1

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.

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.

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).

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.

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.

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.

This talk will contain some remarks on the different aspects of the fractional Laplacian and a derivation of fractional diffusion from Kinetic Models inspired by the work of Mellet and illustrated by an example of Uriel and Helene Frisch on radiative transfert which goes back to 1977.

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.

A minimal representative for a dynamical system is a system that has the simplest possible dynamics in its topological equivalence class. This is very much related to "dynamical forcing": when existence of certain periodic orbits forces existence of others. This is quite useful in the analysis of chaotic systems. I'll give examples of minimal representatives in dimensions one, two and three. In dimension three, I'll show that the minimal representative for the chaotic Lorenz equations (for the correct parameters) is the geodesic flow on the modular surface. This will be an introductory talk.

We consider combustion problems in the presence of complex chemistry and nonlinear diffusion laws for the chemical species. The nonlinear diffusion coefficients are obtained by resolution of the so-called Stefan-Maxwell equations. We prove the existence of weak solutions for the corresponding system of equations which involves coupling between the incompressible Navier-Stokes and equations for temperature and species concentrations.

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.

 

Submonolayer deposition (SD) is a blanket term used to describe the initial stages of processes, such as molecular beam epitaxy, in which material is deposited onto a surface, diffuses and forms large-scale structures. It is easy to simulate using Monte Carlo methods, but theoretical results are few and far between. I will discuss various approaches to SD in the 1-dimensional situation, focusing on open mathematical problems and the difficulty of passing to the 2-dimensional case, which is of most applied interest. This is mainly joint work with Paul Mulheran.

Transport of material inside long cells (e.g. up to meters in the case of neuronal cells) requires active processes other than simple diffusion. Molecular motors (such as kinesin and dynein) that "walk" along microtubules (long structural biopolymers) are important in such transport. In this talk I will describe some recent work on the dynamics of these proteins in simple cells: the filamentous hyphae of a fungus (Ustilago maydis). We find that quasi-steady state (QSS) reduction of the model to a Fokker-Plank equation, as well as simulations of the original model provide insight into the behavior of the system for various parameter regimes. I will conclude with a brief survey of other recent work on cellular and multi-cellular dynamics in my group.

In homogeneous materials, discrete and continuous distributions of dislocations are often modeled by different geometric objects - typically, a body with a finite number of dislocations is modeled as a Riemannian manifold with singularities, while a body with a continuous distribution of defects is modeled as a smooth manifold with a non-Riemannian affine-connection (e.g. a metric connection with a non-zero torsion tensor). There are several approaches to how does this connection (or torsion tensor) manifests in the mechanical behavior of a body -- in some works it appears as part of the elastic energy associated with it, and in some it is related only to plastic deformations. In this talk I will present a rigorous homogenization theorem for distributed dislocations, thus bridging between the different approaches modeling them. This will be achieved by introducing a new notion of convergence of manifolds, which applies to this class of homogenization problems. Then I will present a Gamma-convergence result for elastic energies of converging elastic bodies, from which we will deduce that the torsion tensor can appear in the mechanical modeling of the body only when considering plastic deformations. Based on a joint work with Raz Kupferman.

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.

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].

For most realistic Hamiltonian systems the phase space contains both chaotic and regular orbits, mixed in a complex, fractal pattern in which islands of regular motion are surrounded by a chaotic sea. The Henon map is an example of such a system. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach that is the subject of the talk. In particular transport in phase space is of great interest for dynamics, therefore the distributions of fluxes through island chains were computed. evidence for their universality was given. The relation to a model proposed by Meiss and Ott will be discussed. Also the statistics of the boundary circle winding numbers were calculated, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. In particular results that contradict conjectures that were made in the past were found.

In the 1960's, Roger Penrose noted that the Cosmic Censorship Conjecture for solutions of the Einstein equations, or more specifically the standard picture of gravitational collapse, heuristically imply lower bounds on the total energy of initial data in terms of geometric quantities such as the area of the outermost horizon. Any counter-example would strongly suggest that the conjecture fails, while proofs of the inequality, or any extensions, lend indirect support to the conjecture. The time symmetric case was established, first for a single black hole by Huisken-Ilmanen, then for multiple black holes, by Bray. In this talk, I will discuss the extension of these results to include charge and other matter models.

Motivated by the study of transport processes in some classes of billiard models, we wish to characterize a limiting regime of higher-dimensional billiards such that interaction between some of their degrees of freedom occurs only rarely while others mix fast. Under such conditions, the dynamics of the slow degrees of freedom can be approximated by a stochastic process with exponentially distributed waiting times. These times correspond to the times separating interactions among the slow degrees of freedom and we pro- pose to call them conditional return times. The definition extends beyond the rare interaction regime and some universal formulas apply.

I plan to talk about a construction of two different solutions to an elliptic system defined on the two dimensional torus. The system can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual properties of a linear operator. Roughly speaking a term effects in a special stabilization of particular norms of the operator. The proof is valid for a particular large force. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. The talk is based on the results of the paper: Jacek Cyranka, Piotr B Mucha : A construction of two different solutions to an elliptic system. arXiv:1502.03363.

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.

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.

We discuss what can be main ingredients of chaotic behaviour in mechanical systems with non-holonomic constraints: attractor-repeller mergers, elliptic points and non-conservative resonances, solenoids, heterodimensional cycles, and universality.

The standard map is a measure-preserving map of the torus; the dynamics generated by it is the subject of numerous conjectures. One of the approaches to the standard map leads to the study of a certain Schroedinger operator. I will start with a brief introduction to discrete Schroedinger operators, and present two results: one pertaining to a general class of discrete Schroedinger operators, and another one -- pertaining to the operator arising from the standard map. Time permitting, I will explain some of the elements of the proof. [Based on joint work with T. Spencer]

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.

The vortex-wave system is a mathematical model for two-dimensional incompressible flows with small regions of concentrated vorticity imbedded upon a continuously distributed background vorticity. The system consists of a coupleing of the 2D vorticity equation with the point-vortex system. In this talk we survey known results and recent progress on mathematical analysis of this system

Little progress was achieved in finding solutions to Laplace Tidal Equations (LTE) over a sphere since these equations were properly formulated by Jean-Pierre Laplace in 1976. For zonally propagating waves the LTE set of Partial Differential Equations was first formulated as an eigenvalue equation by Michael Selwyn Longuet-Higgins in 1968 but this formulation has not yielded explicit expressions for either the phase speeds or the latitude-dependent amplitudes of the waves. In recent years I've developed an exact Schrodinger equation formulation for wave solutions of LTE in Cartesian Coordinates and this formulation could also be applied to spherical coordinates where it yields an approximate Schrodinger eigenvalue equation in. The solutions of this approximate equation yields highly accurate explicit expressions for the zonally propagating waves solutions of LTE. The new wave solutions can be applied in various areas of Dynamical Meteorology and Physical Oceanography, including the construction of new bases for spherical global scale models and the analysis of satellite derived data on the variation of Sea Surface Height Anomalies.