourierg series, we leverage insights from complex analysis and the Discrete Poisson Summation Formula. We showcase the applicability of our method through two examples: firstly, verifying the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, proving the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

 

24 April
Zehui Zhou, Rutgers University
“TBA.”
Abstract