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**SPRING 2013**

**Tuesday May 7**, 2:00-3:00pm in CR 5127

Speaker: Oren Louidor, Department of Mathematics, UCLA

Title: Isoperimetry in Supercritical Percolation

Abstract: We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every realization of the infinite connected component we prove that, as the volume of the isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular norm. As an application we then show that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane. Joint work with M. Biskup, E. Procaccia and R. Rosenthal.

Coffee and cookies at 1:45pm

**Wednesday May 1**, 3:45-4:45pm in CR 5208

Speaker: Guihong Fan, Arizona State University

TITLE: The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons

ABSTRACT: We consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show that a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov-Takens bifurcation with Z_2 symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a â€œfigure-eightâ€ homoclinic orbit. This is a joint work with Sue Ann Campbell (University of Waterloo), Gail Wolkowicz (McMaster University), and Huaiping Zhu (York University).

Coffee/tea and cookies at 3:30pm.

**FALL 2012**

**December 4, 2012
2:00-3:00pm, Location: CR 5208**

Henry Boateng

University of Michigan

Title: *Quasi-Equilibrium Kinetic Monte-Carlo*

Abstract: We present an off-lattice kinetic Monte Carlo algorithm in (1+1)-dimensions that drives surface diffusion by a chemical potential gradient. Interactions between atoms are defined by the Lennard-Jones potential which removes the restriction on atomic positions to lattice points enforced by lattice based models. The method is validated by simulations of hetero- epitaxial growth, annealing of strained bilayer systems and a qualitative verification of StoneyÃ¢Â€Â™s formula. The algorithm captures the effect of misfit and deposition flux on island formation, the formation of vacancies and edge dislocations unlike lattice and continuum models, and naturally incorporates intermixing.

**November 27, 2012
2:00-3:00pm, Location: CR 5208**

Pablo Seleson

ICES, University of Texas at Austin

Title: *Bridging scales with nonlocal continuum models for
applications to material failure and damage*

Abstract: Nonlocal models are used nowadays in many fields, including image processing, solid mechanics, heat conduction, and diffusion, among others. A particular type of nonlocal models is characterized by integral equations, depending on differences of field variables and not on their spatial derivatives, in contrast to classical continuum models based upon partial differential equations; this feature allows nonlocal models to be applicable to a larger class of problems. This is the case of peridynamics, a nonlocal generalization of classical continuum mechanics, which has been used for the study of material failure and damage. We will provide an overview of the peridynamics theory of solid mechanics, its applications, and concurrent multiscale approaches.

**November 20, 2012
2:00-3:00pm, Location: CR 5208**

Christoph Ortner

Mathematics Institute, University of Warwick

Title: *From atomistic to continuum descriptions of
crystalline solids: validity and failure of the Cauchy-Born
rule*

Abstract: Multi-scale modelling has become a paradigm that transcends all scientific disciplines. A key challenge that arises in many scenarios is the connection between discrete atomistic and continuum descriptions of matter. The Cauchy-Born rule postulates such a connection for crystal elasticity, which seems almost naive at first glance. Nevertheless, the Cauchy-Born model has been found to provide an accurate description of crystal elasticity even at the sub-grain scale. In this talk, I will describe various attempts to derive and understand the Cauchy--Born model and its limitations: Cauchy's derivation of linearized elasticity, the variational approach of the 90's, and most recently the point of view of approximation theory. Time permitting, I will mention some interesting concepts to make the connection between molecular mechanics and Cauchy-Born elasticity rigorous.

**SPRING 2012**

**February 9, 2012
11:00-12:00, Location: EH 2028**

Andrew Bernoff

Harvey Mudd College

Title: *Langmuir Layers: Exploring a (nearly)
Two-dimensional Fluid Experiment*

Abstract: A Langmuir Layer is a molecularly thin layer of a polymer, lipid or liquid crystal on the surface of another fluid. In this (nearly) two-dimensional layer, we can observe bubbles of a fluid phase that even when stretched or highly contorted always appear to return to a circular shape. The force driving these evolutions is line tension, a two-dimensional analog of surface tension. We report on a combined experimental, theoretical, and numerical study of Langmuir layers and show how we can deduce the strength of the line tension in the system by comparing theory and experiment. As time permits we will also report on some other phenomena observed in Langmuir systems, including collapse of gas phase bubbles, co-existence of three or more fluid phases, and formation of dogbone and labyrinth patterns due to dipolar repulsion in the layer. This work is collaborative with Jacob Wintersmith (HMC 2006), George Tucker (HMC 2008), Elizabeth Mann (Kent State), Lu Zou (Kent State), J. Adin Mann, Jr. (CWRU) and James Alexander (CWRU).

**February 14, 2012
IRIS Colloquium: 2:00-3:00pm, Location: LO 1124**

Bjorn Birnir

University of California, Santa Barbara

Title: *The Kolmogorov-Obukhov Theory of
Turbulence*

Abstract: The Kolmogorov-Obukhov statistical theory of turbulence, with intermittency corrections, is derived from a stochastic Navier-Stokes equation with generic noise. In this talk we discuss how the laminar solution of the Navier-Stokes equation becomes unstable for large Reynolds numbers, and the unstable solution is the solution of the stochastic Navier-Stokes equation. This is the unique solution that describes fully developed turbulence. In order to compare with experiments and simulations, we find the solution of the stochastic Hopf's equation for the invariant measure. Gaussian noise and dissipation intermittency produce the Kolmogorov-Obukhov scaling of the structure functions of turbulence. The Feynmann-Kac formula produces log-Poisson processes from the stochastic Navier-Stokes equation. These processes, first found by She, Leveque, Waymire and Dubrulle in 1995, give the intermittency corrections to the structure functions of turbulence, stemming from velocity fluctuations. The probability density function (PDF) of the two-point statistics that can be compared to experiments and simulations turn out to be similar to the generalized hyperbolic distributions first suggested by Barndorff-Nilsen in 1977. We compare the theoretical PDF with PDFs obtained from DNS simulations and wind tunnel experiments.

**February 16, 2012
11:00-12:00, Location: EH 2028**

Pak-Wing Fok

University of Delaware

Title: *Mathematical Models of Atherosclerosis*

Abstract: Atherosclerotic plaques are fatty deposits that grow mainly in arteries and develop as a result of a chronic inflammatory response. Although plaques are frequently thought of as fatty deposits with little or no internal organization, they are actually commonly characterized according to their composition and morphology. In this talk, I will present models for two types of plaque: thin-cap fibroatheromas (TCFAs) and pathological intimal thickenings (PITs). TCFAs are characterized by inflammation and the presence of necrotic cores. By solving coupled reaction-diffusion equations for macrophages and dead cells, we explore the joint effects of hypoxic cell death and chemoattraction to oxidized low-density lipoprotein (Ox-LDL), a molecule that is strongly linked to atherosclerosis. The model predicts cores that have approximately the right size and shape when compared to ultrasound images. Normal mode analysis and calculation of the smallest eigenvalue enable us to compute the formation times of the core. An asymptotic analysis reveals that the distribution of Ox-LDL within the plaque determines not only the placement and size of cores, but their time of formation. PITs are characterized by the absence of endothelial cells and negative remodeling whereby the vessel lumen decreases in size. I will present some work in progress on PIT, described as a free-boundary problem. The model couples the diffusion of Platelet-Derived Growth Factor, governed by a Helmholtz equation, to cell migration and proliferation. The predictions are compared with data from animal studies.

**February 23, 2012
11:00-12:00, Location: EH 2028**

Helene Barucq

INRIA and Université de Pau, France

Title: *Enriched Local Radiation boundary conditions for
the Helmholtz equation*

Abstract: Most of the local radiation boundary conditions that have been designed for the Helmholtz equation are based on an approximation of the Dirichlet-to-Neumann operator inside the hyperbolic region. The conditions are thus representing the propagating waves only. This talk deals with the idea of improving a low-order radiation boundary condition by adding the modelling of both evanescent and glancing waves.