Dia: Dimecres 8 de novembre de 2000.

Lloc: Aula 1, Facultat de Matemàtiques, UB.



Dia: Dimecres 15 de novembre de 2000.

Lloc: Aula 1, Facultat de Matemàtiques, UB.



Dia: Dimecres 29 de novembre de 2000.

Lloc: Aula 1, Facultat de Matemàtiques, UB.



Dia: Dimecres 13 de desembre de 2000.

Lloc: Aula 1, Facultat de Matemàtiques, UB.



Dia: Dimecres 17 de gener de 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.



Dia: Dimecres 24 de gener de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 31 de gener de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 7 de febrer de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 14 de febrer de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 21 de febrer de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 7 de març de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 14 de març de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 21 de març de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 28 de març de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, UB.



Dia: Dimecres 4 d'abril del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



(*)ABSTRACT:
We describe a numerical method for computing the linearized normal behaviour of an invariant curve of a diffeomorphism of $\RR^n$, $n\ge 2$. In the reducible case, the method computes not only the normal eigenvalues --either elliptic or hyperbolic-- but also the corresponding eigendirections, that are the first order approximation to the invariant manifolds (stable, unstable and central) around the curve. Moreover, the method seems to be able to detect the non-reducibility --if this is the case-- of the linearized system.

The input of the method is the invariant curve --including its rotation number-- as well as a numerical procedure for computing the map and its differential. Hence, this method can be easily used on Poincar\'e sections of ODE. Due to the spectral character of the approximations used, the convergence of the process is very fast for sufficiently smooth cases. We note that the method is also valid for computing the normal behaviour of tori of higher dimensions. Finally, as examples, we study the stability of the invariant curves that appear in some concrete problems. In particular, we compute the unstable manifold for a given invariant curve of a 6-D symplectic map.


Dia: Dimecres 18 d'abril del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



SESSIÓ EXTRAORDINÀRIA

Dia: Dilluns 23 d'abril del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.




(*) ABSTRACT:
Collins and Stewart noted that many quadruped gaits can be described by spatio-temporal symmetries. For example, when a horse paces it moves both left legs in unison and then both right legs and so on. The motion is described by two symmetries: Interchange front and back legs, and swap left and right legs with a half-period phase shift.

Biologists postulate the existence of a central pattern generator (CPG) in the neural system that sends periodic signals to the legs. CPGs can be thought of as electrical circuits that produce periodic signals and can be modeled by coupled systems of differential equations with symmetries based on leg permuation.

In this lecture we discuss animal gaits; describe how periodic solutions with prescribed spatio-temporal symmetry can be formed in symmetric systems; construct a CPG architecture that naturally produces quadrupedal gait rhythms; and make several testable predictions about gaits.


(**) ABSTRACT:
A central problem in evolutionary biology is the occurrence in the fossil record of new species of organisms. Darwin's view, in 'The Origin of Species', was that speciation is the result of gradual accumulations of changes in body-plan and behaviour. Mayr asked why gene-flow failed to prevent speciation, and his answer was the classical allopatric theory in which a small founder population becomes geographically isolated and evolves independently of the main group.

An alternative class of mechanisms, sympatric speciation, assumes that no such isolation occurs. These mechanisms overcome the stabilising effect of gene-flow by invoking selection effects, for example sexual selection and assortative mating. We interpret sympatric speciation as a form of symmetry-breaking bifurcation, and model it by a system of nonlinear ODEs that is `all-to-all coupled', that is, equivariant under the action of the symmetric group S_N.

Such bifurcations can be interpreted as speciation events in which the dominant long-term behaviour is divergence into two species. Generically this divergence occurs by jump bifurcation--- `punctuated equilibrium'. Despite the discontinuity of such a bifurcation, mean phenotypes change smoothly during such a speciation event. So do mean-field genotypes related to continous characters.

Our viewpoint is that speciation is driven by natural selection acting on organisms, with the role of the genes being secondary: to ensure plasticity of phenotypes. This view is supported, for example, by the evolutionary history of African lake cichlids, where over 400 species (with less genetic diversity than humans) have arisen over a period of perhaps 200,000 years. Sympatric speciation of the kind we discuss is invisible to classical mean-field genetics, because mean-field genotypes vary smoothly.

Our methods include numerical simulations and analytic techniques from equivariant bifurcation theory. We relate our conclusions to field observations of various organisms, including Darwin's finches.


Dia: Dimecres 25 d'abril del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



Dia: Dimecres 2 de maig del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



Dia: Dimecres 9 de maig del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



Dia: Dimecres 16 de maig del 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



NOTA: El dimecres 23 de maig NO hi haurà seminari.


Dia: Dimecres 6 de juny de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



Dia: Dimecres 13 de juny de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



Dia: Dimecres 20 de juny de 2001.

Lloc: Aula 2, Facultat de Matemàtiques, Universitat de Barcelona.



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