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Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

In this seminar I will discuss current work, joint with AndrewVince and Alex Grant. The goal is to tie together several related areas, namelytiling theory, IFS theory, and NCG, in terms most familiar to fractal geometers.Our focus is on the underlying code space structure. Ideas and a conjecture willbe illustrated using the Golden b tilings of Robert Ammann

Series: Analysis Seminar

A well-known elementary linear algebra fact says that any linear
independent set of vectors in a finite-dimensional vector space cannot
have more elements than any spanning set. One way to obtain an analog of
this result in the infinite
dimensional setting is by replacing the comparison of cardinalities
with a more suitable concept - which is the concept of densities.
Basically one needs to compare the cardinalities locally everywhere and
then take the appropriate limits. We provide a rigorous
way to do this and obtain a universal density theorem that generalizes
many classical density results. I will also discuss the connection
between this result and the uncertainty principle in harmonic analysis.

Series: Analysis Seminar

Finding and understanding patterns in data sets is of significant
importance in many applications. One example of a simple pattern is the
distance between data points, which can be thought of as a 2-point
configuration. Two classic questions, the Erdos distinct
distance problem, which asks about the least number of distinct
distances determined by N points in the plane, and its continuous
analog, the Falconer distance problem, explore that simple pattern.
Questions similar to the Erdos distinct distance problem and
the Falconer distance problem can also be posed for more complicated
patterns such as triangles, which can be viewed as 3-point
configurations. In this talk I will present recent progress on Falconer
type problems for simplices. The main techniques used come
from analysis and geometric measure theory.

Series: Analysis Seminar

It was shown by Keith Ball that the maximal section of an n-dimensional
cube is \sqrt{2}. We show the analogous sharp bound for a maximal
marginal of a product measure with bounded density. We also show an
optimal bound for all k-codimensional marginals in this setting,
conjectured by Rudelson and Vershynin. This bound yields a sharp small
ball inequality for the length of a projection of a random vector. This
talk is based on the joint work with G. Paouris and P. Pivovarov.