Computability Theory, Set Theory and Geometric Measure Theory

Theodore A. Slaman (University of California Berkeley).

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Course summary:

We will explore issues of Geometric Measure Theory using methods from Computability Theory and Set Theory. We will not assume any prior knowledge of these areas.

About the lecturer:

Theodore Allen Slaman is a professor of mathematics at the University of California, Berkeley. He works in the area of computability theory, broadly interpreted. Within the traditional boundaries of the subject, he has worked on the structure of Turing degrees and on its substructures given by the computably enumerable degrees, the $\Delta^0_2$-degrees, and others. Slaman has contributed other areas within mathematical logic: descriptive set theory, models of Peano arithmetic, and subsystems of second order arithmetic. Slaman has also explored other areas of mathematics that are amenable to a computability-theoretic perspective: randomness (where there is a well-established connection with computability theory), the theory of normal numbers and uniform distributions, and most recently Hausdorff dimension.

Selected results:

Computability Theory The Turing jump is definable from the order induced by relative computability (Shore and Slaman 1999). The automorphism group of the Turing degrees is countable (Slaman and Woodin 1986). The extension of embeddings problem for the computably enumerable degree is decidable (Slaman and Soare 2001).

Set Theory
If there is a nonconstructible real then every perfect set has a nonconstructible element (Groszek and Slaman 1998). There is a set $U$ such that every uncountable analytic set is the image of $U$ under a continous injective map (Slaman 1999)

Peano Arithmetic
$\Delta_n$-induction is equivalent to $\Sigma_n$-bounding, over the base theory of $\Sigma_0$-induction plus the exponential function is total. (Slaman 2004)

Subsystems of Second Order Arithmetic
Ramsey's theorem for pairs is $\Pi^1_1$-conservative over $RCA_0$ plus $\Sigma_2$-induction. (Cholak, Jockusch and Slaman 2001) Stable Ramsey's theorem for pairs is stricty weaker than Ramsey's theorem for pairs (Chong, Slaman and Yang (2014)

Normal Numbers
Characterization of sets $M$ such that there is a real number which is simply normal an integer base $b$ if and only if $b$ is an element of $M$ (Becher, Bugeaud and Slaman 2015)

Hausdorff Dimension
It is consistent with ZFC such that there is a coanalytic subset of the reals $P$ which has Hausdorff dimension 1 and every closed subset of $P$ is countable (Slaman to appear)

Thursday, May 26th in 5440
14:15 - 15:45 lecture
16:15 - 17:15 class
Friday, May 27th in 5440
14:15 - 15:45 lecture
16:15 - 17:15 class
Saturday, May 28th in 5440
10:15 - 11:45 lecture
12:15 - 13:15 class

Materials Lecture Exercises