DATE: Tuesday, November 11, 2008

PLACE: Ryerson. University of Chicago.

1100 East 58th Street, Chicago, IL 60637.

Schedule:

Lunch: 12:00 at Ryerson 352 (the Barn), catered from the Medici.

Talks (at Ryerson 251):

- 12:45-1:45: Chris Conidis - U. of Chicago.

New Results in Effective Ring Theory

- 2:00-3:00: Keng Meng
(Selwyn) Ng - U. of Victoria, Wellington.

Jump Inversion for truth table degrees.

- 3:00: coffee break in Ryerson 255.

- 3:40-4:40: Peter Gerdes - U. of Notre Dame.

Computable in Every Majorizing Function.

- 4:40: coffee break in Ryerson 255.

New Results in Effective Ring Theory

Abstract: We will examine ring theory from both the points of view of effective algebra and reverse mathematics. In particular, we shall present new results which shed light on the reverse mathematical strength of the classic theorem from commutative algebra which says that every Artinian ring is Noetherian. We also determine the effective set theoretic complexity of the Baer-McCoy radical (i.e. lower nilradical), and Levitzki radical in noncommutative computable rings.

Keng Meng (Selwyn) Ng - U. of Victoria, Wellington.

Jump Inversion for truth table degrees.

Abstract: By formalizing relative computability one has Turing reducibility, which is the most general effective reducibility between sets of natural numbers. If we place restraints on oracle accessibility, we can also look at various strong reducibilities. The Turing jump operator is also another important concept in modern computability theory, and has been studied widely in conjunction with definability issues. We will prove a result involving these two concepts, starting by reviewing related notions, and the classical jump inversion theorems. We will then discuss a recent result of Anderson, which proves the Friedberg jump inversion for tt-degrees. We will then show that the Sacks' version of jump inversion fails for the tt-degrees.

Peter Gerdes - U. of Notre Dame.

Computable in Every Majorizing Function

Abstract: We characterize of the functions that are uniformly computable in every majorizing function as the \( \Pi^0_1 \) singletons and present some results about these functions. In particular we present a new proof of a corollary of Harrington's showing that there are non-computable \( \Pi^0_1 \) singletons computing no non-computable \( \Delta^0_{\alpha} \) sets. We will also discuss the more general situation of functions that are merely computable in every majorizing function to the extent that time permits.