Keller 303
Introduction for grad students and other beginners, as a prequel to Thursday’s talk.
Speaker: André Nies (University of Auckland), Fellow of the Royal Society of New Zealand
Title: Structure within the class of $K$-trivial sets
Abstract: The $K$-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity $K$ grows as slowly as possible. Since 2002, many alternative characterisations of this class have been found: properties such as low for $K$, low for Martin-Löf (ML) randomness, and basis for ML randomness, which state in one way or the other that the set is close to computable.
Initially the class looked quite amorphous. More recently, internal structure has been found. Bienvenu et al. (JEMS 2016) showed that there is a “smart” $K$-trivial set, in the sense that any ML-random computing it must compute all $K$-trivials. Greenberg, Miller and Nies (submitted) showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches using cost functions.
The talk gives an overview and ends with open questions (of which there remain many).
Location: Keller Hall 303
TITLE: Asymptotic Fixed Points, Part II
ABSTRACT: Continuing the earlier seminar, I will give nonstandard proofs for
one or more (depending on time) results like the
one below (which consolidates and generalizes a number of recent results in
the area).
Suppose
$(X,d)$ is a complete metric space,
$T:Xto X$ is continuous,
$phi, phi_n:[0,infty)to[0,infty)$, and
$phi_n$ converges to $phi$ uniformly on the range of $d$,
$phi$ is semicontinuous and satisfies $phi(s)0$,
$d(T^nx,T^ny)lephi_n(d(x,y))$ for all $x,yin X$ and $ninmathbb N$.
Moreover, suppose that for any $x,y$, $phi(t)lesssim t$ on infinite
elements of
$${^*d(T^nx,T^my) : m, n text{ hyperintegers}}.$$
Then $T$ has a unique fixed point $x_infty$, and for every $xin X, limlimits_{ntoinfty}T^n(x)=x_infty$. Moreover, this convergence is
uniform on bounded subsets of $X$.