Seminars and Colloquia by Series

Monday, February 16, 2015 - 11:00 , Location: Skiles 005 , Lei Zhang , Georgia Institute of Technology , Organizer: Lei Zhang
We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show  that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.
Tuesday, February 10, 2015 - 11:00 , Location: Skiles 005 , Sara Lapan , Northwestern University , , Organizer:
Given a holomorphic map of C^m to itself that fixes a point, what happens to points near that fixed point under iteration?  Are there points attracted to (or repelled from) that fixed point and, if so, how?  We are interested in understanding how a neighborhood of a fixed point behaves under iteration.  In this talk, we will focus on maps tangent to the identity.  In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point.  This theorem from the early 1900s continues to serve as inspiration for this study in higher dimensions.  In dimension 2 our picture of a full neighborhood of a fixed point is still being constructed, but we will discuss some results on what is known, focusing on the existence of a domain of attraction whose points converge to that fixed point.
Monday, October 20, 2014 - 11:00 , Location: Skiles 005 , Fabio Difonzo , School of Mathematics, Georgia Institute of Technology , Organizer:
We consider several possibilities on how to select a Filippov sliding vector field on a co-dimension 2 singularity manifold, intersection of two co-dimension 1 manifolds, under the assumption of general attractivity. Of specific interest is the selection of a smoothly varying Filippov sliding vector field. As a result of our analysis and experiments, the best candidates of the many possibilities explored are based on the so-called barycentric coordinates: in particular, we choose what we call the moments solution. We then examine the behavior of the moments vector field at first order exit points, and show that it aligns smoothly with the exit vector field. Numerical experiments illustrate our results and contrast the present method with other choices of Filippov sliding vector field. We further present some minimum variation properties, related to orbital equivalence, of Filippov solutions for the co-dimension 2 case in \R^{3}.
Monday, October 6, 2014 - 11:00 , Location: Skiles 005 , William Gignac , School of Mathematics Georgia Inst. Technology , Organizer: Rafael de la Llave
In this talk, we will discuss a question posed by Vladimir Arnold some twenty years ago, in a subject he called "dynamics of intersections." In the simplest setting, the question is the following: given a (discrete time) holomorphic dynamical system on a complex manifold X and two holomorphic curves C and D in X which pass through a fixed point P of the system, how quickly can the local intersection multiplicies at P of C with the iterates of D grow in time? Questions like this arise naturally, for instance, when trying to count the periodic points of a dynamical system. Arnold conjectured that this sequence of intersection multiplicities can grow at most exponentially fast, and in fact we can show this conjecture is true if the curves are chosen to be suitably generic. However, as we will see, for some (even very simple) dynamical systems one can choose curves so that the intersection multiplicities grow as fast as desired. We will see how to construct such counterexamples to Arnold's conjecture, using geometric ideas going back to work of Yoshikazu Yamagishi.
Monday, September 29, 2014 - 11:00 , Location: Skiles 005 , Marcel Guardia , Univ. Polit. Catalunya , Organizer: Rafael de la Llave
The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom  possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.
Monday, May 12, 2014 - 11:00 , Location: Skiles 006 , Jordi-Lluis Figueras Romero , Department of Mathematics, Uppsala University , Organizer:
We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with 3 different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.
Friday, April 18, 2014 - 11:00 , Location: Skiles 006 , Professor Joe Auslander , University of Maryland , Organizer: Yingfei Yi
Let (X, T) be a flow, that is a continuous left action of the group T on the compact Hausdorff space X. The proximal P and regionally proximal RP relations are dened, respectively (assuming X is a metric space) by P = {(x; y) | if \epsilon > 0 there is a t \in T such that d(tx, ty) < \epsilon} and RP = {(x; y) | if \epsilon > 0 there are x', y' \in X and t \in T such that d(x; x') < \epsilon, d(y; y') < \epsilon and t \in T such that d(tx'; ty') < \epsilon}. We will discuss properties of P and RP, their similarities and differences, and their connections with the distal and equicontinuous structure relations. We will also consider a relation V defined by Veech, which is a subset of RP and in many cases coincides with RP for minimal flows.
Monday, March 24, 2014 - 15:00 , Location: Skiles 005 , Professor Ken Palmer , Providence University, Taiwan , Organizer: Yingfei Yi
Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).
Monday, March 24, 2014 - 11:00 , Location: Skiles 005 , Professor Michael Li , Univeristy of Alberta , , Organizer: Yingfei Yi
Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on  each vertex, and directed edges represent couplings among vertex  systems. Questions such as stability in the large, synchronization,  and complexity in terms of dynamic clusters are of interest. A more  recent approach is to investigate the connections between network  topology and dynamical behaviours. I will present some recent results  on the construction of global Lyapunov functions for coupled systems  on networks using a graph theoretic approach, and show how such  a construction can help us to establish global behaviours of compelx  models.
Monday, February 24, 2014 - 11:00 , Location: Skiles 005 , Aynur Bulut , Univ. of Michigan , Organizer: Rafael de la Llave
In this talk we will discuss recent work, obtained in collaboration with Jean Bourgain, on new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schr\"odinger equations posed on the unit ball in two and three dimensional Euclidean space, with Dirichlet boundary conditions. We consider initial data chosen according to a Gaussian random process associated to the Gibbs measures which arise from the Hamiltonian structure of the equations, and results are obtained almost surely with respect to these probability measures. In particular, this renders the initial value problem supercritical in the sense that there is no suitable local well-posedness theory for the corresponding deterministic problem, and our results therefore rely essentially on the probabilistic structure of the problem. Our analysis is based on the study of convergence properties of solutions. Essential ingredients include probabilistic a priori bounds, delicate estimates on fine frequency interactions, as well as the use of invariance properties of the Gibbs measure to extend the relevant bounds to arbitrarily long time intervals.