Seminars and Colloquia by Series

Monday, October 1, 2012 - 16:00 , Location: Skiles 06 , Adam Fox , Univ. of Colorado , Organizer: Rafael de la Llave
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. 
Monday, September 24, 2012 - 11:05 , Location: Skiles 006 , Tomas Gedeon , Montana State University , gedeon@math.montana.edu , Organizer: Shui-Nee Chow

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.
Friday, September 7, 2012 - 11:05 , Location: Skiles 006 , Jason Mireles-James , Rutgers University , Organizer: Rafael de la Llave
 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.
Monday, August 27, 2012 - 11:00 , Location: Skiles 06 , M. Capinski , AGH Univ. Krakow and SOM, Gatech , Organizer: Rafael de la Llave
We shall present a method which establishes existence of normally hyperbolic invariant manifolds for maps within a specified domain. The method can be applied in a non-perturbative setting. The required conditions follow from bounds on the first derivative of the map, and are verifiable using rigorous numerics. We show how the method can be applied for a driven logistic map, and also present examples of proofs of invariant manifolds in the restricted three body problem.
Wednesday, July 11, 2012 - 11:00 , Location: Skiles 006 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
This talk is devoted to quasi-periodic Schr\"odinger operators beyond the Almost Mathieu, with more general potentials and interactions, considering the connections between the spectral properties of these operators and the dynamical properties of the asso- ciated quasi-periodic linear skew-products. In par- ticular, we present a Thouless formula and some consequences of Aubry duality. We illustrate the results with numerical computations. This is a join work with Joaquim Puig
Monday, April 23, 2012 - 11:05 , Location: Skiles 006 , Vadim Kaloshin , Univ. of Maryland , Organizer: Rafael de la Llave
Consider a generic perturbation of a nearly integrable system of  {\it arbitrary degrees of freedom $n\ge 2$ system}\[H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,\]with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion \[\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.\]Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion. The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang) 
Friday, April 20, 2012 - 11:10 , Location: Skiles 006 , Timothy Blass , Carnegie Melon University , Organizer: Rafael de la Llave
I will discuss recent work on the stability of linear equations under parametric forcing by colored noise. The noises considered are built from Ornstein-Uhlenbeck vector processes. Stability of the solutions is determined by the boundedness of their second moments. Our approach uses the Fokker-Planck equation and the associated PDE for the marginal moments to determine the growth rate of the moments. This leads to an eigenvalue problem, which is solved using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes into "ladder operators." The results are given in terms of a perturbation expansion in the size of the noise. We have found very good agreement between our results and numerical simulations. This is joint work with L.A. Romero.
Monday, April 16, 2012 - 11:05 , Location: Skiles 006 , Akhtam Djalilov , Univ. of Samarkand and CUNY Stony Brook , Organizer:
An important question in circle dynamics is regarding the absolute continuity of aninvariant measure. We will consider orientation preserving circle homeomorphisms withbreak points, that is, maps that are smooth everywhere except for several singular pointsat which the rst derivative has a jump. It is well known that the invariant measuresof sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesguemeasure. But in the case of homeomorphisms with break points the results are quitedierent. We will discuss conjugacies between two circle homeomorphisms with breakpoints.Consider the class of circle homeomorphisms with one break point b and satisfying theKatznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1]and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza-tion behavior of such circle maps.
Monday, April 9, 2012 - 11:05 , Location: Skiles 006 , Dr. Anton Gorodetski , UC Irvine , asgor@math.uci.edu , Organizer:
The discrete Schrodinger operator with Fibonacci potential is a central model in the study of electronic properties of one-dimensional quasicrystals. Certain renormalization procedure allows to reduce many questions on specral properties of this operator to the questions on dynamical properties of a so called trace map. It turnes out that the trace map is hyperbolic, and its measure of maximal entropy is directly related to the integrated density of states of the Fibonacci Hamiltonian. In particular, this provides the first example of an ergodic family of Schrodinger operators with singular density of states measure for which exact dimensionality can be shown. This is a joint work with D. Damanik.
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

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