Sept 24:
Speaker: Alexander Volberg, Michigan State
Title:
Analytic capacity and Buffon needle landing near Cantor
sets Abstract:
The lecture is devoted to the interplay between the analytic
object: analytic capacity, and the geometric object: Buffon needle
probability. The latter is related to Geometric measure Theory
questions
such as rectifiability in the sense of Besicovitch. The new results in
all
these interrelated areas will be presented.
Oct 1:
Speaker: Keying Guan, Beijing Jiaotong University, China
Title:
An Exact Solution of Euler's Equation with Unsteady Vortices and
Brownian Motions
Abstract:
Based on the conception "pseudo-potential" of the incompressible plane
flow,
an exact solution to the Euler equation is given. With the KAM theory
and
the second order Melnikov function, it is proved that this solution
describes infinitely many unsteady vortices distributed periodically on
the
whole plane and the Brownian motion appearing along the border region
separating different vortices.
Oct 8:
Speaker: Satya Mandal, KU
Title:
Topology in Algebra
Abstract:
Abstract: The set C(M) of all continuous real valued functions f : M →
R
over a topological space M has a structure of a commutative ring. This
fact
was among the early motivations to study the abstract ring theory.
Concepts
in topology, often, have a natural counter part in algebra. For
example, for
finite dimensional connected paracompact (Hausdorff) spaces M, there is
an 1-1 correspondence between the real vector bundles over M and
finitely
generated pro jective modules over C(M). This is why, more often than
not,
progress in topology has influenced developments in algebra.
The vector bundle theory in topology is rich, beautiful and classical.
In
some cases, we are still trying to catch up with such theories in
topology,
for a counter part in algebra.
We will discuss how the theory of vector bundles in topology influenced
developments in algebra, and the correspondences between the classical
theory
in topology and the newly developed theory in algebra.
Oct 14:
Speaker: Terry Lyons, Oxford
Title:
Rough Path Analysis
Abstract:
If Yt and Xt are two Banach space valued Holder continuous functions on
some interval
t ∈ [0, T] and f is also a Holder continuous function on the Banach
space. The theory
of rough path analysis is the study of how to define the integral f OT
f(Yt )dXt and the
study of the differential equation dYt = f(Yt )dXt . The theory is
motivated by problems
in probability and has application to many fields such as stochastic
analysis of Brownian
motion and in particular fractional Brownian motions.
Nov 12:
Speaker: Bjorn Sandstede, Brown
Title:
Snakes and ladders
Abstract:
Localized stationary or time-periodic structures play an important role
in many
applications such as buckling, fluid flows, nonlinear optics, and
chemical reactions.
In this talk, I will give an overview of recent analytical, numerical,
and experimental
work in which these structures have been investigated. In particular, I
will discuss
localized rolls, hexagon patches, and target patterns, and explore how
the spatial width
of these patterns depends on systems parameters.
Nov 19:
Speaker: Jeffrey, Humpherys
Title:
The Newton-Kalman Filter
Abstract:
Since its development in the 1960's, the Kalman filter has enjoyed it's
place as one of the most important theoretical tools of the 20th Century.
This statistical estimation method is used pervasively in virtually every
quantitative field, including computer science, engineering, economics,
and finance. Each year, hundreds of patent applications are filed and
thousands of academic papers written exposing some new variation or
application of the Kalman filter. In this talk, we give an overview
of the Kalman filter--its history, mathematical and statistical formulations,
and its use in applications. We then show it can be formulated and derived as
a straightforward application of Newton's Method for root finding, which goes
back over 300 years. We then discuss why this shouldn't be a big surprise.
