Midwest Computability Seminar

Midwest Computability Seminar

XXXV


The Midwest Computability Seminar is a joint venture between the University of Chicago, the University of Notre Dame, the University of Wisconsin-Madison, and the University of Illinois Chicago. It meets once or twice per semester, and is attended by faculty and students from these universities and others in the area. The seminar started in the fall of 2008.




SLIDES:    Miller    Ravishankar    Turetsky



DATE: Thursday, September 25th, 2025

PLACE: Eckhart Hall 202, The University of Chicago
5734 S. University Ave., Chicago IL


REMOTE ATTENDANCE: https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09
Meeting ID: 997 5433 2165
Passcode: midwest


Speakers:

Schedule:


Abstracts:


Russell Miller

Title: Nice Galois groups and nasty ones

Abstract: The absolute Galois group Gal(F) of a field F is the Galois group of its algebraic closure F relative to F, containing precisely those automorphisms of F that fix F itself pointwise. Even for a field as simple as the rational numbers ℚ, Gal(ℚ) is a complicated object. Indeed (perhaps counterintuitively), Gal(ℚ) is among the thorniest of all absolute Galois groups normally studied.

When F is countable, Gal(F) often has the cardinality of the continuum. However, it can be presented as the set of all paths through an F-computable finite-branching tree TF, built by a procedure uniform in F. After a formal definition of tree presentations of continuum-sized structures in functional signatures, we will consider the basic properties of this tree TF, which depend in some part on F.

Next we will address questions about the subgroup consisting of the computable paths through this tree, along with other subgroups similarly defined by Turing ideals. One naturally asks to what extent these are elementary subgroups of Gal(F) (or at least elementarily equivalent to Gal(F)). This question is connected to the computability of Skolem functions for Gal(F), and also to the arithmetic complexity of definable subsets of Gal(F). When F=ℚ, we have a few answers -- one of them joint with Debanjana Kundu -- and many more questions. In the simpler situations of the absolute Galois group of a finite field, or of the Galois group of the cyclotomic field over ℚ, much more is known, thanks in part to joint work by Jason Block and the speaker.


Karthik Ravishankar

Title: Contrasting the halves of an Ahmad pair

Abstract: We study Ahmad pairs in the Σ02 enumeration degrees. We say (A,B) form an Ahmad pair if A ≰e B and every Z <e A satisfies Z ≤e B. Ahmad pairs have recently drawn interest as they are a key obstacle in solving the AE theory of the local structure.

In this talk we characterize the left halves of an Ahmad pair as precisely the low3 and join irreducible degrees. We also show that right halves cannot be low3 . This is a natural separation between the two halves and is a significant strengthening of previous work.

We then define a hierarchy of join irreducibility notions using which we characterize the left halves of Ahmad n-pairs as those that are low3 and n-join irreducible. This allows us to extend and clarify previous work to show that for any n there is a set A which is the left half of an Ahmad n-pair but not of an Ahmad (n+1)-pair.


Daniel Turestky

Title: Particular lightface and boldface complexities somewhat above Σ11

Abstract: What is the complexity of Muchnik reducibility? At first glance it's Π12, but maybe we can do better? There's a natural complexity class, sitting above Σ11 but within Δ12, which I'll discuss. Muchnik reducibility and another reducibility based on particular embeddings of linear orders both look to sit at this level, and I'll present partial results for this.


Previous Seminars:

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