Analysis & its Applications

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Weekly Seminar

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Singular Limits of Variational Problems
Eigenvalues of Random Matrices and Graphs
Spectral Properties of Elliptic Operators
Stability and Instability of Coherent Structures
Pattern Formation
Quantum Condensed Phases

Weekly Seminar

Spring 2007

• January 16, 2007: Christopher Bergevin, Harvard-MIT Division of Health Sciences
  Listening to Distortion in the Ear
  Abstract: The ear not only responds to sound but also actively emits it as well. These otoacoustic emissions (or OAEs) provide a non-invasive measure of auditory function and reveal important underlying physiological properties of the ear, where direct experimental measurement is difficult. The ear itself is a remarkable biological detector in that it exhibits both sharp frequency selectivity and an enormous dynamic range over which it is sensitive. In order to achieve these properties, the ear shows remarkably nonlinear behavior. Compressive growth in the peripheral response is necessary in order to encode information in auditory nerve fibers, which have a much smaller dynamic range. It is not well understood how the ear produces this nonlinear behavior, which has other consequences such as perceptible distortion and two-tone suppression. OAEs evoked using an external acoustic stimulus sometimes exhibit non-monotonic growth with respect to stimulus level. It is widely believed that these growth functions can be accounted for based upon the behavior of a single cellular component in the ear (which exhibits a nonlinear response approximated by a hyperbolic tangent function). The OAE studies presented here reveal that this single 'source' model is inadequate to describe a wide range of behavior observed. This indicates that alternative approaches are needed to better understand the ear's nonlinearity and the origin of its exquisite sensitivity.



• January 23, 2007: Organizational meeting



• January 30, 2007: Silas Alben, Harvard University
  Flows, Bumps, and Flexibility: Fish Fins, Whale Flippers, and More
  Abstract: I will discuss a few recent studies on how organisms propel themselves through water, focusing on the appendages that allow them to do so efficiently. I will begin with fish fins, which have evolved over millions of years in a convergent fashion, leading to a highly-intricate fin-ray structure that is found in half of all fish species. This fin ray structure gives the fin flexibility plus one degree of freedom for shape control. I will present a linear elasticity model of the fin ray, based on experiments performed in the Lauder Lab in Harvard's Biology Department.
  In conjunction with this work, I will present numerical simulations of a fully-coupled fin-fluid model, based on a new method for computing the dynamics of a flexible bodies and vortex sheets in 2D flows. The simulations are applied to the most common mode of fish swimming, based on tail fin oscillations. In the passive case, an optimal flexibility for thrust is identified, and we consider also the optimal distribution of flexibility, with reference to recent measurements of tapering of insect wings and fish fins. We also briefly present work on fundamental instabilities of a flexible body aligned with a flow (the "flapping flag" problem).
  I will then discuss work on the role of bumps on the leading edge of humback what flippers, in collaboration with Ernst van Nierop and Michael Brenner at Harvard. Bumps have been shown in wind tunnels to increase the angle of attack at which flippers lost lift dramatically, or "stall". This stall-delay is thought to enable greater agility. In this study we propose an aerodynamic mechanism which explains why the lift curve flattens out as the amplitude of the bumps is increased, leading to potentially desirable control properties.
  Finally, I will briefly describe results on a recent problem in self-assembly: the formation of 3D structures from flat elastic sheets with embedded magnets. The ultimate utility of this method for assembly depends on whether it leads to incorrect, metastable structures. We examine how the number of metastable states depends on the sheet shape and thickness. Using simulations and the theory of dislocations in elastic media we identify out-of-plane buckling as the key event leading to metastability. The number of metastable states increases rapidly with increasing variability in the boundary curvature and decreasing thickness.



• February 6, 2007: Ray Mejía, National Heart Lung and Blood Institute
  Mathematical Prediction of High Energy Metabolite Gradients in Mammalian Cells
  Abstract: Adenine nucleotides are high energy compounds used to support work functions in all cells that may also act as second messengers by coupling changes in cell metabolism to a functional response. We describe a mathematical model to evaluate the distribution of cellular adenosine nucleotides, and use the model to test the hypothesis that local changes can modulate function without significant changes in global cytosolic concentrations. The model incorporates knowledge of cell structure to predict the spatial concentration profiles of ATP, ADP and inorganic phosphate. The steady state was perturbed by increasing the activity of membranebound ion transporters including the Na-K ATPase on the cell periphery or the V-type H +-ATPase which serves to acidify intracellular compartments including endosomes/lysosomes. In each model type, this is done for a range of cytosolic diffusivities, including local low diffusivity near the pump sites. Results suggest that local changes in the concentration of ADP, in particular at near membrane sites, may serve to modulate ion transport, and thereby cell behavior.



• February 13, 2007: Veronika Furst
  Wavelets and Multiresolution in Abstract Hilbert Spaces
  Abstract: The complete characterization of orthonormal wavelets has been known in L2(R). G. Gripenperg and X. Wang proved independently that a square-integrable function of norm one is an orthonormal wavelet if and only if two (somewhat unexpected) equations are satisfied.
  We will discuss an analogous characterization of semiorthogonal Parseval wavelets in an abstract Hilbert space. We will examine the structural assumptions that must be made about this space in order to compensate for the loss of known facts about L2(R). More importantly, we will discover that the abstract characteristic equation is the analog not of the two original equations but of their combination with a condition placed on the wavelet dimension function.



• February 20, 2007: Sunhi Choi
  Locating the first nodal set in convex domains
  Abstract: I will talk about the eigenvalue and the nodal set of the first Neumann eigenfunction of a convex domain in Rⁿ. If a convex domain W is contained in a long and thin cylinder [0,N] ´ B_e(0) Ì Rⁿ with nonempty intersections with {x₁ = 0} and {x₁ = N}, then the smallest nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to e, whose coefficients are expressed in terms of the volume of the cross sections of W. Also, the first nodal set is located within a distance comparable to e near the zero of the corresponding ordinary differential equation.



• February 27, 2007: Lotfi Hermi
  On Riesz and Carleman Means of Eigenvalues
  Abstract: Riesz means are generalizations of the Weyl counting function for eigenvalues. In this presentation, we prove a new monotonicity principle for Riesz means of eigenvalues of the Dirichlet Laplacian, and discuss consequences for Riesz and Carleman means. This monotonicity principle complements the famous result of Aizenman-Lieb.
  We also show that the Berezin-Li-Yau inequality arises from a class of universal inequalities for these Riesz means and this new monotonicity result, as well as offer three alternative new routes to this inequality, in addition to the well-known proof by Ari Laptev (by himself, and with T. Weidl).
  Connections between various new and classical results are made via a host of integral transforms such as the Laplace, Weyl, and Riemann-Liouville fractional transforms, adding new tools to an already rich class of convexity and Legendre transform methods.
  At the heart of some of these inequalities are pure commutator techniques and sum rules. In the course of developing these inequalities we prove new Gaussian-type bounds for the spectral zeta function and conjecture new ones.
  This is joint work with Professor Evans Harrell of Georgia Tech.



• March 6, 2007: Sunhi Choi
  The first Neumann eigenfunction of a convex domain: Part II
  Abstract: I will present the proof of the theorem that the first Neumann eigenvalue and the nodal set of a convex domain can be approximated well by the eigenvalue and the zero of an ordinary differential equation whose coefficients are expressed solely in terms of the cross-sectional volume of the domain. If time permits, I will also talk about what is known for the Dirichlet eigenfunction and its open problems.



• March 13, 2007: Spring break, no seminar



• March 20, 2007: Analysis and its Applications and Nonlinear Waves research group meeting.



• March 27, 2007: Yves Pomeau
  Statistical Mechanics of a Gravitational Plasma (with something else if I have time)
  Abstract: Statistical mechanics was developed under the assumption that particles interact with short range forces. It was pointed out early on that the general methods of this theory cannot apply to systems of particles interacting with gravitation. However such gravitational plasmas are actually present in our Galaxy in the form of local clusters of millions of stars.



• April 3, 2007: Jason Newport
  Solutions to the Nonlinear Schrodinger Equation with Dirac mass Initial Data
  Abstract: I am studying the defocussing nonlinear Schrodinger equation with Dirac mass initial data. We will start with a brief introduction to the scattering theory associated with the NLS equation, and a formal extension of the theory to handle Dirac masses. The bulk of the talk will be on the Riemann Hilbert analysis that gives us an asymptotic description of the solutions. Time permitting we will discuss properties of our solutions.



• April 10, 2007: Dorin Dumitrascu
  On the non-invertibility of some extensions
  Abstract: From the 1970's work of Brown, Douglas, and Fillmore, to that of Kasparov, and more recently to that of Manuilov and Thomsen, extensions have played an important role in the various homology-cohomology theories associated to C*-algebras. I will introduce the semigroup of extensions, present the connection with the KK-theory groups, and finally discuss some recent examples, due to Manuilov and Thomsen, of non-invertible extensions associated to groups with property T. This will be an expository talk.



• April 17, 2007: Adam Spiegler
  A topological test for stability of degenerate equilibria on reduced Hamiltonian Systems
  Abstract: The energy-Casimir method is a very powerful, geometric method for determining sufficient conditions for stability of equilibria of Hamiltonian systems with symmetry. The geometry of such systems imposes restrictions on the application of this method, namely it only determines stability of certain generic equilibria of the system. In this talk I will present a topological generalization of the energy-Casimir method which can be applied in order to analyze the stability of non-degenerate equilibria of Hamiltonian systems with symmetry as well.



• April 24, 2007: Chris Eilbeck, Heriot-Watt University
  PDEs and addition formulae associated with trigonal curves
  Abstract: I consider generalizations of the elliptic Weierstrass P function to curves of higher genus, especially trigonal curves of the form y³ = x^s + ..., s = 4, 5. In the trigonal case, these functions give exact multi-periodic solutions of the Boussineq equation. The work also leads to new addition formulae for the classical Weierstrass function.



• May 1, 2007: Matt Salomone
  Circle Actions and Quasi-Periodic Continuation
  Abstract: Poincaré showed that if a system of differential equations possesses a periodic trajectory, it also possesses nearby periodic trajectories under certain nondegeneracy conditions. We show how Poincaré's method and conditions directly give nearby quasi-periodic trajectories as well, in the presence of a transverse circle action. Of particular interest is the scope of this quasi-periodic continuation on problems in celestial mechanics. This material contains joint work with Z. Xia.

Fall 2006

• August 29, 2006: Organizational meeting



• September 5, 2006: David Glickenstein
  Proof of the Poincaré conjecture and Ricci flow
  Abstract: The Poincaré conjecture is a question about the shape of space posed by Henri Poincaré in 1904. In the 102 years since, it has/had become one of the most famous open questions in mathematics. Three Fields medals have been awarded for work on the Poincaré conjecture and its generalizations, including one just a few weeks ago (although it was turned down by the awardee). We will look into the developments of the recent proof(s) of the conjecture with an emphasis on recent events. In particular, we will describe the conjecture itself and look at the major contributions of W. Thurston (framing the question as part of a geometrization conjecture), R. Hamilton (applying geometric heat equation arguments via introduction of the Ricci flow), and G. Perelman (applying comparison geometry to analyze singularities of Ricci flow).
  This talk will be intended for a general audience and will have no prerequisites, not even the recent New Yorker article on the subject.



• September 12, 2006: No seminar
  



• September 19, 2006: Bane Vasic
  Error Correction Systems for Nano-Scale Fault-Tolerant Memories
  Abstract: As the demand for higher densities, read/write speeds, and power efficiency continues, a wide range of new nano-scale technologies is being actively investigated for future computer memory systems. The main challenge is that in nano-scale systems both the storage elements and logic gates are inherently unreliable. It is in contrast to the state-of-the-art systems where only the memory elements are considered unreliable while error correction encoders and decoders are assumed to be made of reliable logic gates.
  The set of problems that will be addressed can be condensed into the following question: given n memory cells and m universal logic gates which fail following a known random mechanism, what is the optimal memory architecture which stores the maximum number of information bits for the longest period of time with arbitrary low probability of error? This complex problem can be divided and reformulated in many ways, but, interestingly, even some of the most fundamental questions related to it are still unanswered.
  In this talk we will give an overview of problems related to designing highly reliable memories made of unreliable components and give some new results in characterization of such memories in terms of their complexity and ability to retain stored information.



• September 26, 2006: Assane Lo
  Witten Laplacian Methods in Statistical Mechanics
  Abstract: The methods for investigating the dynamical behavior of certain classical unbounded spin systems took an interesting direction when powerful and sophisticated PDE techniques were introduced in the mathematical technology. The methods are generally based on the analysis of suitable differential operators which are in some sense deformations of the standard Laplace Beltrami operator. These operators commonly called Witten Laplacians were first introduced by Edward Witten (1982) in the context of Morse theory for the study of some topological invariants of compact riemannian manifolds. In 1994, Bernard Helffer and Johannes Sjostrand introduced two other elliptic operators serving to get direct methods for the study of integrals and operators of the type that appear naturally in Statistical Mechanics and Euclidean Field theory. In 1996, J.Sjostrand observed that the Hellfer-Sjostrand operators are in fact equivalent to Witten's Laplacians on zero and one forms. Since then, there has been significant advances in the use of these Laplacians for the study of the thermodyamic behavior of quantities related to the Gibbs measure. In this talk, I will discuss the solvability of the corresponding Witten Laplacian equations and the regularity of their solutions.



• October 3, 2006: David Pritchard, Massachusetts Institute of Technology and Effective Educational Technologies
  Item Response Theory for Learning and Modelling Learning
  Abstract: Item Response Theory is the modern mathematical theory of test construction and analysis: test items are characterized by a difficulty, and students by a skill. Students are assumed to answer each item only once and not to learn during the test. We have extended IRT to a web-based tutoring environment (Mastering Physics) that offers students useful spontaneous responses to their incorrect responses, requestable help consisting of hints and graded subtasks, and an opportunity to answer again. We find dramatic skill changes on different learning paths through this tutoring. We discuss standard IRT, its experimental justification, and a generalization that can quantify learning. In addition, we discuss several dynamical models of learning and show a deep connection between IRT and constructionist models. Opportunities abound for future research collaboration on detailed interactions (40Gig) of 10⁵ students.



• October 10, 2006: Chris Sinclair, Pacific Institute for the Mathematical Sciences, Simon Fraser University / University of British Columbia
  Averages over Ginibre's Real Ensemble
  Abstract: In 1965 J. Ginibre introduced three ensembles of random matrices whose entries are chosen independently with Gaussian density from C, R and Hamilton's Quaternions. Unlike the `classical' ensembles, Ginibre's ensembles have no symmetry constraints placed on the matrices. There are many open questions surrounding these ensembles. Ginibre's real ensemble (GinOE) has been particularly intransigent due to the fact that the eigenvalues come in two flavors: real and complex conjugate pairs. Quantities of interest, such as the joint eigenvalue probability density function and the correlation functions naturally fracture into sums over the number possible decompositions of the eigenvalues into real and complex conjugate pairs. Amazingly, ensemble averages can be written in a way which is independent of this decomposition. In this talk I will explain why this is important, and its potential for resolving some open problems surrounding GinOE.



• October 17, 2006: Leonid Kunyansky
  Thermoacoustic tomography, Helmholtz equation, and inversion of the spherical mean Radon transform
  Abstract: Thermoacoustic tomography is a new kind of tomography that combines the high resolution of acoustic imaging with the high sensitivity of the electromagnetic waves. Mathematically, it is based on the reconstruction of a function from its spherical means (integrals over spheres) whose centers lie outside the object of interest. As it will be shown, certain exact solutions of this problem can be derived from rather basic properties of the Helmholtz equation and eigenfunctions of the Dirichlet Laplacian. The talk will be accesible to students with working knowledge of the Green's formula for the Helmholtz equation.



• October 24, 2006: Ibrahim Fatkullin
  Large Drift Asymptotics for Stochastically Perturbed Gradient Flows
  Abstract: On the macroscopic level many physical phenomena are described by stochastically perturbed gradient flows. The gradient flow evolution corresponds to a steepest descent on a free energy landscape, stochastic perturbations account for the influence of thermal fluctuations. In many problems the macroscopic variables allow for an additional separation of scales which means that the macroscopic equations of motion may be simplified yet further. Mathematically this corresponds to a special kind of asymptotic limit when the free energy is extremely large except on a submanifold of the (macroscopic) phase space. Then in the limit the dynamics becomes constrained to this manifold. I will discuss how one can derive equations governing such asymptotic dynamics and will illustrate the methods on several examples.



• October 31, 2006: Ibrahim Fatkullin
  Large Drift Asymptotics for Stochastically Perturbed Gradient Flows, Part II
  Abstract: On the macroscopic level many physical phenomena are described by stochastically perturbed gradient flows. The gradient flow evolution corresponds to a steepest descent on a free energy landscape, stochastic perturbations account for the influence of thermal fluctuations. In many problems the macroscopic variables allow for an additional separation of scales which means that the macroscopic equations of motion may be simplified yet further. Mathematically this corresponds to a special kind of asymptotic limit when the free energy is extremely large except on a submanifold of the (macroscopic) phase space. Then in the limit the dynamics becomes constrained to this manifold. I will discuss how one can derive equations governing such asymptotic dynamics and will illustrate the methods on several examples.



• November 7, 2006: Misha Stepanov
  Large negative velocity gradients in Burgers turbulence
  Abstract: We consider one-dimensional Burgers equation driven by large-scale white-in-time random force. The tails of the velocity gradients probability distribution function are analyzed by saddle point approximation in the path integral describing the velocity statistics. The structure of the saddle-point (instanton), that is, the velocity field configuration realizing the maximum of probability, is studied numerically. The resulted shape of the tail is consistent with inviscid prediction and with short-time analytical instanton estimations.



• November 14, 2006: No Seminar



• November 21, 2006: Ken McLaughlin
  Some new universality results for random matrices: algebraic curves and dbar problems
  Abstract: I'll describe some recent developments in random matrix theory having to do with the "universality conjecture". Of course, I'll have to explain what all those buzzwords mean. In addition to some recent results, some open research directions will be described, that might appeal to graduate students.



• November 28, 2006: Charles Newman, Courant Institute, New York University
  Scaling Limit of Two-Dimensional Critical Percolation
  Abstract: We introduce and discuss the continuum nonsimple loop process (joint work with Federico Camia). This process, which describes the full scaling limit of 2D critical percolation, consists of countably many noncrossing nonsimple loops in the plane on all spatial scales; it is based on the Schramm Loewner Evolution (with parameter 6) and extends the work of Schramm and Smirnov on the percolation scaling limit. If time permits, we will introduce some ideas associated with the further extension to scaling limits of "near-critical" percolation (joint work with Camia and Renato Fontes).
  References: math.PR/0611116, 0604487, 0605035 (Comm. Math. Phys. 268 (2006) 1-38); cond-mat/0510740, 0604390, J. Stat. Phys. 116 (2004) 157-173.



• December 5, 2006: Nick Ercolani
  Self-Dual Test Functions for an Irrotational Ginzburg-Landau Problem with Twist
  Abstract: We will review some recent results on getting upper bounds on the energy of asymptotic minimizers for the Cross-Newell variational problem which models patterns and defect formation far from threshold. Beyond this we will explore the extent to which "self-dual" solutions capture the structure of minimizers for this class of variational problems. This is joint work with Shankar Venkataramani.

Spring 2006

• January 17, 2006: Organizational meeting



• January 24, 2006: Dorin Dumitrascu
  Some functional analysis aspects of first order differential operators
  Abstract: I shall discuss the "calculus" of first order differential operators. First the main functional analysis constructions and results will be presented: functional calculus, elliptic estimates and Sobolev spaces, Rellich lemma etc. This analysis will then be used to show how such operators generate cycles in the analytic K-homology groups, the dual theory of topological K-theory. Finally I shall indicate the product operations that can be done with these homology classes.
  The main example of 1st order elliptic operator on a spin manifold is the Dirac operator and it is a remarkable fact that often all the K-homology classes are generated from the Dirac operator.
  The talk will be accessible to graduate students.



• January 31, 2006: CANCELLED



• February 7, 2006: Marty Greenlee
  On Green's Theorem and Cauchy's Theorem
  Abstract: Green's Theorem is proved using only the geometric (or physical) definition of curl, without the use of partial derivatives. The curl free (conservative) case can then be used to prove Cauchy's Theorem - arguably the most fundamental theorem of complex analysis. This also leads to a geometric version of the Cauchy-Riemann equations which are both necessary and sufficient for holomorphy, and to a proof that a continuous two dimensional radial vector field is conservative even if nowhere pointwise differentiable (of course such a vector field has distribution derivatives of all orders).
  This talk, or at least most of it, should be accessible to students that have completed courses comparable to Math 424 and Math 425b.



• February 14, 2006: Shankar Venkataramani
  Wavy leaves, torn plastic, Hilbert's theorem and the Monge-Ampere equation
  Abstract: The title says it all.



• February 21, 2006: Recheduled on Friday, February 24th.



• February 24, 2006, at 1 pm in Math 402: Evans Harrell, Georgia Institute of Technology
  Note the unusual date and time
  Some isoperimetric problems arising in the physics of thin structures and in geometry
  Abstract: Some Schrödinger equations that arise in the physics of thin structures will be described. Then some sharp inequalities for eigenvalues will be presented, and in some circumstances these are shown to be saturated in the cases of circles or spheres. One result of this kind, obtained recently in joint work with Loss and Exner, follows from a new "isoperimetric" theorem of a classical type, viz., that for p &le 2, the L^p norm of the chord x(s) - x(s+a) between two points separated by arclength a on a closed curve of specified total length is maximized by the circle. This is false for large p.
  Some non-isoperimetric optimizers and some open questions will also be discussed.



• February 28, 2006: Nick Ercolani and Shankar Venkataramani
  Laplace Growth Dynamics
  Abstract: We will present an overview of Laplace Growth Dynamics and some of its connections to Integrable Systems Theory. Time permitting we will also discuss some potential extensions to related to problems in optimal disccretization.



• March 7, 2006: Annalisa Calini
  A continuous analogue of iteration of Schwarz reflection
  Abstract: Schwarz reflection takes a pair of (sufficiently close) regular analytic curves and produces a third such curve. Iteration of Schwarz reflection gives a discrete curve dynamics, invariant under conformal transformations. The continuous analogue of this discrete process leads to a partial differential equation easily reducible to a first order ODE.
  I will describe how the continous dynamics can be interpreted as the geodesic flow on the space L of unparametrized analytic curves, endowed with a formal symmetric space structure. I will introduce the Schwarz function of analytic arcs as a natural parameter on L, describe the connection between the continuous curve dynamics and the classical theory of stationary ideal fluid flows, and provide several concrete examples of solutions of the geodesics equation.
  This is joint work with Joel Langer, of Case Western Reserve University.



• March 21, 2006: Luis Garcia-Naranjo
  Integrability of Nonholonomic systems
  Abstract: Mechanical systems with constraints in the velocities are termed nonholonomic. These often arise in rolling systems (balls, disks, etc). The equations of motion for these systems fail to be Hamiltonian. They are Hamiltonian with respect to an "almost-Poisson" bracket that does not satisfy the Jacobi identity. Hence, for nonholonomic systems there is not a general notion of integrability as Liouville's theorem for classical Hamiltonian systems. There are, however, some solvable nonholonomic systems with a preserved measure for which the phase space is foliated by invariant tori, placing these systems together with integrable Hamiltonian systems. In this talk I will present a series of such examples and show that in some cases the equations of motion for such systems can be cast in Hamiltonian form with respect to a nonlinear Poisson bracket after a time change.



• March 28, 2006: Dorin Dumitrascu
  Some functional analysis aspects of first order differential operators, II
  Abstract: This is a continuation of my talk from the end of January. After reviewing the main points behind the construction of K-homology classes of elliptic differential operators, I shall discuss some examples of "products" between K-homology classes and between K-homology and K-theory. These are instances of the more general Kasparov product in KK-theory. A sketch of a proof for an Atiyah-Singer index-type theorem for twisted Dirac operators on even-dimensional hypersurfaces will end the presentation.



• April 4, 2006: Michel Destrade, CNRS / Université Pierre et Marie Curie, Paris
  Vibrations and traveling waves in second-grade solids
  Abstract: We consider special solids with a constitutive equation split into the sum of a purely elastic part and a viscoelastic part. We take the second part to coincide with the Cauchy stress of some dissipative second-grade (non-Newtonian) fluids. For the first part, we consider in turn nonlinear solids with a linear shear response and then with a nonlinear shear response. The implications of these choices are rich and varied. They include oscillatory creep and recovery evolutions, attenuated vibrations of a non-viscous solid, circularly-polarized finite amplitude waves, and for rectilinear shearing motions, solitary and compact travelling waves.
  This is joint work with Giuseppe Saccomandi, Universita di Lecce.



• April 11, 2006: Ibrahim Fatkullin
  Maximal entropy principle for systems of interacting rods
  Abstract: One of the formulations of the second law of thermodynamics postulates that entropy of a closed system always increases, i. e., it is maximal in the equilibrium state. Statements of such kind were made precise and proven for a class of interacting particle systems in the sixties and seventies by Dobrushin, Ruelle, and others. I will present an extension of their results to systems of particles that have orientational degrees of freedom and are suitable for the description of liquid crystals and short polymers.



• April 18, 2006: Lennie Friedlander
  Szego regularized determinants and zeta function regularized determinants
  Abstract: There are different ways of regularizing the determinant of an operator that does not belong to the trace class. Szego studied the operator of multiplying by a positive, periodic function. With respect to the trigonometric basis, one can represent the operator as an infinite matrix. The determinant of finite truncated matrices do not converge as the size grows, but one can extract the divergent term, and it is reasonable to treat the limit of what remains as a regularized determinant of the multiplication operator.
  This procedure can be generalized to some other operators (Guillemin - Okikiolu). On the other hand, there is another way of regularizing the determinant, via the zeta-function. These two methods do not give the same result. In our recent paper, V. Guillemin and I studied the differnce between the corresponding deteminants (more precisely, between logarithms of the determinants). It turns out that the difference is a local, explicitly computable quantity.



• April 25, 2006: Brenton Lemesurier
  Modeling thermalized nonlinear wave motion in large molecular systems: stochastic discrete nonlinear Schroedinger equations with heat loss terms
  Abstract: Wave motion in molecular systems such as long bio-molecules involves nonlinear effects that can stabilize traveling waves and produce energy localization. The small scales mean that thermal effects from Brownian motion in their aqueous medium are also important, along with a thermal balance through mechanisms for loss of that thermal energy.
  Modeling of heat input effects through stochastic discrete nonlinear Schroedinger equations without balancing heat loss mechanisms sometimes leads to far more disorder and disruption of wave phenomena than is observed in physical systems, so modeling heat loss is important. Unfortunately, this leads to highly nonlinear terms, which combined with the stochastic terms modeling heat input produce challenges to analysis and numerical simulation.
  In this talk, I propose to present an update on numerical methods, including some analysis of accuracy and validation of convergence, along with simulation results including a thermalized discrete 1D cubic Schroedinger equation, as a reduced model of DNA and protein molecules.



• May 2, 2006: Erin McNicholas
  Embedded Tree Structures
  Abstract: A map is an embedding of a connected, labeled graph into a compact oriented surface such that the edges of the graph do not intersect and the complement of the graph is a disjoint union of open cells homeomorphic to the disk. These cells are called the faces of the map. In the case of genus zero one-face maps, the embedded graph is a tree. There are various methods of representing a genus zero one-face map. Connections between these representations and the map's embedded tree will be explored, and combinatorial properties of the tree discussed. In particular, we will explore connections between the distribution of vertex degrees, vertex adjacencies, and the spectrum of the embedded tree.

Fall 2005

• August 23, 2005: Organizational meeting



• August 30, 2005: Momar Dieng
  Edge eigenvalues and applications
  Abstract: The past decade has seen exciting developments in random matrix theory, and the realization of many deep connections to other branches of mathematics, as well as outside mathematics. One of the milestones has been the discovery of explicit analytic expressions for the distribution of the largest eigenvalue in the Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic ensembles (denoted GOE, GUE, and GSE respectively) by Tracy and Widom. It is natural to ask about similar results for the m-th largest eigenvalue in general. This is relevant for theoretical reasons as well as for many applications (e. g. multivariate statistics). Tracy and Widom answered this question in the GUE case. They provide a recursive formula which involves a Fredholm determinant D₂(s,λ) whose m-th derivative with respect to λ is evaluated at λ = 1. I will outline the solution to the problem in the GOE and GSE cases. This work generalizes the GOE and GSE Tracy-Widom distributions. Together with the work of Johnstone and Soshnikov, the new formulas give the asymptotic behavior of the m-th largest eigenvalue of the appropriate Wishart distribution, which is of interest in multivariate statistics. They also provide, as a corollary, an alternate proof of an intriguing interlacing property between GOE and GSE eigenvalues in the edge-scaling limit. Finally, if time permits, I will mention another natural application in the area of random walks.



• September 6, 2005: Lennie Friedlander
  Extremal properties of eigenvalues for a metric graph
  Abstract: A metric graph is a graph viewed not as a combinatorial object but as a one-dimensional variety. One can study differential operators on metric graphs.
  I will talk about extremal properties of eigenvalues of the Laplacian.



• September 13, 2005: Karl Glasner
  Grain boundary network dynamics
  Abstract: This talk considers the singular limit of the regularized Cross-Newell phase diffusion equation (equivalently a model for epitaxial growth). Discontinuities in gradients concentrate on curves known as grain boundaries, which form networks that evolve in time. Matched asymptotic analysis is used to derive a free boundary problem for the grain boundary curve motion, which is driven by curvature and variations of line energy coupled along characteristics of the hyperbolic eikonal equation. An intermediate boundary layer near extrema junctions is found, leading to a unique nonlocal junction condition. The limiting dynamics is shown to be a gradient flow of the sharp interface energy on a attracting manifold. Dynamic scaling of the long-time coarsening process is explained by dimensional analysis of the reduced problem.



• September 20, 2005: Brenton Lemesurier
  The perils of continuum limit models for wave motion with spatial noise: vibrations in thin molecular films
  Abstract: Wave self-focusing in thin molecular films with noise and damping acting on internal modes can be modeled by a stochastic discrete nonlinear Schrodinger equation on a lattice of molecules. Continuum limits have been studied leading to modified nonlinear Schrodinger equations. This will be a short, informal talk on the mathematical difficulties of such continuum limits, including issues of possible ill-posedness and evidence for an "ultra-violet catastrophe". This suggests addressing the spatially discrete models directly. If time and interest allows, related issues of numerical methods will be discussed.



• September 27, 2005: Nick Ercolani
  Some Variational Insights into Pattern Formation Far from Threshold
  Abstract: I will be describing some recent work with Shankar Venkataramani which extrends variational models of the regularized Cross-Newell phase-diffusion equation to allow for defect twist. I will review pattern formation in Rayleigh-Bénard convection in order to provide a physical context for this analysis.



• October 4, 2005: Nick Ercolani
  Some Variational Insights into Pattern Formation Far from Threshold. Part II
  Abstract: I will be describing some recent work with Shankar Venkataramani which extrends variational models of the regularized Cross-Newell phase-diffusion equation to allow for defect twist. I will review pattern formation in Rayleigh-Bénard convection in order to provide a physical context for this analysis.



• October 11, 2005: Annalisa Calini
  Investigating knot types of algebro-geometric solutions of the Vortex Filament Equation
  Abstract: Several geometric and topological features of finite-gap solutions of the Vortex Filament Equation can be related to their algebro-geometric description. The extent to which topological information can be "read off" this description is addressed in ongoing work with Tom Ivey. I will present previous results, and then show how the theory of isoperiodic deformations (Krichever, Grinevich and Schmidt) can be used to generate examples of closed finite-gap solutions, formulate conjectures - and obtain some results - on their knot types.



• October 18, 2005: Lotfi Hermi
  On the Factorization Method and the Buslaev-Faddeev-Zakharov Trace Formulae
  Abstract: The Factorization Method (also called Addition or Commutation Method) provides an alternative and efficient route to the sum rules of Buslaev-Faddeev-Zakharov for the bound states of a Schrodinger operator. In this talk, I will describe the method of Benguria and Loss as well as Pavlovic's recent extension and draw connections between these sum rules and the conservation laws of the mKdV equation.
  This is joint work with J. Adrian Espinola-Rocha.



• October 25, 2005: Shankar Venkataramani
  Some Variational Insights into Pattern Formation far from Threshold - part III
  Abstract: I will continue Nick's discussion of a variational apporach to getting defects in the Cross-Newell equation. I will present some rigorous results for the scaling of the energy and the structure of the defects. I will also describe our numerical investigations.



• November 1, 2005: Adrian Espinola Rocha
  Gibbs Phenomenon in the Manakov System: Short-Time Asymptotics for Squared Barrier Potentials.
  Abstract: The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schrödinger and the Gross-Pitaevskii equations.
  Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Tranform (IST) can be exploited to obtain asymptotic information about solutions.
  This talk will briefly describe the IST for the Manakov system, and its asymptotic behavior at short times. We will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, we will show that the continuous spectrum gives the dominant contribution at short-times.



• November 8, 2005: Bernard Deconinck, University of Washington
  Computing Spectra of Linear Operators
  Abstract: Many problems in pure and applied mathematics may be reduced to that of determining the spectrum of a linear operator. This is the case for the linear stability analysis of equilibrium solutions of finite or infinite-dimensional evolution systems, and for the forward scattering problem associated with any integrable system. In this talk, I will show how a method which goes back to Hill (1886) may be used to compute spectra of linear operators with periodic coefficients. It may be extended to problems on an infinite domain. The method is algorithmic in nature and as such its only competitor are finite-difference methods. Hill's method converges exponentially, due to its spectral origins. It also incorporates Floquet theory, allowing for the determination of the entire spectrum, as opposed to isolated elements of it. I will illustrate the method using a variety of examples.



• November 15, 2005: Brenton Lemesurier
  Modeling and simulating self-focusing collapse of waves in thin molecular films: self-trapping in a discrete nonlinear Schroedinger equation with noise and damping in the phase
  Abstract: Wave self-focusing in thin molecular films with noise and damping acting on internal modes can be modeled by a discrete nonlinear Schroedinger equation modified with noise and damping in the phase.
  Previous models and their numerical solutions have lead to conjectures on the inhibition of self-focusing wave collapse by phase noise, and its restoration when phase damping is also added. In other words, questions of if and when wave energy gets localized on one or a few molecules, as in Discrete Self-Trapping models.
  This talk will introduce developments of numerical methods and results of simulations. These generally confirm focusing inhibition by noise and both smoothing and restoration of focusing by damping, and explore the effects of the spatial length scale of the noise.
  Continuum limits in the form of Stochastic Nonlinear Schroedinger Equations have also been introduced, but with problems such as ill-posedness and lack of sufficient smoothness in the ODE solutions: work in progress on possibly more relevant PDE models will be described.
  This is joint work with Barron Whitehead.



• November 22, 2005: Silvia Madrid
  Stability of travelling waves of a nonlinear Klein-Gordon system of equations by means of the Evans function
  Abstract: The Evans function can be used to determine spectral stability of one-dimensional localized solutions. In the first part of this talk, we will discuss how the Evans function of the linearized Klein-Gordon equations about a travelling wave is defined. Since in general, it is difficult to analyze the Evans function analytically, its numerical computation can be a valuable aid. In the second part of the talk, I will summarize different ways of evaluating the Evans function, including the method we proposed. This is joint work with S. Lafortune and J. Lega.



• November 29, 2005: Joe Watkins
  Microsatellite Mutation Models
  Abstract: The mutations one finds in the genome have been extensively employed as a tool for dating genealogical events. For these purposes, one scans for polymorphic pieces of DNA and then develops mathematical models for their evolution. Questions that can be addressed using only neutral mutations over non-recombining regions of the DNA simplify both the model building and the ensuing statistical analysis.
  For the dating of the more recent events, the choice is the relatively rapidly mutating microsatellites.
  After reviewing the classical methods for microsatellite mutation models, this talk moves on to recent biochemical analyses of mutational events and casts their evolution as a Markov process. We obtain exact results for the probability distributions for the change in microsatellite length over any given number of generations.
  If time permits, we will discuss some of the thermodynamic constraints of the models, and the use of these models in dating historical events. In particular, we will talk briefly about the project that provided impetus for this research, the peopling of Indonesia.

Spring 2005

• January 18, 2005: Organizational meeting



• January 25, 2005: Marty Greenlee
  Quadratic Interpolation and Convergence of Rayleigh-Ritz Eigenpairs for Self Adjoint Operators
  Abstract: Estimates of convergence rates for Rayleigh-Ritz eigenvectors in the literature are given in the energy norm, and the norm of the basic Hilbert space. Quadratic interpolation in the Rayleigh quotient leads to eigenvalue and eigenvector rate of convergence estimates in a scale of Hilbert spaces. Weaker norms lead to faster convergence rates, but for possible applications such as bifurcation calculations the appropriate norm is likely to be determined via Sobolev inequalities.
  In this first of two talks a spectral implementation for the Biharmonic Dirichlet problem in a rectangle will be presented. The presence of corners requires new theorems on interpolation spaces to obtain results as sharp as are obtainable for domains with smooth boundaries. We will then indicate how such interpolation theorems can be obtained between Sobolev spaces subject to homogeneous boundary conditions on other polygonal domains in the plane, for possible finite element implementation of the preceding theory.



• February 1, 2005: Marty Greenlee
  Quadratic Interpolation and Convergence of Rayleigh-Ritz Eigenpairs for Self Adjoint Operators. Part II
  Abstract: This continuation of last week's seminar will begin a brief amplification of the previous, in particular dependence on the data of constants in asymptotic formulae. Then characterization of interpolation spaces between Sobolev spaces subject to homogeneous boundary conditions on plane polygonal domains will be sketched. Finally, possible application of the preceding to certain methods for finding bounds complementary to the Rayleigh-Ritz bounds, thus getting at numerical error estimation, will be discussed as time permits.



• February 8, 2005: Jon Wilkening, Courant Institute
  Stress-driven grain boundary diffusion: modeling, analysis and numerical methods
  Abstract: Microchips often fail when the metallic interconnects between transistors and diodes on the chip degrade due to extremely high current densities. The physics of this process is quite interesting; it is a non-local moving interface problem involving elastic deformation and diffusion. Stress singularities can develop which make boundary conditions difficult to understand and numerical simulation difficult to implement reliably.
  After describing the model, I will outline our recent proof of well-posedness, which uses techniques from semigroup theory and requires an analysis of a type of Dirichlet to Neumann map involving the equations of elasticity. I will also briefly describe my recent work on computing stable asymptotics for singularities of Agmon-Douglis-Nirenberg elliptic systems near corners and interface junctions, and show how to adjoin these singular functions to the finite element basis to accurately and efficiently resolve stress singularities without mesh refinement. If time permits, I will also show that my least squares finite element formulation for elasticity transitions gracefully to the Stokes equations in the incompressible limit, and show how to incorporate the convection term to obtain an efficient Navier-Stokes solver for low to moderate Reynolds numbers.



• February 15, 2005: Yves Pomeau
  Poincaré as applied mathematician
  Abstract: At the end of his rather short life (he died in 1912, born in 1845) Poincaré got interested in a mathematical question related to the recent (1901) realization by Marconi of transmission of an electromagnetic signal across Northern Atlantic. The geometry was such that the received signal (at least this is what was believed - wrongly) was due to diffraction effects in the shadow. Poincaré was first (1909) to show that, in the short wave limit, the amplitude of the signal should be proportional to exp(-(k r)^1/3 q), k wavenumber, r radius of the Earth, q angular distance on the Earth between the edge of the shadow and the receiving antenna. I'll show the main ideas of the rather involved calculation by Poincaré, and a much more direct proof of the same result due to Fock (1945). Finally I'll explain what was wrong in the physical assumptions of the model used by Poincaré.



• February 22, 2005: Jean-Marc Fellous, Department of Biomedical Engineering, Duke University
  In search of the neural code: discovering spike patterns in neuronal responses
  Abstract: When a cortical neuron is repeatedly injected with the same fluctuating current stimulus - frozen noise - the timing of the spikes is highly precise from trial to trial and the spike pattern appears to be unique. We show here that the same repeated stimulus can produce more than one reliable temporal pattern of spikes. A new method is introduced to find these patterns in raw multi-trial data and is tested on surrogate datasets. Using it, multiple coexisting spike patterns were discovered in pyramidal cells recorded from rat prefrontal cortex in vitro, in data obtained in vivo from area MT of the monkey (Buracas et al., 1998) and from the cat lateral geniculate nucleus (Reinagel and Reid, 2002). The spike patterns lasted from a few tens of milliseconds in vitro to several seconds in vivo. We conclude that the pre-stimulus history of a neuron may influence the precise timing of the spikes in response to a stimulus over a wide range of time scales.



• March 22, 2005: Thomas Kriecherbauer, Department of Mathematics, Ruhr-Universität Bochum
  Universality for invariant random matrix ensembles
  Abstract: We will review some recent results on the universality conjecture in Random matrix theory, where we will focus on invariant ensembles which appear in mesoscopic physics. In this case the universal behavior of the statistics of the locally rescaled eigenvalues can be established by a detailed asymptotic analysis of associated orthogonal polynomials which are of Laguerre type.



• March 29, 2005: Ken McLaughlin
  Random tilings and random matrices
  Abstract: I will describe some random tiling problems, and some interesting phenomena arising in a scaling limit, involving a connection to eigenvalues of random matrices.



• April 5, 2005: Erin McNicholas
  Eigenvalue statistics of random one-face maps
  Abstract: One-face maps can be uniquely represented by a special class of 3-regular graphs. Using this representation, one can examine the spectral density and scaled eigenvalue spacing distributions for random one-face maps. Numerical experiments have revealed interesting connections between the underlying geometry of the map and universal distributions from Random Matrix Theory. In this talk the graphical representation of one-face maps will be explained, and recent numerical results discussed.



• April 12, 2005: Lay May Yeap
  The elliptical instability and normal forms
  Abstract: The elliptical instability refers to the mechanism by which a planar rotating flow with elliptical streamline becomes three dimensional. We consider an incompressible, inviscid fluid rotating in an ellipsoidal domain. Linear stability for the near-axisymmetric case is well understood. I shall talk about our recent work on linear stability not restricted to near axisymmetric case. We discover an asymptotic regime that relates instabilities in long channel shear flows to the problem of elliptical instability. I will also discuss our weakly non-linear analysis for the near-axisymmetric case using co-dimension two normal forms. The analysis confirms the results of Guckenheimer and Mahalov, among others. Normal forms are simplified dynamical equations that are characterized by the homological equations. I will try to address the issue of finding all polynomial solutions to homological equations using classical invariant theory and Groebner basis methods.



• April 19, 2005: Math Awareness Week: Romeel Davé, Department of Astronomy
  Simulations of Galaxy Formation
  Abstract: As Hubble and other Great Observatories peer further back into the origins of our universe, it seems we find ourselves commensurately further away from a full understanding for the origin and evolution of galaxies. The largest galaxies that we can see to the furthest distances are particularly puzzling, showing abundances and colors that present challenges for current theories of galaxy formation. I will describe recent progress from cosmological hydrodynamic simulations towards unraveling some of these puzzles. While simulations suggest a physical mechanism for the "downsizing" of galaxies, i.e., the reduction in the characteristic mass of actively star forming objects, this mechanism cannot yet explain the strength of downsizing seen in the real universe. I will then go on to describe work towards understanding the progenitors of massive galaxies as observed at z~4 by the GOODS legacy project, and present preliminary comparisons and predictions that will help us constrain the early evolution of massive systems.



• April 26, 2005: Yves Pomeau
  Eigenvalues of the Laplacian to the power n as n goes to infinity
  



• May 3, 2005: Nick Ercolani
  The Dynamics of Modulated Wavetrains
  Abstract: We present an overview of a recent paper by Doelman, Sandstede, Scheel and Schneider which rigorously establishes the validity of modulation equations for slowly varying nonlinear wavetrain solutions in a class nonlinear partial differential equations on the real line. This talk will attempt to explain the issues through fundamental examples such as approximation of weakly modulated wavetrains via Burgers equation in reaction-diffusion systems or the complex Ginzburg-Landau equation.

Fall 2004

• September 7, 2004: Shankar Venkataramani
  Nonconvex variational problems and pattern formation
  Abstract: Nonconvexity is a robust mechanism for generating patterns in equilibrium (minimum energy) states. I will give a gentle introduction to nonconvex variational problems, their occurence in applied problems and some of the analytical tools that are used to study them. Using a few motivating examples, I will then formulate a set of conjectures that attempt to relate "hard" analytical questions about the structures/patterns in equilibrium with easier geometric questions.



• September 14, 2004: Shankar Venkataramani
  Patterns in thin elastic sheets: Geometry and Analysis
  Abstract: I will discuss two problems. A geometric problem about isometric immersions of two manifolds in R³, and a variational problem for the Foppl-von Karman energy for thin sheets. I will then relate thes results of our analysis to a variety of structures that are observed in thin sheets including the crumpling of paper and buckling on multiple scales in a variety of systems.



• September 21, 2004: Shankar Venkataramani
  Patterns in thin elastic sheets: Geometry and Analysis II
  Abstract: I will discuss two problems. A geometric problem about isometric immersions of two manifolds in R³, and a variational problem for the Foppl-von Karman energy for thin sheets. I will then relate thes results of our analysis to a variety of structures that are observed in thin sheets including the crumpling of paper and buckling on multiple scales in a variety of systems.



• September 28, 2004: Ken McLaughlin
  Approximation theory & analysis of integrable systems I: the zeros of exp(z)
  Abstract: It is a well-known fact that on compact sets, the Taylor series associated to an analytic function converges. Perhaps the best known example is exp(z) = 1 + z + (1/2)z²+ ... If you truncate the series, the nth Taylor polynomial has n zeros. How do the zeros fly off to infinity as n grows? Is there a general theory for similar global questions regarding the asymptotic behavior of Taylor polynomials? This will serve as an introduction to some aspects of approximation theory and also Riemann-Hilbert problems.



• October 5, 2004: Nick Ercolani
  Continuum limits of Toda lattices associated to a random matrix partition function
  Abstract: We have been considering the large N expansion of a random matrix partition function which describes the expectations of matrix observables with respect to a conjugation-invariant measure on the given ensemble of matrices. Here the ensemble is the space of N x N Hermitian matrices. The large N expansion gives asymptotic information about the statistics of invariant qualities, such as eigenvalues, as the size of the matrices becomes large. This has many interesting applications to statistical physics, combinatorics and enumerative geometry. This partition function has a natural interpretation in terms of a general solution to a commuting family of integrable ODEs known as the Toda Lattice hierarchy.
  This talk will principally concern a scaling limit associated to the large N expansion which may be understood in terms of a PDE continuum limit of the associated Toda lattice hierarchy. We will show how this continuum limit can provide detailed information concerning the fine structure of the large N expansion of the original partition function. This "fine structure" amounts to detailed information about graphical enumeration on Riemann surfaces of arbitrary finite genus. This is joint work with Ken Mclaughlin and Virgil Pierce.



• October 12, 2004: Maurice Hasson
  Fast growing polynomials and best approximation in the complex plane
  Abstract: The polynomials zⁿ satisfy the following extremal property: Among all polynomials of degree n whose absolute value is bounded by 1 on the unit disk, zⁿ are the fastest growing polynomials in the complex plane, in a precise sense. We will relate this property to the fact that the partial sums of the Taylor expansion of entire function f(z) are the (near) best approximation to f(z) on the unit disk. We will then see how to modify zⁿ in order to obtain good approximation properties on an arbitrary domain of the complex plane.



• October 19, 2004: Ken McLaughlin
  Approximation theory & integrable systems: Analysis of the Taylor approximation of an entire function via Riemann-Hilbert problems
  Abstract: We will give an introduction to the analysis of Riemann-Hilbert problems by describing the behavior of the nth Taylor approximant to an entire function (such as exp(z)) on domains which grow with n. A by-product will be a description of the behavior of the zeros of the Taylor approximants.



• October 26, 2004: Maurice Hasson
  For Growing Polynomials, Best Approximation in the Complex Plane and Matrix Preconditioning
  Abstract: We will review the theory of fast growing polynomials and show how to construct them numerically using the (exterior) conformal mapping. We will expand analytic functions in series of fast growing polynomials and show how to use these expansions to construct the (near) best uniform approximation of these analytic functions on a given curve in the complex plane. We will then concentrate on the function f(z) = 1z and show how the near best approximation is used for the purpose of matrix preconditioning. Numerical experiments will then be presented.



• November 2, 2004: Vladimir Zakharov
  What mathematics is behind the freak wave?
  Abstract: The phenomenon of freak waves is a well-documented hazard to mariners that is responsible for ships as well as human lives losses. We try to explain this phenomenon in terms of soliton and wave collapses. It is known that the Stokes wave of finite amplitude is unstable. In 2-D case this instability is a modulational instability, which leads to excitation of long-scale modulation. Our numerical experiments show that after some period of time, the nonlinear development of modulational instability generates a very steep breaking wave. On our opinion, this process is a pretty complicated one, consisting of several stages. On the first stage, the Stokes wave decomposes to a system of "quasisolitons". Later, quasisolitons of finite amplitude could merge and form a quasisoliton of essentially high amplitude. This quasisoliton is unstable; its instability ends up with the formation of breaking freak wave. Different mathematical models, all of them being generalizations of the focusing Nonlinear Schrodinger equation, are discussed.



• November 9, 2004: Rosangela Sviercoski
  Multiscale analytical solutions for equations in divergence form and an introduction to homogenization theory
  Abstract: In this talk, we will present analytical solutions to equations with variable coefficient including the generalized Laplace's Equation in n-D. We will discuss the functional spaces for such solutions, the conditions on the coefficient to ensure their uniqueness and, explicitly describe the suitable boundary functions g(x), including the periodic case. It is always desirable that we obtain analytical solutions for a given governing equation, but in many cases, this is not still possible. However, if one is interested in analyzing the system from a macrostructure point of view, it is desirable to simplify it in such a way that the phenomena of interest will remain adequately described, while finer details which are not of interest can be disregarded. Simplified equations are called homogenized equations, and the procedure of replacing the original equation is called homogenization. We will introduce the setting of the homogenization theory and apply it to 1-D steady state diffusion equation. We show how this relates to the known results in the literature. In the next talk, we will extend the homogenization theory to n-dimensional problems including the nonlinear case.



• November 16, 2004: Joceline Lega
  New results on the stability of pulse-like solutions to a coupled nonlinear Klein-Gordon system
  Abstract: When subject to sufficient twist, an elastic filament kept under tension typically undergoes a writhing bifurcation. Near threshold, the dynamics of the filament may be modeled by two coupled nonlinear Klein-Gordon equations, which are envelope equations for the amplitudes of the local deformations and twist. I will consider the question of the spectral stability of a two-parameter family of pulse-like solutions of these envelope equations. More precisely, I will explain how to obtain a criterion on the speed of propagation of the pulses, which is a necessary and sufficient condition for their spectral stability. This will involve Evans function techniques as well as Hamiltonian methods. I will also discuss the numerical evaluation of the Evans function.
  This work is joint with Stephane Lafortune and Silvia Madrid-Jaramillo.



• November 23, 2004: Lotfi Hermi
  Improving Some Weyl-Type Lower Bounds for Sums of Eigenvalues of the Clamped Plate Problem
  Abstract: Weyl-type bounds for the eigenvalues of the clamped plate problem give lower estimates for the sum of eigenvalues of the biplacian in terms of the volume of the underlying region. In this work, we report improvements on these bounds with correction terms involving the "moment of inertia" of the domain under consideration. Similar results are proven for higher order operators.



• November 30, 2004: Joanna Monti-Masel
  Prion replication kinetics and the implications for therapeutics
  Abstract: The infectious agent in mad cow disease contains no nucleic acid. The mechanism of prion or protein-only replication is highly controversial, but appears to involve aggregation and misfolding of the PrP protein. I present a novel kinetic model that combines the kinetics of PrP polymer elongation and polymer breakage in a mathematically tractable way. Each size class of PrP polymer is described by a differential equation, leading to an infinitely large equation set. The kinetics can be simplified through a fortuitous moment closure to a closed system of 3 coupled differential equations. The model is in agreement with a wide range of in vivo data, and the parameters of the model can be inferred from the data. A range of otherwise puzzling patterns in the data can be explained with reference to the model. Analysis of the replication mechanism has profound implications for drug design, suggesting that drugs which cap growing polymer ends are the most likely to be effective at low drug doses, while drugs that destroy prions by breaking them up could accelerate diseases if given at too low a dose.