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Series: School of Mathematics Colloquium

How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices, or, more generally, k-dimensional faces and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric, but other metrics can and have been considered. We will present several results about these issues, mostly related to approximation by “random polytopes”.

Series: School of Mathematics Colloquium

Though the modern analytic celestial mechanics has been existing for more
than 300 years since Newton, there are still many basic questions
unanswered, for instance, there is still no rigorous mathematical proof
explaining why our solar system has been stable for such a long time (five
billion years) hence no guarantee that it would remain stable for the next
five billion years. Instead, it is known that there are various instability
behaviors in the Newtonian N-body problem.
In this talk, we mention three types instability behaviors in Newtonian
N-body problem. The first type we will talk about is simply chaotic
motions, which include for instance the oscillatory motions, in which case,
one body travels back and forth between neighborhoods of zero and infinity.
The second type is “organized” chaotic motions, also known as Arnold
diffusion or weak turbulence. Finally, we will talk about our work on the
existence of the most wild unstable behavior, non collision singularities,
also called finite time blow up solution.
The talk is mostly expository. Zero background on celestial mechanism or
dynamical systems is needed to follow the lecture.

Series: School of Mathematics Colloquium

Approximate separable representation of the Green’s functions for
differential operators is a fundamental question in the analysis of
differential equations and development of efficient numerical
algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width
or degrees of freedom of the corresponding differential
equation. Computationally, being able to approximate a Green’s function
as a sum with few separable terms is equivalent to the existence of low
rank approximation of the discretized system which can be explored for
matrix compression and fast solution techniques such as in fast multiple
method and direct matrix inverse solver. In this talk, we will mainly
focus on Helmholtz equation in the high frequency limit for which we
developed a new approach to study the approximate separability of
Green’s function based on an geometric characterization of the relation
between two Green's functions and a tight dimension estimate for the
best linear subspace approximating a set of almost orthogonal vectors.
We derive both lower bounds and upper bounds and show their sharpness
and implications for computation setups that are commonly used
in practice. We will also make comparisons with other types of
differential operators such as coercive elliptic differential operator
with rough coefficients in divergence form and hyperbolic differential
operator. This is a joint work with Bjorn Engquist.

Series: School of Mathematics Colloquium

Special time.

We consider the time-domain boundary element method for exteriorRobin type boundary value problems for the wave equation. We applya space-time Galerkin method, present a priori and a posteriori errorestimates, and derive an h-adaptive algorithm in space and time withmesh renement driven by error indicators of residual and hierarchicaltype.Numerical experiments are also given which underline our theoreticalresults. Special emphasis is given to numerical simulations of the soundradiation of car tyres.

Series: School of Mathematics Colloquium

As the cornerstone of two-fluid models in plasma theory,
Euler-Maxwell (Euler-Poisson) system describes the dynamics of compressible
ion and electron fluids interacting with their own self-consistent electromagnetic field. It is
also the origin of many famous dispersive PDE such as KdV, NLS, Zakharov,
...etc. The electromagnetic interaction produces plasma frequencies which enhance
the dispersive effect, so that smooth initial data with small amplitude
will persist forever for the Euler-Maxwell system, suppressing any possible shock
formation. This is in stark contrast to the classical Euler system for a
compressible neutral fluid, for which shock waves will develop
even for small smooth initial data. A survey along this direction for
various two-fluid models will be given during this talk.

Series: School of Mathematics Colloquium

Bernstein's inequality connecting the norms of a (trigonometric) polynomial
with the norm of its derivative is 100 years old. The talk will discuss some
recent developments concerning Bernstein's inequality: inequalities
with doubling weights, inequalities on general compact subsets of
the real line or on a system of Jordan curves.
The beautiful Szego-Schaake–van der Corput generalization
will also be mentioned along with some of its recent variants.

Series: School of Mathematics Colloquium

Much effort in the past several decades has gone into lifting various algebraic structures into a topological context. I will describe one such lifting: that of the arithmetic theory of elliptic curves. The result is a rich and highly structured family of cohomology theories collectively known as elliptic cohomology. By forming "global sections" one is led to a topological enrichment of the ring of modular forms. Geometric interpretations of these theories are enticing but still conjectural at best.

Series: School of Mathematics Colloquium

Singular and oscillatory integral estimates, such as maximal theorems and restriction estimates for measures on hypersurfaces, have long been a central topic in harmonic analysis. We discuss the recent work by the speaker and her collaborators on the analogues of such results for singular measures supported on fractal sets. The common thread is the use of ideas from additive combinatorics. In particular, the additive-combinatorial notion of "pseudorandomness" for fractals turns out to be an appropriate substitute for the curvature of manifolds.

Series: School of Mathematics Colloquium

Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed:the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss a result showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.

Series: School of Mathematics Colloquium

Kick-off of the <a href="http://ttc.gatech.edu">Tech Topology Conference</a>, December 5-7, 2014

In 1985, Barnsley and Harrington defined a "Mandelbrot Set" M
for pairs of similarities -- this is the set of complex numbers z
with norm less than 1 for which the limit set of the semigroup
generated by the similarities x -> zx and x -> z(x-1)+1 is
connected. Equivalently, M is the closure of the set of roots of
polynomials with coefficients in {-1,0,1}. Barnsley and Harrington
already noted the (numerically apparent) existence of infinitely
many small "holes" in M, and conjectured that these holes were
genuine. These holes are very interesting, since they are "exotic"
components of the space of (2 generator) Schottky semigroups. The
existence of at least one hole was rigorously confirmed by Bandt in
2002, but his methods were not strong enough to show the existence
of infinitely many holes; one difficulty with his approach was that
he was not able to understand the interior points of M, and on the
basis of numerical evidence he conjectured that the interior points
are dense away from the real axis. We introduce the technique of
traps to construct and certify interior points of M, and use
them to prove Bandt's Conjecture. Furthermore, our techniques let
us certify the existence of infinitely many holes in M. This is
joint work with Sarah Koch and Alden Walker.