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Alfonso Castro, Harvey Mudd College, USA
Superlinear boundary problems
Recent developments and open questions on superlinear boundary values problems will be presented. Emphasis will be placed on cases where critical point theory is applicable.

Shui-Nee Chow, Georgia Tech, USA
Spike solutions for a singularly perturbed differential equation modeling an electrical circuit
We consider a singularly perturbed system of differential equations with periodic forcing which is derived from an electrical circuit used in modulation and demodulation schemes. The system presents the spiking phenomena over a one time period that has important application in signal processing and in the technology in communication. In this talk we are interested in the number of cycles a solution completes in one time period (which is precisely the number of spikes) and the stability of a spiking solution. We will show rigorous mathematical analysis can be developed to enable us to give a complete identification of subregions Vn, n=1,2,..., in the parameter space so that in each Vn the system produces stable spiking solutions with precisely n spikes.

Djairo De Figueiredo, Universidade Estadual de Campinas, Brazil
An Orlicz approach to elliptic systems
The lecture will concern the existence of positive solutions for systems of coupled semilinear elliptic systems. 
Luca Dieci, Georgia Tech, USA
Approximation of Lyapunov and Dichotomy Spectra
In this talk we discuss the numerical approximation of Lyapunov and Dichotomy spectra for continuous dynamical systems. Our purpose is three-fold:
(1) To discuss how and when the so--called QR and SVD techniques can be be used to approximate the spectra;
(2) To highlight the key features of a suite of codes which we developed for approximating the spectra;
(3) To give illustrative numerical results, also for ODEs resulting from spectral approximation of dissipative PDEs.
Work done in collaboration with Cinzia Elia (Univ. of Bari, Italy), Mike Jolly (Indiana Univ.), and Erik Van Vleck (Univ. of Kansas).

Patricio Felmer, University of Chile, Chile
Critical Exponent and other related Questions for the Pucci's Extremal Operator
In this talk we present some results on the existence of a critical exponent for the Pucci's extremal operator in the radial case. We present some existence results for 'semilinear' equations as well as for the eigenvalue problem.

Irene Fonseca, Carnegie Mellon University, USA
Variational Methods in the Study of Imaging, Foams, Quantum Dots ... and more
Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order variational problems and to models involving lower order density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics and thin films will be addressed.

Nassif Ghoussoub, PIMS, Canada
Anti-selfdual Lagrangians and variational resolutions of non self-adjoint equations and dissipative evolutions
We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seem inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are still minima of action functionals. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functionals, but because they are also zeroes of the Lagrangian itself. The approach has many advantages: It solves variationally many equations and systems that cannot be obtained as Euler-Lagrange equations of action functionals, since they can involve non self-adjoint or other non-potential operators; It also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. Most remarkable are the permanence properties that ASD Lagrangians possess making them more pervasive than expected and quite easy to construct.

Marty Golubitsky, University of Houston, USA
Coupled Systems: Theory and Examples
A coupled cell system is a collection of individual, but interacting, dynamical systems. Coupled cell models assume that the output from each cell is important - not just the dynamics considered as a whole. In these systems the signals from two or more cells can be compared and patterns of activity can emerge. We ask when can the cell dynamics in a subset of cells be identical (synchrony) or differ by a phase shift.
In particular: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much is related to the specific dynamics of cells and the way they are coupled?
We illustrate the ideas through a series of examples and discuss three theorems. The first theorem classifies spatio-temporal symmetries of periodic solutions; the second gives necessary and sufficient conditions for synchrony in terms of network architecture and its symmetry groupoid; and the third shows that synchronous dynamics may itself be viewed as a coupled cell system through a quotient construction. This research is joint with Ian Stewart.

Celso Grebogi, Universidade de Sao Paulo, Brazil
Obstruction to modeling and shadowing
Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one fundamental goal is to develop models whose solutions accurately reflect the nature of the physical process. A dynamical model uses simplifying assumptions and approximations in the hope of capturing the essential characteristics of how a physical system evolves with time. The question of whether a model accurately reflects nature is one constantly faced by scientists. Recently we have discovered that there exists a new level of mathematical difficulty, brought from the theory of dynamics systems, that can limit our ability to represent nature using deterministic models. Specifically, we have discovered that certain chaotic systems found in nature cannot be modeled, particularly higher-dimensional chaotic systems. No model of such a system produces solutions that are realized by nature. (Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.)

James M. Greenberg, U.S. Office of Naval Research-Global, European Office of Aerospace Research and Development, and Carnegie Mellon University, USA
Congestion Redux
In this talk I’ll focus on a class of 2nd order traffic models and show that these models support stable oscillatory traveling waves typical of the waves observed on a congested roadway. The basic model has trivial or constant solutions where cars are uniformly spaced and travel at a constant equilibrium velocity that is determined by the car spacing. The stable traveling waves arise because there is an interval of car spacing for which the constant solutions are unstable. These waves consist of a smooth part where both the velocity and spacing between successive cars are increasing functions of a Lagrange mass index. These smooth portions are separated by shock waves that travel at computable negative velocity.

Barbara Keyfitz, University of Houston, USA
Self-similar solutions of two-dimensional conservation laws
This is a report of research carried out jointly with Suncica Canic, Eun Heui Kim and Gary Lieberman. We describe how self-similar reduction of two-dimensional conservation laws leads to boundary value problems for equations which change type, and outline what is known at this point about solving these problems. In particular, we have established a method for solving free boundary problems for degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution, and we summarize the principal features of this method. Some interesting features of our research include the appearance of weak shocks (continuous but non-Lipschitz compression waves) in equations similar to the gas dynamics equations; bifurcation criteria given by a priori estimates; and a conjectured scenario for resolution of the triple-point paradox.

David Kinderlehrer, Carnegie Mellon University, USA
Remarks about diffusion mediated transport
Diffusion mediated transport is implicated in the operation of many molecular level systems. These include some liquid crystal and lipid bilayer systems, and, especially, the motor proteins responsible for eukaryotic cellular traffic. All of these systems are extremely complex and involve subtle interactions on varying scales. The chemical/mechanical transduction in motor proteins is, by contrast to many materials microstructure situations, quite distant from equilibrium. These systems function in a dynamically metastable range .
Our plan is to explain the relationship of the Monge-Kantorovich mass transfer problem to models for conventional kinesin type motors and their relations. These concepts permit us to establish consistent thermodynamical dissipation principles from which evolutionary equations follow. What properties are necessary for transport? What is the role of diffusion? What is the role of other elements of the system and how can dissipation be exploited to understand this? How successful are we? The opportunity to discover the interplay between chemistry and mechanics could not offer a more exciting venue.
We are reporting here on joint work with Michel Chipot, Jean Dolbeault, Stuart Hastings, and Michal Kowalczyk which has also attracted the interest of Bryce McLeod and Xinfu Chen.

Yanyan Li, Rutgers University, USA
On the Yamabe problem and a fully nonlinear version of it
We present some joint work with Lei Zhang on the compactness of solutions to the Yamabe problem. We also present some joint work with Aobing Li on the existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem as well as some Liouville type theorems on the associated conformally invariant fully nonlinear equations.

Stig-Olof Londen, Helsinki University of Technology, Finland
Quasilinear Evolutionary Equations And Continuous Interpolation Spaces
We analyze the abstract parabolic quasilinear evolutionary equation $$ D_t^{\alpha}(u_t-z)+A(u)u=f(u)+h(t); \ u(0)=x, \ u_t(0)=z, $$ in continuous interpolation spaces allowing a singularity as $t\down 0$. Here $D_t^{\alpha}$ denotes the time-derivative of order $\alpha \in (0,1)$. First, fractional derivatives in the space $L^p((0,T);X)$ are briefly considered; next these derivatives are examined in spaces of continuous functions having (at most) a prescribed singularity as $t\down 0$. We study the related trace spaces and by using maximal regularity results for the linearized equation we formulate existence and uniqueness results for the quasilinear equation.

Yuan Lou, Ohio State University, USA
Convergence in competition models with small diffusion coefficients
It is well known that for reaction-diffusion 2-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the reaction system implies global stability for the reaction-diffusion system. This is not in general true for spatially inhomogeneous models. We show here that for an important range of such models, for small enough diffusion coefficients, global convergence to an equilibrium holds for the reaction-diffusion system, if for each point in space the reaction system has a globally attracting hyperbolic equilibrium. This work is planned as an initial step toward understanding the connection between the asymptotics of reaction-diffusion systems with small diffusion coefficients and that of the corresponding reaction systems. This is a joint work with V. Hutson and K. Mischaikow.

Juan Manfredi, University of Pittsburg, USA
Convexity from the PDE point of view
Convex functions in Euclidean space play an important role in the regularity theory of non-linear elliptic partial differential equations. They can be characterized as universal subsolutions of homogeneous fully nonlinear second order elliptic partial differential equations. In the first part of the talk, we will first show that this PDE definition is equivalent to the usual one. Then we will use well-known estimates for subsolutions of familiar PDEs to derive estimates for convex functions.
Another advantage of the PDE definition of convexity is that it can be considered in the case of Carnot groups. In the second part of the talk we will present the theory of convex functions on Carnot groups. Our approach is based on the viscosity theory of subsolutions for subelliptic equations and the geometric role played by -harmonic functions.
[JLMS] P. Juutinen, G. Lu, J. Manfredi and B. Stroffolini, Convex functions on Carnot groups, submitted for publication.
[LMS] G. Lu, J. Manfredi and B. Stroffolini, Convex functions on the Heisenberg group, Cal. Var. and PDE 19 (2004), pp. 1--22.

Hildebrando Munhoz Rodrigues, Universidade de Sao Paulo, Brazil
Smooth linearization in infinite dimensional Banach spaces
This is a joint work with J. Solá-Morales, from Universitat Politécnica de Cataluña, Spain. In this talk we will discuss some new results on C1-linearization for contraction diffeomorphisms, near a fixed point, valid in infinite dimensional Banach spaces. We also will present a new result for a saddle case. As an intermediate step, we prove specific results of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorems. Our results essentially generalize some classical results by P. Hartman and G. R. Belitskii in infinite dimensions, and a result of X. Mora-J. Sola-Morales in the infinite dimensional case. It is shown that the result can be applied to some abstract systems of semilinear wave equations.

George Sell, University of Minnesota, USA
Global climate modeling and longtime dynamics
This talk is based on joint work with Victor Pliss.

Michael Sullivan, Southern Illinois University, USA
Weighted Flow Equivalence of Shifts of Finite Type
Let G be a finite group. We classify G-equivariant flow equivalence of nontrivial irreducible shifts of finite type in terms of (i) elementary equivalence of matrices over ZG and (ii) the conjugacy class in ZG of the group of G-weights of cycles based at a fixed vertex. In the case G = Z/2, we have the classification for ''twistwise flow equivalence''. We include some algebraic results and examples related to the determination of E(ZG) equivalence, which involves K_1(ZG). This research is joint with Mike Boyle.

Gunther Uhlmann, University of Washington, USA
Travel Time Tomography and the Dirichlet-to-Neumann Map
In inverse boundary problems one attempts to determine properties of a medium by making measurements at the boundary of the medium. In the lecture we will concentrate on two inverse boundary problems, Electrical Impedance Tomography and Travel Tomography, which arise in medical imaging, geophysics and other fields. We will also discuss a surprising connection between these two inverse problems.
Travel Time Tomography, consists in determining the index of refraction or sound speed of a medium by measuring the travel times of waves going through the medium. In differential geometry this is known as the the boundary rigidity problem. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points of the boundary of a compact Riemannian manifold with boundary. The inverse boundary problem consists in determining the Riemannian metric from the boundary distance function.
Calderón's inverse boundary value problem consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. This inverse problem is also called Electrical Impedance Tomography (EIT). The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map.
A connection between these two inverse problems has led to a solution of the boundary rigidity problem in two dimensions for simple Riemannian manifolds. We will describe this solution in the lecture.

Jianhong Wu, York University, Canada
Delay differential equations arising from dynamic memory and pattern recognition of neural networks
We describe a few classes of delay differential equations arising from dynamic memory storage and retrieval, and from clustering data sets in high dimensional spaces. We illustrate the connection between the abstract formulation of concepts in nonlinear dynamical systems (attractors, invariant manifolds and bifurcations, for example) and cognitive tasks such as memory storage, learning and pattern recognition in neural networks, and show how delays are adapted to increase the networks's capacity for memory storage and to improve the network's efficiency for pattern recognition.

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