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Analysis, Geometry, and Dynamical Systems Seminar   RSS

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14/12/2012, 11:00 — 12:00 — Room P3.10, Mathematics Building
Rafael de la Llave, Georgia Tech.

Manifolds on the verge of a regularity breakdown

There are two main stabibility arguments for solutions in dynamical systems: the theory of normal hyperbolicity and the Kolmogorov Arnold Moser theory of perturbations. In recent times, there has been progress in developing versions of the theory that are well suited for computations. The theory does not require that the system is close to integrable, but rather uses geometric identities. The theorems prove that approximate solutions satisfying some non-degeneracy assumptions correspond to a true solution. Furthermore, the proofs lead at the same time to very efficient algorithms. Implementing these algorithms, leads to some conjectural insights on the phenomena that happen at breakdown. They turn out to be remarkably similar to phenomena that were observed in phase transition and "renormalization group" provides a unifying point of view. Nevertheless, many questions remain open.

27/11/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jinjun Li, Instituto Superior Técnico

Topological entropy of refined irregular sets

16/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
David Owen, Carnegie Mellon University

Multiscale Geometrical Changes in Continua: Structured Deformations and Some Questions in Analysis

Classes of injective deformations of a continuous body that are stable under composition and under the taking inverses are central to the description of geometrical changes of a continuous body at both the macroscopic and microscopic levels. Classical, simple, and structured deformations are described and assessed from the point of view of continuum mechanics and of variational problems. The natural way of assigning an energy to a structured deformation, through relaxation of an energy assigned to a smaller class of deformations, leads to various alternative, "variationally friendly" notions of structured deformations. Several of these alternative notions are examined from the point of view of approximation by deformations in a smaller class and of relaxation of an initial energy defined on the smaller class of deformations. Questions are raised as to whether or not different notions of structured deformations lead to different relaxed energies for a structured deformation that satisfies the defining requirements of two or more notions. A recent example of an explicit formula for a relaxed energy is used to illustrate these questions.

10/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roman Hric, Matej Bel University, Banska Bystrica, Slovakia

Dense orbits of homeomorphisms, flows and their time maps

24/04/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Henrique Oliveira, Instituto Superior Técnico

New Results on the Collatz Problem

03/04/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building
Saber Elaydi, Trinity University, San Antonio, USA

Application of singularity theory in planar discrete dynamical systems and applications to competition models

13/03/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
George Sell, University of Minnesota

How to use numerical techniques to study the dynamics inside the attractor

16/02/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
José Ferreira Alves, Universidade do Porto

Gibbs-Markov structures vs statistical properties in dynamical systems

In a classical approach to dynamical systems one frequently uses certain geometric structures of the system to deduce statistical properties, such as invariant measures with stochastic-like behaviour, large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. We show that in many cases stochastic-like behaviour itself implies that the system has certain geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration.

18/10/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Bernold Fiedler, Freie Universität Berlin

Schoenflies spheres in Sturm attractors

22/06/2011, 16:00 — 17:00 — Room P3.10, Mathematics Building
Diogo Oliveira e Silva, University of California, Berkeley

On trilinear oscillatory integrals

We examine a certain class of trilinear integral operators which incorporate oscillatory factors e iP, where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations. Tools include sublevel set estimates, higher dimensional versions of van der Corput's lemma and corresponding multilinear analogues. This is joint work with Michael Christ.

24/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Felipe Rivero, Universidad de Sevilla

Structure stability of pullback attractors

Attractors help us to understand the dynamics of a system thanks to the asymptotic information given about the solutions. But any mathematical object needs a robust definition, that is, needs consistence under perturbations, because any physical model might experiment any kind of natural or artificial changes. Our aim in this talk is to show how a gradient-like pullback attractor has a stable structure under small perturbations and show some nontrivial examples where the perturbation can be found in the time dependent operator or in the nonlinearity.

15/03/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Bernold Fiedler, Institute of Mathematics, Free University of Berlin

Negative Bendixson-Dulac criteria in several dimensions? A chemostat example

The negative criterium of Bendixson and Dulac states that area contracting planar vector fields cannot possess periodic orbits. A physics student asked in class why this would not generalize to three dimensions, obviously, using the curl.

We give a higher-dimensional example, arising from chemostats, where a suitable differential form excludes periodicity. This is joint work with Sze-Bi Hsu.

07/12/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Manuel de León, ICMAT-Consejo Superior de Investigaciones Cientificas

Generalized Hamiltonian systems and applications to nonholonomic mechanics

Over twenty years ago Alan Weinstein proposed to formulate mechanics in the framework of Lie algebroids. The use of Lie algebroids in mechanics has recently permitted two unexpected applications: the development of a Hamilton--Jacobi theory and a simple reduced scheme for nonholonomic mechanical systems.

08/06/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Joana Oliveira dos Santos, Université Paris Dauphine

A geometric definition of the Aubry-Mather set

Given an optical Hamiltonian H:T *M m on the cotangent bundle of a compact manifold M without boundary its dynamics can be studied with the use of the Lagrangian action functional. Using this approach, Mather defined an important compact invariant set, that he called the Aubry set, and which has the distinguished property of being contained in a Lipschitz Lagrangian graph. In the present seminar, we will study it from a geometric point of view and give the lines of new proofs of some of its most important properties: dynamical invariance, symplectic invariance, and the graph property (which is tautological for the definition we use).

18/05/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Enrico Valdinoci, Università di Roma Tor Vergata

Geometric properties of elliptic PDEs arising from a variational problem

We will discuss some rigidity, monotonicity and symmetry properties for elliptic PDEs in relation with minimal surfaces and dynamical systems. We will discuss the relation between a problem posed by De Giorgi for the Allen-Cahn equation and a question raised by Bangert in a variational context studied by Moser and related to the Aubry-Mather Theory.

18/05/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Saber Elaydi, Trinity University, San Antonio, Texas

Invariant Manifolds in Competition Models

07/05/2010, 11:30 — 12:30 — Room P3.10, Mathematics Building
Martin Burns, University of Strathclyde, Glasgow

Steady state solutions for a quasilinear parabolic bistable equation

06/05/2010, 11:00 — 12:00 — Room P3.10, Mathematics Building
Michael Grinfeld, University of Strathclyde, Glasgow

Modelling submonolayer deposition

30/03/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Josep Sardanyés, Complex Systems Lab-Universitat Pompeu Fabra, Barcelona

Error threshold in RNA quasispecies models with complementation

12/01/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Rachid El Harti, University Hassan I, Morocco

Thompson group and Dixmier problem

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