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Series: CDSNS Colloquium

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

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.)

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.