DEPARTMENT OF MATHEMATICS AND STATISTICS

09/29/14 Sebastian Furthauer, NYU, How cells tell left from right: Active Chiral Processes in Soft Biological Matter.

Many processes in developmental biology break left-right symmetry with a consistent handedness. In many species the primary determinant of this asymmetry has been linked to the actin cytoskeleton. We hypothesized that active chiral processes in the cell cortex might be at the origin of this symmetry breakage. To test our idea we developed a thin-film active chiral fluid theory that describes the cell cortex as a thin film of fluid that is kept out of equilibrium by molecular scale force and torque dipoles. We combine this theory with experimental analysis of the C. elegans embryo and uncover that the actomyosin cortex generates active chiral torques that facilitate chiral symmetry breakage.

10/06/14 Francesc Martinez, Barcelona, Mathematical modelling of special solid-liquid phase transitions.

Part I: Materials made from supercooled melts can have markedly different properties to the standard form of the material. Such materials are currently used in medicine, defence, electronics and sports. The solidification of these melts can be modelled by means of the supercooled Stefan problem (SSP) with a variable phase change temperature (TI). Previous works in the topic assume a linear relation between TI and the velocity of the freezing front (v). In this part of the talk, I present solutions for the SSP with a more physically realistic nonlinear relation between TI and v.

Part II: Nanoparticles have generated tremendous interest in the scientific community in the last few decades: they exhibit unique optical, electrical, and magnetic properties not seen at the bulk scale. These properties and their size make them attractive for industrial and biomedical applications. One form of application, based on the melting process of nanoparticles, is used in drug delivery or microelectronics. In this part of the talk, we present a mathematical model describing the melting of nanoparticles that takes into account the Gibbs-Thomson effect and the thermal expansion of the nanoparticles, which turns out to be non-negligible at that scale. The melting times obtained from our results are in good agreement with experimental observations.

10/20/14 Mark Lyon, University of New Hampshire, Fourier Continuation and Accurate Approximation .

Fourier Continuation/Extension methods and variations on theses techniques have been introduced as a resolution to the Gibbs phenomenon, thus allowing fast and accurate Fourier representation of non-periodic functions. Current research with respect to these Fourier methods and their applications will be presented, including the solution of PDEs where these methods can be particularly advantageous.

10/27/14 Frederic Dias, University College Dublin, Extreme waves: their observation and their generation.

The study of extreme wave events on the ocean is a rapidly expanding area of research worldwide. Although much work in this area is based on modeling and experiments in controlled wave tanks, the starting point of all studies is of course observation in the natural world. During this talk, we will provide some evidence of extreme wave events and describe the main mechanisms for their generation.

11/03/14 Jun Lai, NYU, A fast solver for multi-particle scattering in a layered medium.

Consider acoustic or electromagnetic scattering in two dimensions from an infinite or periodic three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. Numerical experiments show the algorithm is very efficient.

11/10/14 Pierre-David Letourneau, Columbia, Resolution and De-aliasing in elastography .

We present an imaging technique for the recovery of the displacement field of an ensemble of random point scatterers, as is commonly needed in the medical imaging method called elastography. We show that by using a small number of well-localized sources and a small acquisition aperture, it is possible to recover the displacement through the construction of an imaging functional based on the ratio of the correlations of the signals received. We perform a resolution analysis of the method and show that displacements are well-reconstructed only when their spatial variations are sufficiently small.

11/17/14 Emil Prodan, Yeshiva, C*-Algebras for Research and Discovery in Materials Science.

C*-Algebras have been advocated by Jean Bellissard as an essential tool for the study of complex aperiodic materials such as liquids, glasses, amorphous solids, disordered solids and quasi-crystals. In essence, the C*-algebras give back the Fourier transform, which is lost when the periodicity of a crystal is disturbed. Based on this observation, Bellissard constructed an entire non-commutative geometry formalism for aperiodic systems, which produced closed-form expressions for virtually any physical quantity of interest. These expressions appear as the non-commutative counterparts of the familiar expressions written in the momentum space (when periodicity permits that). Recently, our research group has demonstrated that these non-commutative expressions accept cannonical finite-volume approximations amenable on computers. They were shown to converge exponentially fast to the thermodynamic limit, hence providing one of the most efficient numerical tool to analyze aperiodic quantum systems. In this talk I will give a brief review of the formalism and show several applications, ranging from quantum transport in disordered systems under magnetic fields, to computations of topological invariants for the so-called topological insulators (in the presence of strong disorder).

12/01/14 Carlos Borges, NYU, Inverse obstacle scattering in two dimenisons with multiple frequency data and multiple angles of incidence .

We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill posed and nonlinear. It is common practice to overcome the ill posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton's method. When multiple frequency data is available, we supplement Newton's method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

12/08/14 Gilou Agbaglah, Cornell, Impact of a single drop on the same liquid: formation, growth and disintegration of jets .

One of the simplest splashing scenarios results from the impact of a single drop on on the same liquid. The traditional understanding of this process is that the impact generates a jet that later breaks up into secondary droplets. Recently it was shown that even this simplest of scenarios is more complicated than expected because multiple jets can be generated from a single impact event and there are bifurcations in the multiplicity of jets. First, we study the formation, growth and disintegration of jets following the impact of a drop on a thin film of the same liquid using a combination of numerical simulations and linear stability theory. We obtain scaling relations from our simulations and use these as inputs to our stability analysis. We also use experiments and numerical simulations of a single drop impacting on a deep pool to examine the bifurcation from a single jet into two jets. Using high speed X-ray imaging methods we show that vortex separation within the drop leads to the formation of a second jet long after the formation of the ejecta sheet.

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