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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.