Monday, March 26, 2012 - 11:00 , Location: Skiles 006 , Predrag Cvitanovic , Georgia Tech, Physics , Organizer: Rafael de la Llave
With recent advances in experimental imaging, computational methods, and dynamics insights it is now possible to start charting out the terra incognita explored by turbulence in strongly nonlinear classical field theories, such as fluid flows. In presence of continuous symmetries these solutions sweep out 2- and higher-dimensional manifolds (group orbits) of physically equivalent states, interconnected by a web of still higher-dimensional stable/unstable manifolds, all embedded in the PDE infinite-dimensional state spaces. In order to chart such invariant manifolds, one must first quotient the symmetries, i.e. replace the dynamics on M by an equivalent, symmetry reduced flow on M/G, in which each group orbit of symmetry-related states is replaced by a single representative.Happy news: The problem has been solved often, first by Jacobi (1846), then by Hilbert and Weyl (1921), then by Cartan (1924), then by [...], then in this week's arXiv [...]. Turns out, it's not as easy as it looks.Still, every unhappy family is unhappy in its own way: The Hilbert's solution (invariant polynomial bases) is useless. The way we do this in quantum field theory (gauge fixing) is not right either. Currently only the "method of slices" does the job: it slices the state space by a set of hyperplanes in such a way that each group orbit manifold of symmetry-equivalent points is represented by a single point, but as slices are only local, tangent charts, an atlas comprised from a set of charts is needed to capture the flow globally. Lots of work and not a pretty sight (Nature does not like symmetries), but one is rewarded by much deeper insights into turbulent dynamics; without this atlas you will not get anywhere.This is not a fluid dynamics talk. If you care about atomic, nuclear or celestial physics, general relativity or quantum field theory you might be interested and perhaps help us do this better.You can take part in this seminar from wherever you are by clicking onevo.caltech.edu/evoNext/koala.jnlp?meeting=M2MvMB2M2IDsDs9I9lDM92
Wednesday, March 14, 2012 - 11:00 , Location: Skiles 006 , Boris Kalinin , Univ. of Southern Alabama , Organizer: Rafael de la Llave
Hyperbolic actions of Z^k and R^k arise naturally in algebraic and geometric context. Algebraic examples include actions by commuting automorphisms of tori or nilmanifolds and, more generally, affine and homogeneous actions on cosets of Lie groups. In contrast to hyperbolic actions of Z and R, i.e. Anosov diffeomorphisms and flows, higher rank actions exhibit remarkable rigidity properties, such as scarcity of invariant measures and smooth conjugacy to a small perturbation. I will give an overview of results in this area and discuss recent progress.
Monday, March 12, 2012 - 11:00 , Location: Skiles 006 , Victoria Sadovskaya , Univ. of Southern Alabama , Organizer: Rafael de la Llave
A linear cocycle over a diffeomorphism f of a manifold M is an automorphism of a vector bundle over M that projects to f. An important example is given by the differential Df or its restriction to an invariant sub-bundle. We consider a Holder continuous linear cocycle over a hyperbolic system and explore what conclusions can be made based on its properties at the periodic points of f. In particular, we obtain criteria for a cocycle to be isometric or conformal and discuss applications and further developments.
Monday, February 27, 2012 - 11:05 , Location: Skiles 006 , Andrew Torok , Univ. of Houston , Organizer: Rafael de la Llave
Consider a hyperbolic basic set of a smooth diffeomorphism. We are interested in the transitivity of Holder skew-extensions with fiber a non-compact connected Lie group. In the case of compact fibers, the transitive extensions contain an open and dense set. For the non-compact case, we conjectured that this is still true within the set of extensions that avoid the obvious obstructions to transitivity. Within this class of cocycles, we proved generic transitivity for extensions with fiber the special Euclidean group SE(2n+1) (the case SE(2n) was known earlier), general Euclidean-type groups, and some nilpotent groups. We will discuss the "correct" result for extensions by the Heisenberg group: if the induced extension into its abelinization is transitive, then so is the original extension. Based on earlier results, this implies the conjecture for Heisenberg groups. The results for nilpotent groups involve questions about Diophantine approximations. This is joint work with Ian Melbourne and Viorel Nitica.
Monday, February 20, 2012 - 11:05 , Location: Skiles 006 , Diego Del Castillo-Negrete , Oak Ridge National Lab , Organizer: Rafael de la Llave
The study of transport is an active area of applied mathematics of interest to fluid mechanics, plasma physics, geophysics, engineering, and biology among other areas. A considerable amount of work has been done in the context of diffusion models in which, according to the Fourier-‐Fick’s prescription, the flux is assumed to depend on the instantaneous, local spatial gradient of the transported field. However, despiteits relative success, experimental, numerical, and theoretical results indicate that the diffusion paradigm fails to apply in the case of anomalous transport. Following an overview of anomalous transport we present an alternative(non-‐diffusive) class of models in which the flux and the gradient are related non-‐locally through integro-differential operators, of which fractional Laplacians are a particularly important special case. We discuss the statistical foundations of these models in the context of generalized random walks with memory (modeling non-‐locality in time) and jump statistics corresponding to general Levy processes (modeling non-‐locality in space). We discuss several applications including: (i) Turbulent transport in the presence of coherent structures; (ii) chaotic transport in rapidly rotating fluids; (iii) non-‐local fast heat transport in high temperature plasmas; (iv) front acceleration in the non-‐local Fisher-‐Kolmogorov equation, and (v) non-‐Gaussian fluctuation-‐driven transport in the non-‐local Fokker-‐Planck equation.
Monday, February 13, 2012 - 11:05 , Location: Skiles 006 , Hector Lomeli , Univ. of Texas at Austin/ITAM , Organizer: Rafael de la Llave
We generalize some notions that have played an important role in dynamics, namely invariant manifolds, to the more general context of difference equations. In particular, we study Lagrangian systems in discrete time. We define invariant manifolds, even if the corresponding difference equations can not be transformed in a dynamical system. The results apply to several examples in the Physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We use a modification of the parametrization method to show the existence of Lagrangian stable manifolds. This method also leads to efficient algorithms that we present with their implementations. (Joint work with Rafael de la Llave.)
Monday, January 30, 2012 - 11:05 , Location: Skiles 006 , Alex Haro , Univ.. of Barcelona , Organizer: Rafael de la Llave
We present a novel method to find KAM tori in degenerate (nontwist) cases. We also require that the tori thus constructed have a singular Birkhoff normal form. The method provides a natural classification of KAM tori which is based on Singularity Theory.The method also leads to effective algorithms of computation, and we present some preliminary numerical results. This work is in collaboration with R. de la Llave and A. Gonzalez.
Monday, January 23, 2012 - 11:05 , Location: Skiles 006 , Jordi Lluis Figueras , Uppsala University , Organizer: Rafael de la Llave
In this talk we will present a numerical algorithm for the computation of (hyperbolic) periodic orbits of the 1-D K-S equation u_t+v*u_xxxx+u_xx+u*u_x = 0, with v>0. This numerical algorithm consists on apply a suitable Newton scheme for a given approximate solution. In order to do this, we need to rewrite the invariance equation that must satisfy a periodic orbit in a form that its linearization around an approximate solution is a bounded operator. We will show also how this methodology can be used to compute rigorous estimates of the errors of the solutions computed.
Monday, January 9, 2012 - 11:05 , Location: Skiles 006 , Marcel Guardia , Institute for Advanced Studies , Organizer: Rafael de la Llave
We consider the restricted planar elliptic 3 body problem, which models the Sun, Jupiter and an Asteroid (which we assume that has negligible mass). We take a realistic value of the mass ratio between Jupiter and the Sun and their eccentricity arbitrarily small and we study the regime of the mean motion resonance 1:7, namely when the period of the Asteroid is approximately seven times the period of Jupiter. It is well known that if one neglects the influence of Jupiter on the Asteroid, the orbit of the latter is an ellipse. In this talk we will show how the influence of Jupiter may cause a substantial change on the shape of Asteriod's orbit. This instability mechanism may give an explanation of the existence of the Kirkwood gaps in the Asteroid belt. This is a joint work with J. Fejoz, V. Kaloshin and P. Roldan.
Monday, December 5, 2011 - 11:00 , Location: Skiles 006 , John Mallet-Paret , Division of Applied Mathematics, Brown University , email@example.com , Organizer: Shui-Nee Chow
We study a class of linear delay-differential equations, with a singledelay, of the form$$\dot x(t) = -a(t) x(t-1).\eqno(*)$$Such equations occur as linearizations of the nonlinear delay equation$\dot x(t) = -f(x(t-1))$ around certain solutions (often around periodicsolutions), and are key for understanding the stability of such solutions.Such nonlinear equations occur in a variety of scientific models, anddespite their simple appearance, can lead to a rather difficultmathematical analysis.We develop an associated linear theory to equation (*) by taking the$m$-fold wedge product (in the infinite dimensional sense of tensorproducts) of the dynamical system generated by (*). Remarkably, in the caseof a ``signed feedback'' where $(-1)^m a(t) > 0$ for some integer $m$, theassociated linear system is given by an operator which is positive withrespect to a certain cone in a Banach space. This leads to very detailedinformation about stability properties of (*), in particular, informationabout characteristic multipliers.