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Past talks for Spring 2010

**January 25**

Speaker: Joey Wilson, Founder of Xandem Technology, Salt Lake City, UT

**Title: ** Radio Tomographic Imaging: Using Simple Wireless Networks to See Through Walls and Locate People

**Abstract: ** Radio Tomographic Imaging (RTI) is an emerging technology that is capable of detecting and locating humans behind walls and through obstructions. This presentation will discuss how RTI works, specifically detailing the inverse mathematical models and solutions. Current research and ideas for future mathematical research will be discussed.

Joey Wilson and Dr. Neal Patwari of the University of Utah were recently featured in many international media outlets for their work in RTI, including The Economist, MSNBC, Wired.com, and many others. Please see the following websites for more information:

http://span.ece.utah.edu/radio-tomographic-imaging

http://span.ece.utah.edu/joey-wilson

http://www.xandem.com

**February 1**

Speaker: Valery Smyshlyaev, University of Bath, Department of Mathematical Sciences

**Title: ** "Non-classical" homogenization and localization and dispersion of waves

**Abstract: ** We review a "non-classical" homogenization which, in contrast to its classical counterpart, deals with explicit limit asymptotic descriptions which remain multiscale ones. This leads to interesting effects physically (e.g. frequency or "directional" localisation, high dispersion etc), and mathematically allows treating from a unified perspectives the "classical" homogenization, the high-contrasts one, and the intermediate cases, via developing new versions of e.g. asympotic expansions with error bounds, two-scale convergence, spectral and operator convergence, and of two-scale compactness.

**February 5 (Friday, 4:15pm, LCB 215)**

Speaker: Pierre Seppecher, Université de Toulon et du Var

**Title: ** Some homogenization results for viscoelastic bodies at fixed frequency

**Abstract: ** Our aim is to characterize the set of all materials which can be obtained by homogenizing high contrasted viscoelastic bodies. We use a variational framework. Indeed lossy elastodynamic problems can be seen as the search of saddle points of convex-concave functionnals (see Milton et al. Proc. R. Soc. A 465, 367-396, 2009). A suitable notion of convergence of convex-concave functions and associated saddle points is the epi-hypo-convergence (see Attouch et al., Trans. Amer. Math. Soc. 280, 1-44, 1983.)

We follow, for elastodynamics at fixed frequency, a scheme which has proved to be efficient in the elastostatic case. In this scheme different steps are necessary. (i) A first homogenization result shows that materials exhibiting simple non local interactions can be obtained. These interactions are simple as they are two-points interactions with a fixed range and direction. (ii) An addivity property allows to reach multiple interactions : truss-like interactions. (iii) A second homogenization result proves that some nodes of the previous trusses can be set free. Hence the truss-like interactions become mechanisms. (iv) the possible responses of such mechanisms are characterized. This step has already been studied (Milton et al. Proc. R. Soc. Lond. A, 464, 967-986, 2008) at a fixed frequency and, very recently, the dependence with respect to the frequency has been investigated (Vasquez et al., arXiv:0911.1501v1). (v) The last step is the approximation of any realizable functional by mechanisms using regularization and discretization procedures.

In this contribution some crucial homogenization steps will be presented.

**February 19 (Friday 4:15pm. LCB 323)**

Speaker: Yekaterina Epshteyn, Carnegie Mellon University, Dept. of Mathematical Sciences and Center for Nonlinear Analysis

**Title: ** Theory for the grain boundary character distribution

**Abstract: ** Most technologically useful materials are polycrystalline,
comprised of many small grains separated by interfaces, called grain
boundaries. The energetics and connectivity of this network of interfaces plays
a role in many material properties and across many scales of use. A central
problem in material science is to develop technologies capable of producing an
arrangement of grains---a texture---that provides for a desired set of material
properties. Since the dawn of man, coarsening has been the principal feature
of these technologies. Cellular structures coarsen according to a local
evolution law, a gradient flow or curvature driven growth, for example, limited
by space filling constraints, which give rise to changes in the configuration.
There are two aspects of coarsening, geometric growth and texture development.
In this presentation we will focus on texture development. We will consider
the grain boundary character distribution, the GBCD, a basic texture measure,
and will establish an entropy based theory for it which suggests that GBCD
satisfies Fokker-Planck type kinetics. For this, we will introduce and will
study a simplified critical event model.

**February 22**

Speaker: Andrew Belmonte, Pennsylvania State University, Department of Mathematics

**Title: **Breaking away: buckling spaghetti and the fragmentation problem

**Abstract: ** What happens to uncooked pasta if you hit it at 80 km/hr? Answering this question challenges the limits of elasticity and material science, from nonequilibrium Euler buckling to the dramatic failure of brittle solids. I will present an experimental and mathematical study of the dynamic buckling of slender rods - including spaghetti, teflon, glass, and steel - due to rapid impact. We calculate the buckling wavelength from a stability analysis of the coupled equations for stress and deformation, supported by full numerical simulations. Experimentally, brittle rods often break, with peaks in the fragment length distribution which depend on the buckling wavelength. This indicates the influence of this deterministic process on the more random processes of fragmentation. We present a general framework for 1D fragmentation, and derive an explicit formula for the fragment distribution in terms of a space-dependent breaking probability, corresponding to a nonhomogeneous Poisson process.

**March 8**

Speaker: Graeme Milton, University of Utah, Mathematics Dept.

**Title: ** Where Science meets Science Fiction.

**Abstract: ** This is a trial run of my talk for the Frontiers of Science
Lecture. Come for an entertaining survey of cloaking. I'll also discuss our new
work on exterior broadband cloaking, joint work with Fernando Guevara Vasquez
and Daniel Onofrei.

**Cancelled.** **March 11 (Joint with Dept. Colloquium, Thursday, 4:15pm JWB 335)**

Speaker: Gunther Uhlmann, University of Washington

**Title: ** 30 Years of Calderón's Problem

**Abstract: **
In 1980 A. P. Calderón wrote a short paper entitled "On an inverse boundary value problem". In this seminal contribution he initiated the mathematical study of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? There has been substantial progress in understanding this inverse problem in the last 30 years or so. In this lecture we will survey some of the most important developments.

**March 12 (Friday 4:15pm. LCB 323)**

Speaker: Marian Bocea, North Dakota State University

**Title: ** Gamma-convergence of Power-law Functionals with Variable
Exponents and Applications

**Abstract: ** Motivated by the analysis of various models related to
polycrystal plasticity, the asymptotic behavior of several classes of
power-law functionals acting on fields belonging to variable exponent
Lebesgue spaces and which are subject to constant rank differential
constraints is studied via Gamma-convergence. The effective yield set
of a polycrystal is characterized in several model cases by means of
variational principles associated to the limiting functionals. Some
applications to PDEs will also be discussed.

**March 15**

Speaker: Gregory J. Rodin, The
University of Texas at Austin, Institute for Computational Engineering and
Sciences

**Title: ** Behavior of cracks near thresholds

**Abstract: ** Thermodynamic analysis of brittle fracture specimens near the
threshold developed by J. R. Rice (Thermodynamics of quasi-static growth of
Griffith cracks, *J. Mech. Phys. Solid*, 26, pp. 61-78, 1978) is
extended to specimens undergoing microstructural changes. The proposed
extension gives rise to a generalization of the threshold concept that mirrors
the way the R-curve generalizes the fracture toughness concept. In the absence
of experimental data, that curve is constructed using a basic lattice
model.

**March 29**

Speaker: Marc Briane, INSA de Rennes

**Title: ** Hall effect and magneto-resistance in composites: 2D positivity results and 3D pathologies

**Abstract: ** In a homogeneous conductor a low magnetic field induces a transversal electric field both orthogonal to the current and to the magnetic field. The perturbed resistivity is characterized by the Hall coefficient at the first-order and by the magneto-resistance at the second-order. In a two-dimensional composite the effective Hall coefficient preserves the bounds of the local Hall coefficient. In dimension three the situation is radically different. Three composites illustrate various pathologies which are inconsistent with classical physics: reversal of the sign of the Hall coefficient, arbitrary large Hall coefficients, and an effective Hall field parallel to the magnetic field. On the other hand, in dimension two the effective magneto-resistance is shown to satisfy a positivity property.

**April 5**

Speaker: Blaise Bourdin, Louisiana State University, Department of Mathematics.

**Title: ** Modeling cracks formation in Enhanced Geothermal Systems: An
approach based on variational fracture

**Abstract: ** Enhanced Geothermal Systems (EGS) represent a virtually
untapped, clean, renewable, economically viable and widely available source of
energy. They rely on harvesting heat by circulating water through artificially
stimulated, highly connected fracture systems in deep hot dry rocks. The goal
of this talk is to present a first step towards the predictive understanding of
the mechanisms used in the creation of these highly connected crack networks. I
will focus on secondary thermal crack growth, where thermal stress induced by
the cold fluid circulating through the hot reservoir lead to nucleation of many
short cracks. I will consider the limiting cases of purely diffusive and purely
advective heat transfer, corresponding to extreme porosity limits in the
reservoir. I will present a mechanistically faithful yet mathematically sound
model, based on Francfort and Marigo's generalization of Griffith's idea of
competition between bulk and surface energies. I will discuss the virtues of
the model, its approximation, and its numerical implementation. Finally, I will
present some numerical experiments in 2 and 3 dimensions.

This work is a collaboration with Corrado Maurini (Institut Jean le Rond
d'Alembert, Université Pierre et Marie Curie, France) and Matthew Knepley
(Computational Institute, University of Chicago).

CANCELLED**April 16 (Friday, 4:15pm. LCB 323)**

Speaker: Hongyu Liu, University of Washington, Department of Mathematics

**Title: ** TBA

**Abstract: ** TBA

**April 19 (student talk)**

Speaker: Loc Nguyen, University of Utah, Mathematics Dept.

**Title:
** Existence of solutions to singular elliptic equations

**Abstract: ** We study the existence of positive solutions to singular
elliptic boundary value problems involving the p-Laplace operator by
establishing a new sub-supersolution theorem and using an eigenfunction
of the p-Laplacian to construct sub- and super-solutions. Our
assumptions on the singular term are more relaxed than in previous work
for the 2-Laplacian, as we allow for non-monotone singular terms with
blowup controlled by a power. We also allow for a parameter dependent
term and study how its growth affects our existence result.

**July 19 **

Speaker: Elena Vilchevskaya, Institute of Mechanics
of the Russian Academy of Sciences

**Title:
** Front propagation of the diffusion-controlled chemical reaction

**Abstract: ** We consider propagation of a chemical reaction front in an elastic
solid. Particularly, we study an oxidation of a ball under external elastic loading. For simplicity
we use a linearized chemical affinity tensor in a small strain approach. It is shown, that at some conditions, the reaction may be
blocked by the oxide growth caused of internal stresses.

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