Departmental Colloquium
Spring 2010
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 14 |
Christel Hohenegger
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One of the challenges in modeling the transport properties of complex fluids (e.g. many biofluids, polymer solutions, particle suspensions) is describing the interaction between the suspended micro-structure with the fluid itself. Here I will focus on my work in understanding the dynamics of active suspensions -- motile bacterial baths are an important example -- and also overview my work on characterization and modeling of complex materials such as lung mucus. Suspensions of active particles, like swimming bacteria or artificial micro-swimmers, have been studied intensely over the past few years. Using a recently derived kinetic model, I have investigated the linearized structure of such an active system near a state of uniformity and isotropy. I show that system instability can arise only from the dynamics of the first azimuthal mode in swimmer orientation, that the growth of fluctuations for a suspension of anterior actuated swimmers is associated with a proliferation of oscillations in swimmer orientation, and that at small-scales the system is controlled independently of the nature of the suspension. Finally a prediction about the onset of the instability as a function of the volume fraction of anterior actuated swimmers can be made. Einstein showed that the thermal fluctuations of tracer particles in a fluid can be related to its bulk viscosity. In recent times this observation has been extended and forms the basis of the field of microrheology, which seeks to use statistical quantities to estimate the viscous and elastic properties of materials from very small volume samples. Following the basic model of two-point microrheology, I have developed a Langevin-based model of the coupled fluctuations of two beads in a viscoelastic liquid and from this derive new relations between measurable quantities and fluid response properties. This approach provides a new interpretation to memory response functions, which play a dominant role in numerical simulation of such systems.
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January 26 TUESDAY |
Juan Souto
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By Mostow's rigidity theorem, geometric invariants of hyperbolic 3-manifolds are in fact topological invariants. On the other hand, it follows from the work of Thurston and Perelman that a 3-manifold is hyperbolic if and only if it satisfies some rather mild conditions. In light of these results, it is an interesting question to try to understand how topological conditions on a 3-manifold M which admits a hyperbolic metric affect the geometry of the hyperbolic metric. This question is rather imprecise. In other words, it has many different incarnations. In this talk I will describe a few results on some concrete formulations of the question above.
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January 28 |
Alexandra Pettet
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The outer automorphism group Out(F) of a free group of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F), previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S). This is joint work with Matt Clay.
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February 9 TUESDAY |
Sunder Sethuraman
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The 'zero-range' interacting particle system, introduced in the 70's, is a formal model of certain queues, traffic, and other physical phenomena. The system follows a collection of random walks on a lattice which interact in that the rate at which a particle jumps depends only on the occupation number of its location. Of interest is the asymptotic behavior of a distinguished, or tagged particle in the system. In this talk, we discuss a nonequilibrium limit for the tagged particle in one dimension when the transition probabilities are mean-zero. The limit process is a diffusion whose coefficients depend on the underlying hydrodynamic evolution of the mass density.
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February 11 |
Hai Long Dao
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To understand an algebraic object (groups, rings, etc.) it often pays to understand some category such object acts on. For example, when our object is a commutative ring R, we wish to explore the category mod(R) of (finitely generated) modules over R. Two fundamental operations in mod(R) are tensor product and Hom. Studying these operations naturally leads to their respective derived functors, namely Tor and Ext. Despite the relatively simple definitions involving these functors, their actual behaviour remain quite mysterious. In this talk I will describe how understanding simple questions such as when the Ext and Tor modules vanish can lead to concrete results on seemingly unrelated topics such as non-commutative crepant resolutions or torsions in Picard groups of certain singularities. Most of the talk will be accessible to non-experts.
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February 16 TUESDAY |
Cecilia Diniz Behn
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Microdialysis and microinjection experiments suggest that neurotransmitter dynamics play an important role in the initiation and maintenance of sleep/wake states. However, the synaptic coupling in traditional population firing rate models does not explicitly incorporate the dynamics of neurotransmitter concentrations. We introduce a novel network modeling framework that includes neurotransmitter concentrations, and we use this framework to model interactions among primary brainstem nuclei involved in rat sleep/wake regulation. Analysis of the bifurcation structure underlying model dynamics provides insights into the mechanisms governing REM sleep, particularly those associated with circadian modulation.
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February 18 |
Yekaterina Epshteyn
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In this talk, I will first discuss several chemotaxis models including the classical Keller-Segel model. Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. In our work we propose a family of stable (even at times near blow up) and highly accurate numerical methods, based on interior penalty discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. This model is the basic step in the modeling of many real biological processes and it is described by a system of a convection-diffusion equation for the cell density, coupled with a reaction-diffusion equation for the chemoattractant concentration. We prove theoretical hp error estimates for the proposed discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume some regularity of the exact solution. Numerical experiments to demonstrate the stability and high accuracy of the proposed methods for chemotaxis models and comparison with other methods will be presented. Ongoing research projects will be discussed as well.
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February 23 TUESDAY |
Fernando Guevara Vasquez
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I present my work on network based methods to reconstruct the parameter of a differential equation inside a body from measurements taken at its surface. Two inverse problems are considered: the electric impedance tomography problem where the conductivity is to be estimated from current and voltage measurements at the surface and the problem of estimating the (linear) elastic properties of a body from displacement and force measurements at its surface. These are severely ill-posed problems requiring some form of regularization. Our inversion strategy is to (1) Find a reduced model of the problem that can be recovered from the data, with deliberately few parameters to cope with the ill-posedness. (2) Interpret the reduced model as a discretization of the underlying PDE. (3) Use the interpretation to estimate the unknown parameter in the PDE. For electric impedance tomography, the reduced model is a resistor network arising from a finite volumes discretization with the number of parameters (resistors) determined by the quantity and quality of the measurements. We show that the model reduction problem of finding the smallest resistor network (of fixed topology) that can predict meaningful boundary measurements is uniquely solvable for a broad class of measurements. We propose a simple inversion method based on an interpretation of the resistors as conductivity averages over the cells of a predetermined grid that is adapted to the measurements. The reconstructions can be further improved with an iterative procedure ensuring that the reconstructed conductivity fits the measurements and incorporating a priori information. Our reconstruction method is well-suited to situations where measurements are only available on a portion of the surface. For the inverse linear elasticity problem we outline the results that would be needed for applying this inversion strategy. Here the reduced models are networks of springs and masses. We present a first step towards a network based inversion method for this problem, namely, a complete characterization of the response function of networks of springs and masses.
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February 25 |
Charles Favre
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Suppose we are given a rational map f : P^d -> P^d on a projective space. The most basic algebraic invariant that one can attach to this map is its degree defined as the degree of the preimage of a generic hyperplane. In this talk, we shall discuss the problem of describing the behaviour of the sequence deg(f^k) when k tends to infinity. The growth of degrees is an important issue in the study of the dynamics of rational maps on projective spaces.
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March 4 |
John R. Parker
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A complex hyperbolic lattice is a discrete group of isometries of the unit complex ball whose quotient has finite volume (with respect to the Bergman metric). There are relatively few examples of complex hyperbolic lattices known, but these examples may be described from several different points of view. Namely, one may use hyperbolic geometry and a fundamental polyhedron; one may use methods of algebraic geometry, in particular line arrangements; one may describe many of them as monodromy groups of hypergeometric functions and one may use techniques from number theory to give many of them as arithmetic groups. In this talk I will explain the relation between these points of view for a family of lattices constructed by Deligne and Mostow in the 1980s.
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March 11 |
Gunther Uhlmann
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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. -->
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March 18 |
Anne Thomas
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Lattices in Lie groups are well-studied, but little is known about lattices in other locally compact groups. Examples of such "exotic" groups include isometry groups of trees, polygonal complexes, and CAT(0) spaces, and Kac-Moody groups. We will survey known results, which include both rigidity and surprising examples of flexibility, and discuss the wide range of tools used to investigate lattices in these non-classical settings.
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April 8 |
Paolo Stellari
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We will review some classical constructions related to the geometry of varieties of dimension 4 described by polynomial equations of degree 3, called cubic fourfolds. The appearance of many analogies with the geometry of some varieties with trivial canonical bundle will naturally lead us to study the interplay between the geometry of cubic fourfolds and the properties of their derived categories.
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