Calendar

Jan
24
Thu
Kameryn Williams: Logic seminar @ Keller 313
Jan 24 @ 2:30 pm – 3:20 pm

Title: Amalgamating generic reals, a surgical approach
Location: Keller Hall 313
Speaker: Kameryn Williams, UHM

The material in this talk is an adaptation of joint work with Miha Habič, Joel David Hamkins, Lukas Daniel Klausner, and Jonathan Verner, transforming set theoretic results into a computability theoretic context.

Let $\mathcal D$ be the collection of dense subsets of the full binary tree coming from a fixed countable Turing ideal. In this talk we are interested in properties of $\mathcal D$-generic reals, those reals $x$ so that every $D \in \mathcal D$ is met by an initial segment of $x$. To be more specific the main question is the following. Fix a real $z$ which cannot be computed by any $\mathcal D$-generic. Can we craft a family of $\mathcal D$-generic reals so that we have precise control over which subfamilies of generic reals together compute $z$?

I will illustrate a specific of this phenomenon as a warm up. I will show that given any $\mathcal D$-generic $x$ there is another $\mathcal D$-generic $y$ so that $x \oplus y$ can compute $z$. That is, neither $x$ nor $y$ can compute $z$ on their own, but together they can.

The main result for the talk then gives a uniform affirmative answer for finite families. Namely, I will show that for any finite set $I = \{0, \ldots, n-1\}$ there are mutual $\mathcal D$-generic reals $x_0, \ldots, x_{n-1}$ which can be surgically modified to witness any desired pattern for computing $z$. More formally, there is a real $y$ so that given any $\mathcal A \subseteq \mathcal P(I)$ which is closed under superset and contains no singletons, that there is a single real $w_\mathcal{A}$ so that the family of grafts $x_k \wr_y w_\mathcal{A}$ for $k \in A \subseteq I$ can compute $z$ if and only if $A \in \mathcal A$. Here, $x \wr_y w$ is a surgical modification of $x$, using $y$ to guide where to replace bits from $x$ with those from $w$.

Mar
5
Tue
Logic seminar: Quinn Culver
Mar 5 @ 2:45 pm – 3:35 pm

Keller Hall 301

Abstract: My plan is to go through (as much as time will allow of) Measure and Integrals in Conditional Set Theory by Jamneshan et al. with the goal of getting to at least one theorem there that witnesses the merits of conditional set theory.

Mar
12
Tue
Logic seminar: Quinn Culver
Mar 12 @ 2:45 pm – 3:35 pm

Keller Hall 301

Abstract: My plan is to go through (as much as time will allow of) Measure and Integrals in Conditional Set Theory by Jamneshan et al. with the goal of getting to at least one theorem there that witnesses the merits of conditional set theory.

Mar
22
Fri
Kenshi Miyabe: Muchnik degrees and Medvedev degrees of the randomness notions
Mar 22 @ 3:00 pm – 3:20 pm
Jan Reimann: Turing Degrees and Randomness for Continuous Measures
Mar 22 @ 3:30 pm – 3:50 pm
Denis Hirschfeldt: Open Questions on Notions of Dense Computability
Mar 22 @ 4:00 pm – 4:20 pm
Mar
23
Sat
Douglas Cenzer
Mar 23 @ 9:00 am – 9:20 am
Thomas Zeugmann
Mar 23 @ 9:30 am – 9:50 am