Calendar

Jan
23
Wed
6.7
Jan 23 @ 9:30 am – 10:30 am
Jan
24
Thu
Kameryn Williams: Logic seminar
Jan 24 @ 2:30 pm – 3:20 pm

Amalgamating generic reals, a surgical approach

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 $w$.

Jan
25
Fri
6.8
Jan 25 @ 9:30 am – 10:30 am
Jan
28
Mon
7.1
Jan 28 @ 9:30 am – 10:30 am
Jan
30
Wed
Meeting #10
Jan 30 all-day
Review
Jan 30 @ 9:30 am – 10:00 am
Feb
1
Fri
Midterm 1
Feb 1 @ 9:30 am – 10:00 am
Feb
4
Mon
7.2
Feb 4 @ 9:30 am – 10:30 am
Feb
6
Wed
7.3
Feb 6 @ 9:30 am – 10:30 am
Feb
8
Fri
7.4
Feb 8 @ 9:30 am – 10:30 am
Feb
11
Mon
7.5
Feb 11 @ 9:30 am – 10:30 am
Feb
13
Wed
7.7
Feb 13 @ 9:30 am – 10:30 am
Feb
15
Fri
7.8
Feb 15 @ 9:30 am – 10:30 am
Feb
20
Wed
8.4 Econ/bio (skip)
Feb 20 @ 9:30 am – 10:30 am
8.5 Probability (can skip, won’t)
Feb 20 @ 9:30 am – 10:30 am
Feb
22
Fri
9.1
Feb 22 @ 9:30 am – 10:30 am
Feb
25
Mon
Meeting #20
Feb 25 all-day
9.2
Feb 25 @ 9:30 am – 10:30 am
Feb
27
Wed
Review
Feb 27 @ 9:30 am – 10:30 am
Mar
1
Fri
Midterm 2
Mar 1 @ 9:30 am – 10:30 am
Mar
4
Mon
9.3
Mar 4 @ 9:30 am – 10:30 am
Mar
6
Wed
9.4 Population (won’t skip)
Mar 6 @ 9:30 am – 10:30 am
Mar
8
Fri
9.5 Linear diff. equations
Mar 8 @ 9:30 am – 10:30 am
Mar
11
Mon
10.1
Mar 11 @ 9:30 am – 10:30 am
Mar
13
Wed
10.2
Mar 13 @ 9:30 am – 10:30 am
Mar
15
Fri
10.3
Mar 15 @ 9:30 am – 10:30 am
Mar
22
Fri
Kenshi Miyabe
Mar 22 @ 3:00 pm – 3:20 pm
Samuel Birns
Mar 22 @ 3:30 pm – 3:50 pm
NN2
Mar 22 @ 4:00 pm – 4:20 pm
NN3
Mar 22 @ 4:30 pm – 4:50 pm