Graduate Logic Seminar

The Graduate Logic Seminar is an informal space where graduate students and professors present topics related to logic which are not necessarily original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.

• When: Mondays 3:30-4:30 PM
• Where: Van Vleck B211
• Organizer: Joseph Miller

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact the organizers.

Fall 2025

The seminar will be run as a 1-credit seminar Math 975. In Fall 2025 students will present a logic topic of their choice (it could be original work, but does not have to be). If you are not enrolled but would like to audit it, please contact Joe Miller.

Presentation Schedule: Sign up here.

September 8 - Organizational Meeting

We will meet to arrange the schedule

September 15 - Contrasting the halves of an Ahmad pair

Abstract: We study Ahmad pairs in the $\Sigma^0_2$ enumeration degrees. We say $(A,B)$ form an Ahmad pair if $A \not \leq_e B$ and every $Z <_e A$ satisfies $Z \leq_e B$.  Ahmad pairs have recently drawn interest as they are a key obstacle in solving the $\forall\exists$ theory of the local structure.

In this talk we characterize the left halves of an Ahmad pair as precisely the low$_3$ and join irreducible degrees. We also show that right halves cannot be low$_3$. 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 low$_3$ 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.

Previous Years

The schedule of talks from past semesters can be found here.