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Series: Research Horizons Seminar

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equation (PDE).This connection is the content of weak KAM Theory.The talk should be accessible to the “generic” mathematician. No a priori knowledge of any of the two subjects is assumed.The set-up of this theory is classical mechanical systems, in its Lagrangian formulation to take advantage of the action principle. This is the natural setting for Celestial Mechanics. Today it is also the setting for motions of satellites in the solar system.Hamilton found a reformulation of Lagrangian mechanics in terms of position and momentum instead of position and speed. In this formulation appears the Hamilton-Jacobi equation. Although this is a partial differential equation, its solutions allow to find solutions of the Hamiltonian (or Lagrangian) systems which are, in fact, governed by an ordinary differential equation.KAM (Kolmogorov-Arnold-Moser) theorem addressed at its beginning (Kolomogorov) the problem of stability of the solar system. It came as a surprise, since Poincare ́’s earlier work pointed to instability. In fact, some initial conditions lead to instability (Poincare ́) and some others lead to stability(Kolomogorov).Aubry-Mather theory finds some more substantial stable motion that survives outside the region where KAM theorem applies.The KAM theorem also provides global differentiable solutions to the Hamilton-Jacobi equation.It is known that the Hamilton-Jacobi equation usually does not have smooth global solutions. Lions & Crandall developed a theory of weak solutions of the Hamilton-Jacobi equation.Weak KAM theory explains how the Aubry-Mather sets can be obtained from the points where weak solutions of the Hamilton-Jacobi equation are differentiable.

Series: Research Horizons Seminar

Many
data sets in image analysis and signal processing are in a
high-dimensional space
but exhibit a low-dimensional structure. We are interested in building
efficient representations of these data for the purpose of compression
and inference. In the setting where a data set in $R^D$ consists of
samples from a probability measure concentrated
on or near an unknown $d$-dimensional manifold with
$d$ much smaller than $D$, we consider
two sets of problems: low-dimensional geometric approximations to the
manifold and regression of a function on the manifold. In the first
case, we construct multiscale low-dimensional empirical approximations
to the manifold and give finite-sample performance
guarantees. In the second case, we exploit these empirical geometric
approximations of the manifold and construct multiscale approximations
to the function. We prove finite-sample guarantees showing that we
attain the same learning rates as if the function
was defined on a Euclidean domain of dimension $d$. In both cases our
approximations can adapt to the regularity of the manifold or the
function even when this varies at different scales or locations.

Series: Research Horizons Seminar

There
is so much that the GT library can do for you, from providing research
materials to assistance with data visualization to patent guidance.
However, rather than trying to guess what you want from us, this year we
asked!
Based on the response to a short ranking survey I sent out last month, this session will cover: 1. How to find grants, fellowships, and travel money with the sponsorship database, Pivot. There are opportunities for postdocs and non US citizens too!2. How to use MathSciNet. We will cover navigating its classification index to actually getting the article
you want. 3. How to find and download articles from our systems, Google Scholar, and from other libraries. And if we have time: 4. How to make a poster and cheaply print it.

Series: Research Horizons Seminar

Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.

Series: Research Horizons Seminar

Series: Research Horizons Seminar

In recent years the problem
of low-rank matrix completion received a tremendous amount of
attention. I will consider the problem of exact low-rank matrix
completion for generic data. Concretely, we start with a
partially-filled matrix M, with real
or complex entries, with the goal of finding the unspecified entries
(completing M) in such a way that the completed matrix has the lowest
possible rank, called the completion rank of M. We will be interested in
how this minimal completion rank depends on the
known entries, while keeping the locations of specified and unspecified
entries fixed. Generic data means that we only consider partial fillings
of M where a small perturbation of the entries does not change the
completion rank of M.

Series: Research Horizons Seminar

The Institute for Defense Analyses - Center for Computing Sciences is a
nonprofit research center that works closely with the NSA. Our center
has around 60 researchers (roughly 30 mathematicians and 30 computer
scientists) that work on interesting
and hard problems. The plan for the seminar is to begin with a short
mathematics talk on a project that was completed at IDA-CCS and
declassified, then tell you a little about what we do, and end with your
questions. The math that we will discuss involves
symbolic dynamics and automata theory. Specifically we will develop a
metric on the space of regular languages using topological entropy.
This work was completed during a summer SCAMP at IDA-CCS. SCAMP is a
summer program where researchers from academia
(professors and students), the national labs, and the intelligence
community come to IDA-CCS to work on the agency's hard problems for 11
weeks.

Series: Research Horizons Seminar

Four
dimensions is unique in many ways. For example $n$-dimensional
Euclidean space has a unique smooth structure if and only if $n$ is not
equal to four. In other words, there is only one way to understand
smooth functions on $R^n$ if and only if
$n$ is not 4. There are many other way that smooth structures on
4-dimensional manifolds behave in surprising ways. In this talk I will
discuss this and I will sketch the beautiful interplay of ideas (you got
algebra, analysis and topology, a little something
for everyone!) that go into proving $R^4$ has more that one smooth
structure (actually it has uncountably many different smooth structures
but that that would take longer to explain).

Series: Research Horizons Seminar

In
this talk, we consider the structure of a real $n \times n$ matrix in
the form of $A=JL$, where $J$ is anti-symmetric and $L$ is symmetric.
Such a matrix comes from a linear Hamiltonian ODE system with $J$ from
the symplectic structure and the Hamiltonian
energy given by the quadratic form $\frac 12\langle Lx, x\rangle$. We
will discuss the distribution of the eigenvalues of $A$, the
relationship between the canonical form of $A$ and the structure of the
quadratic form $L$, Pontryagin invariant subspace theorem,
etc. Finally, some extension to infinite dimensions will be mentioned.

Series: Research Horizons Seminar

A motivating problem in number theory and algebraic geometry is to find
all integer-valued solutions of a polynomial equation. For example,
Fermat's Last Theorem asks for all integer solutions to x^n + y^n = z^n,
for n >= 3. This kind of problem is easy
to state, but notoriously difficult to solve. I'll explain a p-adic
method for attacking Diophantine equations, namely, p-adic integration
and the Chabauty--Coleman method. Then I'll talk about some recent
joint work on the topic.