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Mathematical Analysis and Applications Seminar

TuesdayJul 04, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Claude BardosTitle:Some remarks about Fractional Laplacian in connection with kinetic theoryAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayMar 07, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Leonid Parnovski Title:Local density of states and the spectral function for almost-periodic operators.Abstract:opens in new windowin html    pdfopens in new window

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.

TuesdayFeb 14, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Dimitry Turaev Title:Averaging over a non-ergodic systemAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayJan 31, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Iosif Polterovich Title:Sloshing, Steklov and cornersAbstract:opens in new windowin html    pdfopens in new window
The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.
TuesdayJan 17, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Tali Pinsky Title:Minimal representatives and the Lorenz equationsAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayJan 03, 201711:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Varga KalantarovTitle:On blow up and preventing of blow up of solutions of nonlinear dissipative PDE’sAbstract:opens in new windowin html    pdfopens in new window
We are going to discuss the impact of convective terms on the global solvability or finite time blow up of solutions of initial boundary value problems for nonlinear dissipative PDEs. We will consider the model examples of 1D Burger's type equation, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation, generalized KdV type equations, and establish that sufficiently strong convective terms prevent solutions from blowing up in a finite time and make the considered systems globally well-posed and dissipative. We will also show that solutions of corresponding equations with weak enough convective terms may blow up in a finite time.
TuesdayDec 27, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Martine MarionTitle:Global existence for systems describing multicomponent reactive flowAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayDec 20, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Emanuel A. Lazar Title:Dynamical Cell Complexes: Evolution, Universality, and StatisticsAbstract:opens in new windowin html    pdfopens in new window
Many physical and biological systems are cellular in nature -- soap foams, biological tissue, and polycrystalline metals are but a few examples that we encounter in everyday life. Many of these systems evolve in a manner that changes their geometries and topologies to lower some global energy. We use computer simulations to study how mean curvature flow shapes cellular structures in two and three dimensions. This research touches on discrete geometric flows, combinatorial polyhedra and their symmetries, and the quantification of topological features of large cellular systems. If time permits, I will also describe some exact results in 1 dimension.
TuesdayDec 13, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Cy MaorTitle:Non-Euclidean elasticity and asymptotic rigidity of manifoldsAbstract:opens in new windowin html    pdfopens in new window

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.

 

TuesdayNov 29, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Volodymyr Derkach Title:Boundary triples and Weyl functions of symmetric operatorsAbstract:opens in new windowin html    pdfopens in new window

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. 

TuesdayNov 22, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Michael Grinfeld Title:Mathematical Challenges in Submonolayer DepositionAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayMay 03, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Victor Ivrii Title:Spectral asymptotics for fractional LaplacianAbstract:opens in new windowin html    pdfopens in new window
Consider a compact domain with the smooth boundary in the Euclidean space. Fractional Laplacian is defined on functions supported in this domain as a (non-integer) power of the positive Laplacian on the whole space restricted then to this domain. Such operators appear in the theory of stochastic processes. It turns out that the standard results about distribution of eigenvalues (including two-term asymptotics) remain true for fractional Laplacians. There are however some unsolved problems.
TuesdayMar 29, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Leah Edelstein-KeshetTitle:Mathematical models of molecular motors and other cellular processesAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayJan 26, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Dima DolgopyatTitle:Discrepancy of multidimensional Kronicker sequencesAbstract:opens in new windowin html    pdfopens in new window

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)

TuesdayJan 19, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Cy MaorTitle:Continuous distribution of dislocations -- homogenization and elastic energyAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayJan 05, 201611:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Victor IvriiTitle:Eigenvalue Asymptotics for Dirichlet-to-Neumann OperatorAbstract:opens in new windowin html    pdfopens in new window

Let X  be a compact manifold with the boundary ∂ X and  (λ) be a Dirichlet-to-Neumann operator: (λ): fu|∂X  where u solves ( The Actual Formul+ λ2= 0,  u|∂X = f . We establish asymptotics as λ→ + ∞ of the number of eigenvalues of  λ-1 R (λ) between s1 and s2.

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

TuesdayDec 29, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Amit AcharyaTitle:Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvementAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayDec 22, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Avner Peleg Title:Coupled nonlinear Schrödinger equations, Lotka-Volterra models, and control of soliton collisions in broadband optical waveguide systems. Abstract:opens in new windowin html    pdfopens in new window

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.

TuesdayDec 15, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Kushal Shah Title:Particle dynamics in periodically driven systems : Fermi Accelerators and Paul TrapsAbstract:opens in new windowin html    pdfopens in new window

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].

TuesdayDec 01, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Konstantinos KourliourosTitle:Powers of Volume Forms on Manifolds with BoundaryAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayNov 24, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Or AlusTitle:Statistical properties of Henon mapsAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayNov 17, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Shiri ArtsteinTitle:Billiard dynamics, a symplectic approachAbstract:opens in new windowin html    pdfopens in new window
We will discuss billiard dynamics in convex domains. After some background we shall describe the symplectic geometry approach using capacities, and show various results on minimal lengths of billiards (both Euclidean and the more general Minkowski billiards) and connections with other questions in geometry.
TuesdayNov 10, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Gilbert Weinstein Title:The Riemannian Penrose Inequality with Charge for Multiple Black HolesAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayNov 03, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Dmitry Turaev Title:On Bonatti-Diaz cyclesAbstract:opens in new windowin html    pdfopens in new window
We consider a partially-hyperbolic system with a heteroclinic cycle which contains a pair of saddles with different dimensions of the unstable manifold. We show that an arbitrary small perturbation of any such system creates a Bonatti-Diaz blender that leads to the emergence of persistent heterodimensional cycles. We also show that C1-generic, C2-generic, and C3- generic properties of systems in this class are different, while the higher order derivatives seem to have no effect on the generic dynamics.
TuesdayOct 27, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Thomas GilbertTitle:Average conditional return times to rare events in billiard modelsAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayOct 20, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:S.V.GonchenkoTitle:Reversed mixed dynamicsAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayAug 04, 201511:15
Mathematical Analysis and Applications SeminarRoom 208
Speaker:Piotr B. MuchaTitle:Two different solutions to a Burgers type systemAbstract:opens in new windowin html    pdfopens in new windowplease note change in room
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.
TuesdayJul 28, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Yanqiu GuoTitle:Backward Behavior of Nonlinear Parabolic and Dissipative Evolution EquationsAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayJul 21, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Edriss S. TitiTitle:Finite Number of Determining Parameters for the Navier-Stokes Equations with Applications into Feedback Control and Data AssimilationAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayJul 14, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Eitan TadmorTitle:Taking tendency into account: a new paradigm for collective dynamics Abstract:opens in new windowin html    pdfopens in new window

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”.

TuesdayMay 26, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Jair KoillerTitle:Computational anatomy and splines on manifolds Abstract:opens in new windowin html    pdfopens in new window

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 The Actual Formula 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 The Actual Formula  norm).   We suggest that both splines problems on The Actual Formulaare 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.

TuesdayMay 19, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Peter J. OlverTitle:Dispersive Quantization of Linear and Nonlinear WavesAbstract:opens in new windowin html    pdfopens in new window
The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.
TuesdayApr 14, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Robert Krasny Title:Lagrangian Particle Methods for Vortex DynamicsAbstract:opens in new windowin html    pdfopens in new window

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 The Actual Formula to The Actual Formula, where The Actual Formula 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.

TuesdayMar 24, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Daniel ReemTitle:Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking MethodsAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayFeb 17, 201511:15
Mathematical Analysis and Applications SeminarRoom 261
Speaker:D. TuraevTitle:Chaotic dynamics in systems of non-holonomic mechanicsAbstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL ROOM
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.
TuesdayFeb 10, 201511:15
Mathematical Analysis and Applications SeminarRoom 261
Speaker:Vitaly MorozTitle:Grounds states of the stationary Choquard equationsAbstract:opens in new windowin html    pdfopens in new window
The Choquard equation, also known as the nonlinear Schrodinger-Newton equation is a nonlinear Schrodinger type equation where the nonlinearity is coupled with a nonlocal convolution term given by an attractive gravitational potential. We present recent results on the existence, positivity, symmetry and optimal decay properties of ground state solutions of stationary Choquard type equations under various assumptions on the decay of the external potential and the shape of the nonlinearity. This is a joint work with Jean Van Schaftingen (Louvain-la-Neuve, Belgium)
TuesdayJan 27, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Mira ShamisTitle:The standard map, and discrete Schroedinger operatorsAbstract:opens in new windowin html    pdfopens in new window
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]
TuesdayJan 20, 201511:15
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Dmitry DolgopyatTitle:Piecewise linear Fermi-Ulam pingpongsAbstract:opens in new windowin html    pdfopens in new windowNote the new time at which the MAAA seminars begin
We consider a particle moving freely between two periodically moving infinitely heavy walls. We assume that one wall is fixed and the second one moves with piecewise linear velocities. We study the question about existence and abundance of accelerating orbits for that model. This is a joint work with Jacopo de Simoi.
TuesdayJan 06, 201511:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:David DynermanTitle:Describing geometry and symmetry of cryo-EM datasets using algebraAbstract:opens in new windowin html    pdfopens in new window

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.

TuesdayDec 23, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Sylvia SerfatyTitle:Microscopic behavior of systems with Coulomb and Riesz interactionsAbstract:opens in new windowin html    pdfopens in new window
We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas and beta ensembles, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. We describe tools to study such systems and derive a next order (beyond mean field limit) "renormalized energy" that governs microscopic patterns of points. We present the derivation of the limiting problem and the question of its minimization and its link with the Abrikosov lattice and crystallization questions. In the statistical mechanics context, we obtain a large deviation principle on the "empirical fields." This is based on joint works with Etienne Sandier, Nicolas Rougerie, Simona Rota Nodari, Mircea Petrache, and Thomas Lebl\'{e}.
TuesdayDec 16, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Milton Lopes FilhoTitle:On the vortex-wave systemAbstract:opens in new windowin html    pdfopens in new window
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
TuesdayDec 09, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Helena Nussenzveig LopesTitle:Vortex sheets in domains with boundaryAbstract:opens in new windowin html    pdfopens in new window
Vortex sheets are idealized models for flows undergoing intense shear in a thin region. They are an ubitiquous phenomena in incompressible fluid dynamics. Mathematically, two-dimensional vortex sheets correspond to solutions of the incompressible 2D Euler equations with locally square-integrable velocity and whose vorticity is a bounded Radon measure. Existence of weak solutions with this regularity has been established when the singular part of vorticity has a distinguished sign, however there is very little qualitative information about these solutions. In this talk we examine the interaction between vortex sheets and a material boundary, namely the boundary of the fluid domain. We discuss the behavior of circulation, net force and torque across this material boundary, for vortex sheet flows.
TuesdayNov 18, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Nathan PaldonTitle:Laplace Tidal Equation over a sphere: New solutions derived from an approximate Schrodinger equationAbstract:opens in new windowin html    pdfopens in new window
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.
TuesdayNov 18, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Nathan PaldonTitle:Laplace tidal equation over a sphere: New solutions derived from an approximate Schr\H{o}dinger equationAbstract:opens in new windowin html    pdfopens in new window
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 Schr\H{o}dinger 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 Schr\H{o}dinger 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.
TuesdayNov 11, 201411:00
Mathematical Analysis and Applications SeminarRoom 1
Speaker:Prof. Matania Ben-Artzi Title: Spectral Theory And Spacetime Estimates Of Divergence-Type OperatorsAbstract:opens in new windowin html    pdfopens in new window