# Calendar

Oct
18
Tue
André Nies: A gentle introduction to randomness and $K$-triviality
Oct 18 @ 1:30 pm – Oct 18 @ 2:20 pm

Keller 303
Introduction for grad students and other beginners, as a prequel to Thursday’s talk.

Oct
20
Thu
Logic Seminar: André Nies
Oct 20 @ 1:30 pm – Oct 20 @ 2:30 pm

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

Oct
27
Thu
Logic Seminar: David Ross part II
Oct 27 @ 1:30 pm – Oct 27 @ 2:30 pm

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 $X$.

Feb
10
Fri
Logic seminar: Mushfeq Khan
Feb 10 @ 2:30 pm – 3:30 pm

The Sporadic Logic Seminar returns this week at a new place and time
(Fridays, 2:30, K404). This week Mushfeq Khan will speak:

Title: “The Homogeneity Conjecture”

Abstract: It is often said that the theorems and methods of recursion theory
relativize. One might go as far as to say that much of its analytical power
derives from this feature. However, this power is accompanied by definite
drawbacks: There are important examples of theorems and open questions
whose statements are non-relativizing, i.e., they have been shown to be
true relative to some oracles, and false relative to others. It follows
that these questions cannot be settled purely through relativizing methods.
A famous example of such a negative result is Baker, Gill, and Solovay’s
theorem on the P vs. NP question.

The observation that techniques based on diagonalization, effective
numbering, and simulation relativize led some recursion theorists (notably
Hartley Rogers, Jr) to formulate what became known as the “Homogeneity
Conjecture”. It said that for any Turing degree d, the partial order of
degrees that are above d is isomorphic to the entire partial order of the
Turing degrees. In 1979, Richard Shore refuted it in an elegant, one-page
article which will be the subject of this talk.

Feb
17
Fri
Mushfeq Khan: The Homogeneity Conjecture II
Feb 17 @ 2:30 pm – 3:30 pm
Feb
24
Fri
Logic seminar: David Ross @ Keller Hall 404
Feb 24 @ 2:30 pm – 3:30 pm

Title: Some applications of logic to additive number theory

Abstract: I will review the Loeb measure construction; I will
assume some exposure to nonstandard analysis, or at least 1st order logic,
comparable to the review I gave last semester in my seminars on fixed
points. Time permitting I will give the Loeb-measure proof of Szemeredi’s
Theorem.

Mar
3
Fri
Logic seminar: David Ross
Mar 3 @ 2:30 pm – 3:20 pm

Logic seminar: David Ross
Title: Some applications of logic to additive number theory (cont.)
Room: Keller 404.

Abstract:
I will continue with some examples of results about sets of positive upper Banach density proved using Loeb measures.

Mar
10
Fri
Logic seminar: David Ross (III)
Mar 10 @ 2:30 pm – 3:30 pm