A trajectory is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus for which F has the form F(t) = t + rho (mod 1) for all points in the torus, and for some rho in the torus. There is an extensive literature on determining the coordinates of the vector rho, called the rotation numbers of F. However, even in the one-dimensional case there has been no general method for computing the vector rho given only the trajectory (u_n), though there are plenty of special cases. I will present a computational method called the Embedding Continuation Method for computing some components of r from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. There is however a caveat; the coordinates of the rotation vector depend on the choice of coordinates of the torus. I will give a statement of the various sets of possible rotation numbers that rho can yield. I will illustrate these ideas with one- and two-dimensional examples.
We present and analyze a novel sparse polynomial approximation method
for the solution of PDEs with stochastic and parametric inputs. Our
approach treats the parameterized problem as a problem of joint-sparse
signal reconstruction, i.e.,
the simultaneous reconstruction of a set of signals sharing a common
sparsity pattern from a countable, possibly infinite, set of
measurements. Combined with the standard measurement scheme developed
for compressed sensing-based polynomial approximation, this
approach allows for global approximations of the solution over both
physical and parametric domains. In addition, we are able to show that,
with minimal sample complexity, error estimates comparable to the best
s-term approximation, in energy norms, are achievable,
while requiring only a priori bounds on polynomial truncation error. We
perform extensive numerical experiments on several high-dimensional
parameterized elliptic PDE models to demonstrate the superior recovery
properties of the proposed approach.
How many triangles are needed to make the new graphs not look like random graphs?
I am trying to answer this question.
(The talk will be during 12:05-1:15pm; please note the room is *Skiles 256*)
The restricted three body problem models the motion of a body of zero mass under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe Keplerian orbits. In 1922, Chazy conjectured that this model had oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Its existence was not proven until 1960 by Sitnikov in a extremely symmetric and carefully chosen configuration. In 1973, Moser related oscillatory motions to the existence of chaotic orbits given by a horseshoe and thus associated to certain transversal homoclinic points. Since then, there has been many atempts to generalize their result to more general settings in the restricted three body problem.In 1980, J. Llibre and C. Sim\'o, using Moser ideas, proved the existence of oscillatory motions for the restricted planar circular three body problem provided that the ratio between the masses of the two primaries was arbitrarily small. In this talk I will explain how to generalize their result to any value of the mass ratio. I will also explain how to generalize the result to the restricted planar elliptic three body problem. This is based on joint works with P. Martin, T. M. Seara. and L. Sabbagh.
I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the "method of interlacing polynomials") and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.
An overarching problem in matrix weighted theory is the so-called A2 conjecture, namely the question of whether the norm of a Calderón-Zygmund operator acting on a matrix weighted L2 space depends linearly on the A2 characteristic of the weight. In this talk, I will discuss the history of this problem and provide a survey of recent results with an emphasis on the challenges that arise within the setup.
Three dimensional lens spaces L(p,q) are well known as the first examples of 3-manifolds that were not known by their homology or fundamental group alone. The complete classification of L(p,q), upto homeomorphism, was an important result, the first proof of which was given by Reidemeister in the 1930s. In the 1980s, a more topological proof was given by Bonahon and Hodgson. This talk will discuss two equivalent definitions of Lens spaces, some of their well known properties, and then sketch the idea of Bonahon and Hodgson's proof. Time permitting, we shall also see Bonahon's result about the mapping class group of L(p,q).
The KLS conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) and present recent progress resulting in the current best bound, as well as a tight bound for the log-Sobolev constant (both with Yin Tat Lee). The conjecture has led to several techniques of general interest.
For a graph G, a set of subtrees of G are edge-independent with
root r ∈ V(G) if, for every vertex v ∈ V(G), the paths between v and r
in each tree are edge-disjoint. A set of k such trees represent a set of
broadcasts from r which can withstand k-1 edge failures. It is easy to
see that k-edge-connectivity is a necessary condition for the existence
of a set of k edge-independent spanning trees for all possible roots.
Itai and Rodeh have conjectured that this condition
is also sufficient. This had previously been proven for k=2, 3. We
prove the case k=4 using a decomposition of the graph similar to an ear
decomposition. Joint work with Robin Thomas.
We obtain an extension of
the Ito-Nisio theorem to certain non separable Banach spaces and apply
it to the continuity of the Ito map and Levy processes. The Ito map
assigns a rough path input of an ODE to its solution (output).
Continuity of this map usually
requires strong, non separable, Banach space norms on the path space.
We consider as an input to this map a series expansion a Levy process
and study the mode of convergence of the corresponding series of
outputs. The key to this approach is the validity of
Ito-Nisio theorem in non separable Wiener spaces of certain functions
of bounded p-variation.
This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.
This is an intoductory talk for the currently using methods for certifying roots for system of equations. First we discuss about alpha-theory which was constructed by Smale and Shub, and explain how this theory could be modified in order to apply in actual problems. In this step, we point out that alpha theory is still restricted only into polynomial systems and polynomial-exponential systems. After that as a remedy for this problem, we will introduce an interval arithmetic, and the Krawczyk method. We will end the talk with a discussion about how these current methods could be used in more general setting.
In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module. This seminar will be an hour long to allow for the GT-MAP seminar at 3 pm.
In this talk I will present an information theoretic approach to
stochastic optimal control and inference that has advantages over
classical methodologies and theories for decision making under
uncertainty. The main idea is that there are certain connections
between optimality principles in control and information theoretic
inequalities in statistical physics that allow us to solve
hard decision making problems in robotics, autonomous systems and
beyond. There are essentially two different points of view of the same
"thing" and these two different points of view overlap for a fairly
general class of dynamical systems that undergo stochastic effects. I
will also present a holistic view of autonomy that collapses planning,
perception and control into one computational engine, and ask questions
such as how organization and structure relates to computation and
performance. The last part of my talk
includes computational frameworks for uncertainty representation and
suggests ways to incorporate these representations within decision
making and control.
This is an Atlanta Science Festival performance in which mathematicians team up with dancers to give an artistic interpretation to the public of some mathematicians and some mathematical concepts. This year's show will have an emphasis on graph theory. There will be two performances at Drew Charter School in East Atlanta. For tickets go to https://www.freshtix.com/events/mathematics-in-motion---4pm-showing or https://www.freshtix.com/events/mathematics-in-motion---7pm-showing .