Adam Day - Victoria
University of Wellington, New Zealand.
Title: Indifferent sets for
genericity
Abstract: Given a class C of
{0,1} sequences and a set of natural numbers I, we say that I is
indifferent for some A in C if any other sequence that differs
from A only on the elements of I is also in C. The idea of an
indifferent set was introduced by Figueira, Miller and Nies, who
investigated indifferent sets for different notions of randomness.
Similar ideas were used by Barmpalias, Lewis and Ng to show that every
p.a. degree is the join of two randoms. We will look at indifferent
sets when C is the class of generic, or weak generic
sequences. We will show that for all generic sequences G, there is some
infinite set I such that I is indifferent for G. We will consider the
computational complexity of indifferent sets as well as the question as
to whether a generic sequence can compute an infinite set to which it
is indifferent.
(joint work with Andrew Fitzgerald)
Liang Yu - Nanjing
University, Nanjing, China.
Title: On strong
Pi^1_1-ML-randomness
Abstract: we show that strong
\Pi^1_1-randomness is strictly stronger than \Pi^1_1-ML-randomness.
Rod Downey - Victoria
University of Wellington, New Zealand.
Title: Euclidean Function of
Computable Euclidean Domains.
Abstract: One of the first
algorithms discussed in almost any elementary algebra course is
Euclid’s algorithm for computing the greatest common divisor of
two integers. In a first course in abstract algebra, this idea is
explained by describing both Z and Q[X] as Euclidean domains. We
recall the definition of a Euclidean domain.
Definition 0.1. A commutative ring R is a Euclidean domain
if it is an integral domain (i.e., there are no zero divisors)
and there is a function φ : R \{0} → ω satisfying
( ∀a, d ∈ R)(∃q ∈ R) [d = 0 or a + qd = 0 or φ(a + qd) < φ(d)] .
If there is such a function φ : R \{0} → ON (where ON is the
class of ordinals),
then R is a transfinite Euclidean domain.
In the former case, we say the function φ is a (finitely-valued)
Euclidean function for R; in the latter case, we say the
function φ is a transfinitely-valued Euclidean function for
R.
It is still a forty year old open question (implicitly a sixty
year old open question) in algebra whether there exists a
transfinite Euclidean domain having no finitely-valued Euclidean
function.
Less well known is that we can define a Euclidian domain via a
hierarchy of sets with the property that it exhausts the set R
\{0} of nonzero elements if and only if R is a (transfinite)
Euclidean domain. At the bottom level R0 of this hierarchy, we
have the units. At the next level R1 , we have all those elements
which either exactly divide all elements or give remainder a unit upon
division. More generally, at level Rα , we have all those elements
which either exactly divide all elements or give remainder in
R<α upon division.
This defines a minimal Euclidian function. In this talk I will
look at the effective content and the reverse mathematics of
Euclidian domains and in particular the existence of minimal
Euclidian functions. Lots of open questions remain. The talk will
be at a reasonably elementary level, notable more for wonderful
open questions rather than the depth of the theorems.
(Joint work with Asher Kach.)
Boris Zilber
Title: Some issues between
Model Theory and Physics.
Abstract: I am going to talk
about Zariski structures arising in the context of noncommutative
geometry and physics and report on an ongoing project, initiated by
physicists, that relates topos theory to physics and model theory.