Trujillo's main square and Huanchaco Beach.

 

The Americas Conference on Differential Equations is a biennial meeting which started in 1994, and it gathers leading experts in the areas of differential equations, dynamical systems and nonlinear analysis from the American continent, as well as special guests from Europe, to exchange ideas and results on cutting-edge research in these very active areas of science. The conference also helps initiating and enhancing ongoing research collaborations, opportunities for new collaborations, and opportunities for graduate studies and postdoctoral positions. Previous editions of this conference have been held in Mexico, Brazil, USA, Venezuela, Canada, Chile, and Colombia. The X Americas Conference is scheduled for August 2014 in Buenos Aires, Argentina. The impact of this Americas Conference is very broad, especially because new generations of young students and faculty are being recruited from all corners of the continent to join the very solid group of mathematicians formed by the top researchers who started this international event.

The IX Americas Conference on Differential Equations hosted at the Universidad Nacional de Trujillo, offered four mini-courses, and several outstanding plenary, invited and contributed talks, as well as several social activities. We had hundreds of participants from USA, Canada, Spain, France, and from several countries in Latin America; in particular, we had several faculty and students coming from Argentina, Brazil, Chile, Colombia, Mexico, Venezuela, as well as participants from 15 Peruvian universities from across the country. Everyone enjoyed Trujillo for the very warm hospitality of its people, great weather, beautiful beaches, and the spectacular Peruvian food.

            Left: the opening day of the conference. Right: traditional Peruvian dance.

 

The following mini-courses were organised:

Ekeland’s Variational Principle and Analysis in Frechet Spaces

Ivar Ekeland, University of British Columbia

Among other things, local surjection theorems and implicit function theorems in Banach spaces and Frechet spaces were presented. In contrast to the classical Nash-Moser approach, Ekeland did not use the Newton procedure, but his own Ekeland's variational principle. As a result, the function to be inverted is not required to be C2, or even C1, or even Frechet-differentiable.

 

Nonnegative Solutions of Elliptic and Parabolic Equations: Symmetry, the Nodal Structure, and Large-Time Behavior

Peter Polacik, University of Minnesota

Nonlinear elliptic and parabolic equations on symmetric domains were discussed. First, symmetry properties of nonnegative solutions were discussed. For solutions which are not strictly positive, the talk also focused on the structure of their nodal sets. It was also shown how the symmetry is used in some results on the asymptotic behavior of nonnegative solutions of parabolic equations.

 

Structural Stability and Bifurcations in the Differential Equations of Classical Geometry

Ronaldo Garcia & Jorge Sotomayor, Universidade F. de Goias,  & Universidade Sao Paulo

The topics discussed include: Generalities about  ODEs, differential  geometry, and  historical perspectives; Lines of principal curvature on surfaces immersed in Euclidean 3-D space; Lines of axial curvature on  surfaces immersed in Euclidean 4-D space; Hypersurfaces immersed in Euclidean 4-D space, and several open problems.

 

Partial Differential Equations: Control and Numerics

Enrique Zuazua, Basque Center for Applied Mathematics

Topics discussed included modeling, analysis, numerical simulation and control of PDEs arising in various contexts of science and technology. In a problem of controllability, it was analyzed whether by means of a suitable (and feasible!) controller the solution can be driven to a desired final configuration. Clasically, the dual notion of controllability is the so called observability problem. This problem is relevant for control purposes but also in other contexts like inverse problems and identification issues. There are different degrees of observability and this was particularly important in the genuinely infinite-dimensional problems considered.  A review of some of the most relevant applications to science, engineering and technology in which these problems arise was given. Then followed a discussion of the basic theory for the wave and heat equation to later address some important coupled models, as the system of thermoelasticity. These problems were also analyzed for networks of flexible strings and beams and, in particular, how the geometry of the network and the mutual lengths of the various strings/beams entering on it may influence the properties of the system in what concerns control and observation. The problem of  the numerical approximation of   control problems was also discussed. It was pointed out that when controlling a finite-dimensional approximation of  the continuous model, one does not actually compute an approximation of the control  one is looking for. The possible remedies to these pathologies are: space discretization, numerical damping, filtering of high frequencies, multi-grid algorithms,  etc.  The latter fact is of great impact form a modeling point view. Indeed, numerical approximation schemes may also be used as discrete models. The analysis showed that these two modeling approaches yield different results from a control theoretical point of view and this should be taken into account. Finally, the switching strategy, which provides a natural and efficient wat of adapting classical splitting strategies for the control of multiphysics systems was introduced.  A list of open problems and directions of possible future research was also given.

Some of the speakers at the conference.

 

The slides of the following plenary presentations can be found on the conference website:

1. Peter Bates, Michigan State University

Persistence of Normally Hyperbolic Invariant Manifolds Under Stochastic Perturbation.

2. Alexandre Carvalho, Universidade Sao Paulo

Nonautomous Dynamics of PDEs: Gradient Evolution Processes

3. Shui-Nee Chow, Georgia Tech

Fokker-Planck Equations on a Graph with Finite Vertices.

4. Charles Doering, Universtiy of Michigan

Ultimate State of 2-D Rayleigh-Benard Convection Between Free-Slip Fixed Temperature Boundaries.

5. Duvan Henao, Pontificia Universidad Catolica de Chile

Variational Models for Cavitation in Solids.

6. John Mallet-Paret, Brown University

Dynamics of Functional Differential Equations.

7. Konstantin Mischaikow, Rutgers University

Rigorous Computation of Dynamics in Infinite Dimensions.

8. Panayotis Panayotaros, I.I.M.A.S., Universidad Nacional Autonoma de Mexico.

Nonlinear Waves in Lattices.

9. Nicolas Saintier, Universidad de Buenos Aires

Geometric Elliptic Problems.

10. Carlos Tomei, Pontificia Universidad Catolica do Rio de Janeiro

Numerical Aspects of Nonlinear Elliptic Equations with Finite Spectral Interaction.

11. Yingfei Yi, Georgia Tech

Noise Perturbation in Finite Dimension.

There were also invited talks by Antonio Capella, Jorge Cossio, Isabel Flores, Luz de Teresa, Hugo Leiva, Jorge Rebaza, Raul Manasevich, Nelson Merentes, Ernesto Perez Chavela, Ramon Plaza, Gustavo Ponce, Hildebrando Rodriguez-Munoz, Carlos Alberto Raposo de Cunha, Noemi Wolanski, Maria Schonbek, Ronaldo Alves-Garcia, Julio C. Ruiz Claeyssen, Jose Raul Quintero, Pedro Isaza, Jose Arzola, Fabian Flores Bazan, Marian Gidea and Roxana Lopez Cruz.