DATE: Tuesday, September 23, 2008

PLACE: Ryerson Annex 277. University of Chicago.

1100 East 58th Street, Chicago, IL 60637.

Schedule:

Lunch: 12:00 at the Medici. 1327 E. 57th St, Hyde Park.

Talks:

- 1:30-2:30: Antonio Montalbán - U. of Chicago.

When low information is no information.

- 3:00-4:00: Logan
Axon - U. of Notre Dame.

Random closed sets and probability. - 4:30-5:30: Joe Miller - U. of Wisconsin.

What good is always guessing wrong?

Title: When low information is no information.

Abstract: It is been conjectured that if a Boolean algebra has a low_n

presentation, it has a computable presentation. We look at another

class of structures where this behavior does occur, namely, the class

of linear orderings with infinitely many descending cuts. This is

joint work with Asher Kach and Joe Miller.

We will also look at the atom relation of Boolean algebras. Here, we

have that complete information implies high information. We show that

every high_3 computably enumerable degree appears in the spectrum of

the atom relation of any computable Boolean algebra with infinitely

many atoms.

3:00-4:00: Logan Axon - U. of Notre Dame.

Title: Random closed sets and probability.

Abstract: Barmpalias, Broadhead, Cenzer, Dashti, and Weber defined a notion of

randomness for closed subsets of Cantor space by coding each infinite

binary tree without dead ends as a ternary real. The set of paths

through such a tree is said to be random if the ternary code for the

tree is (Martin-Loef) random. Every closed set of Cantor space is the

set of paths through a unique binary tree without dead ends and so we

have a well founded definition of random closed set. Probability

theorists, on the other hand, have defined a random closed set to be

something quite different. A random closed set as defined in the

literature of probability theory is a measurable map from a

probability space to the space of closed sets of a topological space

(where this space is equipped with the Fell topology and the

corresponding Borel sigma-algebra). In particular a random closed set

of Cantor space is simply a measurable map from any probability space

to the space of closed sets of Cantor space.

This talk addresses one way of reconciling these two notions of random

closed set. First we use the probability theory framework to develop a

measure-based theory of randomness for the space of closed sets of

Cantor space. Then we show that random closed sets in the sense of

Barmpalias et. al. are a specific example of randomness with respect

to this new framework. To conclude we will look at some other examples

of randomness for closed sets that arise from this framework.

4:30-5:30: Joe Miller - U. of Wisconsin.

Title: What good is always guessing wrong?

Abstract: We'll discuss connections between diagonally noncomputable functions and algorithmic randomness.