In this talk I'll describe recent work with Henry Wilton (UCL) in which
we prove that there does not exist an algorithm that can determine which
finitely presented groups have a non-trivial finite quotient.
In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H=C(Z/p) of complex valued functions on Z/p={0,...,p-1}, the integers modulo a prime number p>>0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form
R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t),
where W(t) in H is a white noise, and τ,ω in ℤ/p, encode the distance from, and velocity of, the object.
Problem (digital radar problem) Extract τ,ω from R and S.
I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^2⋅log(p) operations. I will then explain how to use techniques from group representation theory to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS).
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley).
What is a phase transition?
The first thing that comes to mind is boiling and freezing of water. The material clearly changes its behaviour without any chemical reaction. One way to arrive at a mathematical model is to associate different material behavior, ie., constitutive laws, to different phases. This is a continuum physics viewpoint, and when a law for the switching between phases is specified, we arrive at pde problems. The oldest paper on such a problem by Clapeyron and Lame is nearly 200 years old; it is basically on what has later been called the Stefan problem for the heat equation.
The law for switching is given e.g. by the melting temperature. This can be taken to be a phenomenological law or thermodynamically justified as an equilibrium condition.
The theory does not explain delayed switching (undercooling) and it does not give insight in structural differences between the phases.
To some extent the first can be explained with the help of a free energy associated with the interface between different phases. This was proposed by Gibbs, is relevant on small space scales, and leads to mean curvature equations for the interface – the so-called Gibbs Thompson condition.
The equations do not by themselves lead to a unique evolution. Indeed to close the resulting pde’s with a reasonable switching or nucleation law is an open problem.
Based on atomistic concepts, making use of surface energy in a purely phenomenological way, Becker and Döring developed a model for nucleation as a kinetic theory for size distributions of nuclei. The internal structure of each phase is still not considered in this ansatz.
An easier problem concerns solid-solid phase transitions. The theory is furthest developped in the context of equilibrium statistical mechanics on lattices, starting with the Ising model for ferromagnets. In this context phases correspond to (extremal) equilibrium Gibbs measures in infinite volume. Interfacial free energy appears as a finite volume correction to free energy.
The drawback is that the theory is still basically equilibrium and isothermal. There is no satisfactory theory of metastable states and of local kinetic energy in this framework.
Free groups, free abelian groups and fundamental groups of
closed orientable surfaces are the most basic and well-understood examples
of infinite discrete groups. The automorphism groups of these groups, in
contrast, are some of the most complex and intriguing groups in all of
mathematics. I will give some general comments about geometric group
theory and then describe the basic geometric object, called Outer space,
associated to automorphism groups of free groups.
This Colloquium talk is the first of a series of three lectures given by
Professor Vogtmann, who is the European Mathematical Society Lecturer. In
this series of three lectures, she will discuss groups of automorphisms
of free groups, while drawing analogies with the general linear group over
the integers and surface mapping class groups. She will explain modern
techniques for studying automorphism groups of free groups, which include
a mixture of topological, algebraic and geometric methods.