### 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 $H:{T}^{*}M\to {\mathbb{R}}^{m}$ 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