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# Mathematical Analysis and Applications Seminar

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

Inspired by empirical observations on honey bee colonies, we analyze the emergence of task differentiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coordinated behavior and collective function. The analysis considers the steady-state response of a mechanical interaction network to exogenous resonant excitation. It demonstrates how an optimal allocation of excitation sensitivities across the network nodes that results in a maximal response may be constructed either using global information about the network topology or, alternatively, through iterated dynamics of an intuitive learning paradigm that only relies on local information within the network. Importantly, the analysis derives explicit conditions on the topology that guarantee optimal selection using only local dynamics, but also describes circumstances when the system may naturally evolve to a condition of collapse, in which all the excitation sensitivities equal zero, at least over intermediate times. The discussion considers the implications to other network designs, including naturally occurring networks, as well as the effects of noise and nonlinearity.

The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization. I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth. I will also discuss optimal recovery algorithms, and some open questions.

generation of neuronal network oscillations are not well understood. We view this process as the individual neurons' oscillations being communicated among the nodes in the network, mediated by the impedance profiles of the isolated (uncoupled) individual neurons. In order to test this idea, we developed a mathematical tool that we refer to as the Frequency Response Alternating Map (FRAM). The FRAM describes how the impedances of the individual oscillators interact to generate network responses to oscillatory inputs. It consists of decoupling the non-autonomous part of the coupling term and substituting the reciprocal coupling by a sequence of alternating one-directional forcing effects (cell 1 forces cell 2, which in turn forces cell 1 and so on and so forth). The end result is an expression of the network impedance for each node (in the network) as power series, each term involving the product of the impedances of the autonomous part of the individual oscillators. For linear systems we provide analytical expressions of the FRAM and we show that its convergence properties and limitations. We illustrate numerically that this convergence is relatively fast. We apply the FRAM to the phenomenon of network resonance to the simplest type of oscillatory network: two non-oscillatory nodes receiving oscillatory inputs in one or the two nodes. We discuss extensions of the FRAM to include non-linear systems and other types of network architectures.

(joint with Noam Aigerman, Raz Sluzky and Yaron Lipman)

Computing homeomorphisms between surfaces is an important task in shape analysis fields such as computer graphics, medical imaging and morphology. A fundamental tool for these tasks is solving Dirichlet's problem on an arbitrary Jordan domain with disc topology, where the boundary of the domain is mapped homeomorphically to the boundary of a specific target domain: A convex polygon. By the Rado-Kneser-Choquet Theorem such harmonic mappings are homeomorphisms onto the convex polygon. Standard finite element approximations of harmonic mappings lead to discrete harmonic mappings, which have been proven to be homeomorphisms as well. Computing the discrete harmonic mappings is very efficient and reliable as the mappings are obtained as the solution of a sparse linear system of equations.

In this talk we show that the methodology above, can be used to compute *conformal* homeomorphisms, for domains with either disc or sphere topology:

By solving Dirichlet's problem with correct boundary conditions, we can compute conformal homeomorphisms from arbitrary Jordan domains to a specific canonical domain- a triangle. The discrete conformal mappings we compute are homeomorphisms, and approximate the conformal homeomorphism uniformly and in H^1. Similar methodology can also be used to conformally map a sphere type surface to a planar Jordan domain, whose edges are identified so that the planar domain has the topology of a sphere.

Typically, when semi-discrete approximations to time-dependent partial differential equations (PDE) or schemes for ordinary differential equation (ODE) are constructed they are derived such that they are stable and have a specified truncation error $\tau$. Under these conditions, the Lax-Richtmyer equivalence theorem assures that the scheme converges and that the error is, at most, of the order of $|| \tau ||$. In most cases, the error is in indeed of the order of $|| \tau ||$. We demonstrate that schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This error reduction is made by constructing the schemes such that they inhibit the accumulation of the local errors; therefore they are called Error Inhibiting Schemes (EIS).

Single cells and cells in a tissue respond to stimuli by deforming, changing their shape, and/or moving. Some of these responses can be understood from the underlying biochemical signaling, a topic that has been of interest to both biologists and modelers. In our recent work, my group has studied the link between mechanical tension on cells and their internal chemical signaling. (Our primary focus has been, and remains, that of Rho proteins.) Here I will describe a simple "toy model" that captures key aspects of what is known biologically. The model is simple enough to understand mathematically, and yet capable fo displaying several regimes of behavior consistent with experimental observations. I describe how we investigated the model in a single cell, and how this was then used to capture multiple cells that interact with one another mechanically. We find waves of expansion and contraction that sweep through the model "tissue" is certain parameter regimes. This work is joint with Cole Zmurchok and Dhanajay Bhaskar.

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.

Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them.

In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.

Based on a joint work with Asaf Shachar.

I will describe work with C.E.Wayne in which we study how dissipation leads to a very slow drift in secular parameters for the Non-linear Schroedinger equation. The nice thing about this model is that one can describe a relatively complex phenomenon in almost explicit terms as a perturbation from an integrable Hamiltonian system

We show that any area-preserving C^{r}-diffeomorphism of a two-dimensional surface displaying an elliptic fxed point can be C^{r}-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every 1 ≤ r≤ 1. This proves a conjecture of Herman stating that the identity map of the disk can be C^{ ∞}-perturbed to a conservative di eomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be C^{∞}- approximated by a conservative diffeomorphisms displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of C^{r}-conservative di eomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.

This is a joint work with Pierre Berger.

Multiple wireless sensing tasks, e.g. radar detection for driver safety, involve estimating the "channel" or relationship between signal transmitted and received.

In this talk I will tell about the standard math model for the radar channel. In the case where the channel is sparse, I will demonstrate a channel estimation algorithm that is sub-linear in sampling and arithmetic complexity (and convince you of the need for such).

The main ingredients in the algorithm will be the use of an intrinsic algebraic structure known as the Heisenberg group and recent developments in the theory of the sparse Fast Fourier Transform (sFFT, due to Indyk et al.)

The talk will assume minimal background knowledge.

The problem of computational super-resolution asks to recover high-frequency features of an object from the noisy and blurred/band-limited samples of its Fourier transform, based on some a-priori information about the object class. A core theoretical question is to quantify the possible amount of bandwidth extension and the associated stability of recovering the missing frequency components, as a function of the sample perturbation level and the object prior.

In this work we assume that the object has a compact space/time extent in one dimension (but otherwise can be fairly arbitrary), while the low-pass window can have a super-exponentially decaying "soft" shape (such as a Gaussian). In contrast, previously known results considered only the ideal "hard" window (a characteristic function of the band) and objects of finite energy. The super-resolution problem in this case is equivalent to a stable analytic continuation of an entire function of finite exponential type. We show that a weighted least-squares polynomial approximation with equispaced samples and a judiciously chosen number of terms allows one to have a super-resolution factor which scales logarithmically with the noise level, while the pointwise extrapolation error exhibits a Holder-type continuity with an exponent derived from weighted potential theory. The algorithm is asymptotically minimax, in the sense that there is essentially no better algorithm yielding meaningfully lower error over the same smoothness class.

The results can be considered as a first step towards analyzing the much more realistic model of a sparse combination of compactly-supported "approximate spikes", which appears in applications such as synthetic aperture radar, seismic imaging and direction of arrival estimation, and for which only limited special cases are well-understood.

Joint work with L.Demanet and H.Mhaskar.

I will discuss the asymptotic behaviour (both on and off the diagonal) of the spectral function of a Schroedinger operator with smooth bounded potential when energy becomes large. I formulate the conjecture that the local density of states (i.e. the spectral function on the diagonal) admits the complete asymptotic expansion and discuss the known results, mostly for almost-periodic potentials.

A classical theorem by Anosov states that the slow motion of a slow-fast system where the fast subsystem is ergodic with respect to a smooth invariant measure can be approximated, in a well-defined sense, by the slow subsystem averaged over the fast variables. We address the question of what happens if the fast system is not ergodic. We discuss a theory which is developing in joint works with V. Gelfreich, T. Pereira, V. Rom-Kedar and K. Shah, and suggest that in the non-ergodic case the behavior of the slow variables is approximated by a random process, and not a single, deterministic averaged system. We also discuss the question of the relevance of ergodicity to the foundations of statistical mechanics.

Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset R^d \to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak (1967) generalized this result and showed that if a sequence $f_n$ satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine map.

In this talk I will discuss generalizations of these theorems to mappings between manifolds, present some open questions, and describe how these rigidity questions arise in the theory of elasticity of pre-stressed materials (non-Euclidean elasticity).

If time permits, I will sketch the main ideas of the proof, using Young measures and harmonic analysis techniques, adapted to Riemannian settings.

Based on a joint work with Asaf Shachar and Raz Kupferman.

Selfadjoint extensions of a closed symmetric operator A in a Hilbert space with equal deficiency indices were described by in the 30's by J. von Neumann. Another approach, based on the notion of abstract boundary triple originates in the work of J.W. Calkin and was developed by M. I. Visik, G. Grubb, F. S. Rofe-Beketov, M. L. Gorbachuck, A .N. Kochubei and others.

By Calkin's approach, all selfadjoint extensions of the symmetric operator A can be parametrized via "multivalued" selfadjoint operators in an auxiliary Hilbert space. Spectral properties of these extensions can be characterized in terms of the abstract Weyl function, associated to the boundary triple. In the present talk some recent developments in the theory of boundary triples will be presented. Applications to boundary value problems for Laplacian operators in bounded domains with smooth and rough boundaries will be discussed.

We study the discrepancy of the number of visits of a Kronicker sequence on a d dimensional torus to nice sets. We are interested in particular in the question how the answer depends on the geometry of the set.

This is a joint work with Bassam Fayad.

(http://arxiv.org/abs/1211.4323 and http://arxiv.org/abs/1206.4853)

Let *X* be a compact manifold with the boundary ∂ *X* and *R *(λ) be a Dirichlet-to-Neumann operator: *R *(λ): *f* → *u*|_{∂X} where *u* solves ( _{}+ λ^{2}) *u *= 0, *u*|_{∂X} = *f* . We establish asymptotics as λ→ + ∞ of the number of eigenvalues of λ^{-1 }*R* (λ) between s_{1} and s_{2}.

This is a joint work with Andrew Hassell, Australian National University.

Line defects appear in the microscopic structure of crystalline materials (e.g. metals) as well as liquid crystals, the latter an intermediate phase of matter between liquids and solids. Mathematically, their study is challenging since they correspond to topological singularities that result in blow-up of total energies of finite bodies when utilizing most commonly used classical models of energy density; as a consequence, formulating nnonlinear dynamical models (especially pde) for the representation and motion of such defects is a challenge as well. I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. The model alleviates the nasty singularities mentioned above and it will also be shown that incorporating a conservation law for the topological charge of line defects allows for the correct prediction of some important features of defect dynamics that would not be possible just with the knowledge of an energy function.

This is joint work with Chiqun Zhang, Dmitry Golovaty, and Noel Walkington.

Transmission rates in broadband optical waveguide systems are enhanced by launching many pulse sequences through the same waveguide. Since pulses from different sequences propagate with different group velocities, intersequence pulse collisions are frequent, and can lead to severe transmission degradation. On the other hand, the energy exchange in pulse collisions can be beneficially used for controlling the transmission.

In this work we show that collision-induced amplitude dynamics of soliton sequences of N perturbed coupled nonlinear Schrödinger (NLS) equations can be described by N-dimensional Lotka-Volterra (LV) models, where the model's form depends on the perturbation. To derive the LV models, we first carry out single-collision analysis, which is based on the method of eigenmode expansion with the eigenmodes of the linear operator describing small perturbations about the fundamental NLS soliton. We use stability and bifurcation analysis for the equilibrium points of the LV models to develop methods for achieving robust transmission stabilization and switching that work well for a variety of waveguides. Further enhancement of transmission stability is obtained in waveguides with a narrowband Ginzburg-Landau gain-loss profile. We also discuss the possibility to use the relation between NLS and LV models to realize transition to spatio-temporal chaos with NLS solitons.

Periodically driven systems are of immense interest in plasma physics both from the point of view of plasma confinement as well as plasma heating.

One of the models to explain plasma heating in capacitive RF discharges is Fermi acceleration, which consists of a particle moving in a dynamical billiard with oscillating boundaries. It is well known that the energy growth rate of an ensemble of particles in a strongly chaotic billiard with moving walls is quadratic-in-time whereas it can be exponential-in-time in billiards with multiple ergodic components. Since a real plasma device allows for an exchange of particles with the surroundings, we have now studied Fermi accelerators with a hole (small enough so as not to disturb the statistics). We find that energy gain is significantly higher in a leaky Fermi accelerators with multiple ergodic components and it can be further increased by shrinking the hole size. In the ergodic case, energy gain is found to be independent of the hole size. Work done jointly with V. Gelfreich, V. Rom-Kedar and D. Turaev [Physical Review E 91, 062920 (2015)].

Paul trap is a device used to confine electrons by using time-periodic spatially non-uniform electric fields and a Nobel Prize as awarded for its discovery in 1989. The time-averaged distribution function of plasma in such devices is usually modelled using the concept of an effective potential (ponderomotive theory). For a specific example of the electric field used in Paul traps, we had shown earlier that the exact solutions of the Vlasov equation (collisionless Boltzmann equation) do not agree with solutions obtained by the effective potential approach. Now we have been able to obtain a perturbative solution of the Vlasov equation for a much more general case and find the same discrepancy with conventional theory. These perturbative solutions represent a non-equilibrium steady state and further work needs to be done to understand their statistical evolution. Work done jointly with B. Srinivasan [arXiv:1510.03974].

In this talk I will consider the problem of local analytic classification of powers of volume forms on manifolds with boundary, i.e. of ordinary volume forms multiplied by the (complex in general) power of a function f, under the action of the group of diffeomorpshims preserving both the boundary and the hypersurface defined by the zero locus of f. In the case where this function defines an isolated boundary singularity in the sense of Arnol'd, I will show how to obtain local normal forms and moduli theorems, analogous to those obtained by Arnol'd, Varchenko, Lando and others for the ordinary, without boundary case. Moreover I will show how these moduli are related to (in fact obtained by) the topological and analytic (Hodge theoretic) invariants of the boundary singularity, such as the relative Picard-Lefschetz monodromy, the relative Brieskorn lattices with their relative Gauss-Manin connection, the relative spectrum and so on, all objects generalising, in the presence of a boundary, the corresponding well known objects already defined for isolated hypersurface singularities.

We say that a system possesses a mixed dynamics if

1) it has infinitely many hyperbolic periodic orbits of all possible types (stable, unstable,saddle) and

2) the closures of the sets of orbits of different types have nonempty intersections.

Recall that Newhouse regions are open domains (from the space of smooth dynamical systems) in which systems with homoclinic tangencies are dense. Newhouse regions in which systems with mixed dynamics are generic (compose residual subsets) are called *absolute Newhouse regions* or *Newhouse regions with mixed dynamics*. Their existence was proved in the paper [1] for the case of 2d diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle containing two fixed (periodic) points with the Jacobians less and greater than 1. Fundamentally, that "mixed dynamics" is the universal property of reversible chaotic systems. Moreover, in this case generic systems from absolute Newhouse regions have infinitely many stable, unstable, saddle and symmetric elliptic periodic orbits [2,3].

As well-known, reversible systems are often met in applications and they can demonstrate a chaotic orbit behavior. However, the phenomenon of mixed dynamics means that this chaos can not be associated with "strange attractor" or "conservative chaos". Attractors and repellers have here a nonempty intersection containing symmetric orbits (elliptic and saddle ones) but do not coincide, since periodic sinks (sources) do not belong to the repeller (attractor). Therefore, " mixed dynamics" should be considered as a new form of dynamical chaos posed between "strange attractor" and "conservative chaos".

These and related questions are discussed in the talk. Moreover, the main attention here is paid to the development of the concept of mixed dynamics for two-dimensional reversible maps. The main elements of this concept are presented in section below.

[1] S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math., 216 (1997), 70-118.

[2] Lamb J.S.W. and Stenkin O.V. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits Nonlinearity, 2004, 17(4), 1217-1244.

[3] Delshams A., Gonchenko S.V., Gonchenko V.S., Lazaro J.T. and Sten'kin O.V. "Abundance of attracting, repelling and elliptic orbits in two-dimensional reversible maps".- Nonlinearity, 2013, v.26(1), 1-35.

In this talk, I will discuss the backward-in-time behaviors of several nonlinear parabolic and dissipative evolution equations. This study is motivated by the investigation of the Bardos-Tartar conjecture on the 2D Navier-Stokes equations. Besides the rigorous mathematical treatment, we provide physical interpretation of the mechanism of singularity formulation, backward in time, for perturbations of the KdV equation. Finally, I will present the connection between the backward behavior and the energy spectra of the solutions.

This is a joint work with E. S. Titi.

In this talk we will implement the notion of finite number of determining parameters for the long-time dynamics of the Navier-Stokes equations (NSE), such as determining modes, nodes, volume elements, and other determining interpolants, to design finite-dimensional feedback control for stabilizing their solutions. The same approach is found to be applicable for data assimilation of weather prediction. In addition, we will show that the long-time dynamics of the NSE can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named *determining form*, which is governed by a globally Lipschitz vector field. The NSE are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs.

This is a joint work with A. Azouani, H. Bessaih, A. Farhat, C. Foias, M. Jolly, R. Kravchenko, E. Lunasin and E. Olson.

We discuss the collective dynamics of systems driven by the “social engagement” of agents with their local neighbors. Canonical models are based on environmental averaging, with prototype examples in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks. The large time behavior of such systems leads to the formation of clusters, and in particular, the emergence of “consensus of opinions”.

We propose an alternative paradigm, arguing that in many relevant scenarios social interactions involve the tendency of agents “to move ahead”. We introduce a new family of models for collective dynamics with tendency. The large time behavior of these new systems leads to the emergence of “leaders”.

Computational anatomy considers spaces of shapes (e.g. medical images) endowed with a Riemannian metric (Sobolev type). The area blends techniques from differential geometry (geometric mechanics), analysis and statistics. The EPDiff equation, which is basically an extension of Euler's equation, without the incompressibility assumption, is often used to match shapes. Time-varying images (4DCA), one of the current research themes in the area. In longitudinal studies (say for Alzheimer's disease) snapshots at given times are interpolated/regressed. The problem arises of comparing two such sequences for classification purposes. For some background, see the recent workshop http://www.mat.univie.ac.at/~shape2015/schedule.html .

In this talk we discuss finite dimensional examples using landmarks. A short process is interpreted as a tangent vector in the space of images. This leads to control problems whose state space is a tangent bundle. Usually, cubic Riemannian splines are taken, ie., minimizing the norm of the acceleration vector, for paths connecting two tangent vectors under a fixed time. We propose as an alternative to cubic splines the time minimal problem under bounded acceleration (morally the norm). We suggest that both splines problems on are completely integrable in the Arnold-Liouville sense. Along the way, we present general technical results about the underlying symplectic structures of control problems whose state space has a bundle structure.

This is joint ongoing work with Paula Balseiro, Alejandro Cabrera, and Teresa Stuchi.

This talk will describe how Lagrangian particle methods are being used to compute the dynamics of fluid vortices. In these methods the flow map is represented by moving particles that carry vorticity, the velocity is recovered by the Biot-Savart integral, and a tree code is used to reduce the computation time from to , where is the number of particles. I'll present vortex sheet computations in 2D with reference to Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and chaotic dynamics. Other examples include vortex rings in 3D, and vortex dynamics on a rotating sphere.

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem’s subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call “zero-convexity”. This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

This is a joint work with Yair Censor.

Cryo-electron microscopy (cryo-EM) is a microscopy technique used to discover the 3D structure of molecules from very noisy images. We discuss how algebra can describe two aspects of cryo-EM datasets. First, we'll describe common lines datasets. Common lines are lines of intersection between cryo-EM images in 3D. They are a crucial ingredient in some 2D3D reconstruction algorithms, and they can be characterized by polynomial equalities and inequalities. Second, we'll discuss how 3D symmetries of a molecule can be detected from only 2D cryo-EM images, without performing full 3D reconstruction.