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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.

Series: Research Horizons Seminar

Series: Research Horizons Seminar

The talk will include a crash course on infinite dimensional
topology, with applications to various topological properties of the
space of congruence classes of convex bodies in the Euclidean space.

Series: Research Horizons Seminar

This
talk will cover some recent and preliminary results in the area of
non-smooth dynamics, with connections to applications that have been
overlooked.
Much of the talk will present open questions for research projects related to this area.