The Midwest Computability Seminar meets twice in the fall and twice in the spring at University of Chicago. Researchers in computability theory and their students and postdocs from University of Chicago, University of Notre Dame, and University of Wisconsin-Madison plus some others throughout the area regularly attend. Normally we have two half hour talks and two 1-hour talks and a few hours to talk and collaborate with each other. The seminar started in the fall of 2008.

DATE: Tuesday, May 11th, 2010.

PLACE: Ryerson, University of Chicago.

1100 East 58th Street, Chicago, IL 60637.

Speakers:

- Adam Day - Victoria Univeristy of Wellington.

- Liang Yu - Nanjing University.

- Rod Downey - Victoria Univeristy of Wellington.

- Boris Zilber - University of Oxford

- 12:00 - 1:00: Lunch. (the Barn Ry 352)
- 1:00 - 1:25: Adam Day (the Barn Ry 352).

- 1:35 - 2:00: Liang Yu (the Barn Ry 352).
- 2:00 - 2:30: Coffee break

- 2:30 - 3:20: Rod Downey (the Barn Ry 352).

- 3:30 - 4:10: Coffee break.
- 4:10 - 5:00: Boris Zilber (the Barn Ry 352).

- 6pm: Dinner. Reza's. 432
W Ontario St

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.

- Sept 23th 2008. Antonio Montalbán - Logan Axon - Joe Miller
- Nov 11th 2008. Chris Conidis - Keng Meng (Selwyn) Ng - Peter Gerdes
- Feb 3rd 2009. David Diamondstone - Bart Kastermans - Richard A. Shore
- April 21th 2009. Dan Turetsky - Julia Knight - Ted Slaman
- Sept 29th 2009. Carl Jockusch - Rachel Epstein - Rebecca Weber
- Jan 26th 2010. Sara Quinn - John Wallbaum - Steffen Lempp - Reed Solomon
- May 11th 2010. Adam Day - Liang Yu - Denis Hirschfeldt - Rod Downey

If you haven't been receiving the announcements, and you would like to be included in the list, send me an email to antonio at uchicago.edu.