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Series: CDSNS Colloquium

We present a method for the detection
of stable and unstable fibers of invariant manifolds of periodic
orbits.
We show how to propagate the fibers to prove transversal
intersections of invariant manifolds. The method can be applied using
interval arithmetic
to produce rigorous, computer assisted estimates
for the manifolds. We apply the method to prove transversal
intersections of stable and unstable manifolds of Lyapunov orbits in
the restricted three body problem.

Series: CDSNS Colloquium

We explain numerical algorithms for the computation of normally hyperbolic invariant manifolds and their invariant bundles, using the parameterization method. The framework leads to solving invariance
equations, for which one uses a Newton method adapted to the dynamics and the geometry of the invariant manifolds. We illustrate the algorithms with several examples. The algorithms are inspired in current work with A. Haro
and R. de la Llave. This is joint work with Alex Haro.

Series: CDSNS Colloquium

Conformally symplectic systems send a symplectic form into a
multiple of itself. They appear in mecanical systems with friction proportional to the velocity and as Euler-Lagrange equations of the time discounted actions common in economics. The conformaly symplectic structure
provides identities that we use to prove "a-posteriori" theorems that
show that if we have an approximate solution which satisfies some non-degeneracy conditions, we can obtain a true solution close to the approximate one. The identities used to prove the theorem, also lead to very efficient algorithms with small
storage and operation counts. We will also present implementations of the algorithms.

Series: CDSNS Colloquium

A very simple model for electric conduction consists of N particles movingin a periodic array of scatterers under the influence of an electric field and of aGaussian thermostat that keeps their energy fixed. I will present analytic result for the behaviourof the steady state of the system at small electric field, where the velocity distribution becomesindependent of the geometry of the scatterers, and at large N, where the system can bedescribed by a linear Boltzmann type equation.

Series: CDSNS Colloquium

A common practice in aerospace engineering has been to carry out deterministicanalysis in the design process. However, due to variations in design condition suchas material properties, physical dimensions and operating conditions; uncertainty isubiquitous to any real engineering system. Even though the use of deterministicapproaches greatly simplifies the design process since any uncertain parameter is setto a nominal value, the final design can have degraded performance if the actualparameter values are slightly different from the nominal ones.Uncertainty is important because designers are concerned about performance risk.One of the major challenges in design under uncertainty is computational efficiency,especially for expensive numerical simulations. Design under uncertainty is composedof two major parts. The first one is the propagation of uncertainties, and the otherone is the optimization method. An efficient approach for design under uncertaintyshould consider improvement in both parts.An approach for robust design based on stochastic expansions is investigated. Theresearch consists of two parts : 1) stochastic expansions for uncertainty propagationand 2) adaptive sampling for Pareto front approximation. For the first part, a strategybased on the generalized polynomial chaos (gPC) expansion method is developed. Acommon limitation in previous gPC-based approaches for robust design is the growthof the computational cost with number of uncertain parameters. In this research,the high computational cost is addressed by using sparse grids as a mean to alleviatethe curse of dimensionality. Second, in order to alleviate the computational cost ofapproximating the Pareto front, two strategies based on adaptive sampling for multi-objective problems are presented. The first one is based on the two aforementionedmethods, whereas the second one considers, in addition, two levels of fidelity of theuncertainty propagation method.The proposed approaches were tested successfully in a low Reynolds number airfoilrobust optimization with uncertain operating conditions, and the robust design of atransonic wing. The gPC based method is able to find the actual Pareto front asa Monte Carlo-based strategy, and the bi-level strategy shows further computationalefficiency.

Series: CDSNS Colloquium

Putting in place the last piece of the big mosaic of the proof of
the Boltzmann-Sinai Ergodic Hypothesis,we consider the billiard
flow of elastically colliding hard balls on the flat $d$-torus ($d>1$),
and prove that no singularity manifold can even locally coincide
with a manifold describing future non-hyperbolicity of the trajectories.
As a corollary, we obtain the ergodicity (actually the Bernoulli mixing
property) of all such systems, i.e. the verification of the Boltzmann-sinai
Ergodic Hypothesis.
The manuscript of the paper can be found at
http://people.cas.uab.edu/~simanyi/transversality-new.pdf

Series: CDSNS Colloquium

For a general time-dependent linear competitive-cooperative
tridiagonal system of differential equations, we obtain canonical
Floquet invariant bundles which are exponentially separated in the
framework of skew-product flows. The obtained Floquet theory is applied
to study the dynamics on the hyperbolic omega-limit sets for the
nonlinear competitive-cooperative tridiagonal systems in time-recurrent
structures including almost periodicity and almost automorphy.

Series: CDSNS Colloquium

Invariant tori play a prominent role in the dynamics of symplectic
maps. These tori are especially important in two dimensional systems
where they form a boundary to transport. Volume preserving maps also
admit families of invariant rotational tori, which will restrict
transport in a d dimensional map with one action and d-1 angles. These
maps most commonly arise in the study of incompressible fluid flows,
however can also be used to model magnetic field-line flows, granular
mixing, and the perturbed motion of comets in near-parabolic orbits.
Although a wealth of theory has been developed describing tori in
symplectic maps, little of this theory extends to the volume preserving
case. In this talk we will explore the invariant tori of a 3
dimensional quadratic, volume preserving map with one action and two
angles. A method will be presented for determining when an invariant
torus with a given frequency is destroyed under perturbation, based on
the stability of approximating periodic orbits.

Series: CDSNS Colloquium

Joint with Applied and Computational Mathematics Seminar

Bio-polymerization processes like transcription and translation are central to a proper function of a cell. The speed at which the bio-polymer grows is affected both by number of pauses of elongation machinery, as well their numbers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is a limit of mean occupancy ODE model. We compare the simulation of this model to a stochastic model and evaluate the combined effect of the polymerase density and the existence of pauses on transcription rate of ribosomal genes.

Series: CDSNS Colloquium

I'll discuss some work on rigorous computation of invariant
manifolds and computer assisted proof of the existence of transverse
connecting orbits for differential equations. I'm also interested in how
these computations can be used to obtain global topological data, such
as the chain groups and boundary maps of Morse Theory.