Tuesday, 1/29:

Wednesday, 1/30:

Tuesday, 2/5:

Wednesday, 2/6:

Tuesday, 2/12:

Wednesday,2/13:

Tuesday, 2/19:

Wednesday, 2/20:

Tuesday, 2/26:

Wednesday, 2/27:

Tuesday, 3/5:

Wednesday, 3/6:

Tuesday, 3/12:

Wednesday, 3/13:

Tuesday, 3/26:

Wednesday, 3/27:

Tuesday, 4/2:

Wednesday, 4/3:

Tuesday, 4/9:

A Borel equivalence relation is hyperfinite if it is the increasing union of Borel equivalence relations having finite classes. A long-standing open problem in descriptive set theory asks if Borel actions of countable amenable groups always induce (via their orbits) hyperfinite equivalence relations. In this talk I will discuss joint work with Scott Schneider which shows that this question has a positive answer for free Borel actions of countable locally nilpotent groups.

Wednesday, 4/10:

Group-theoretic rigidity techniques such as Zimmer and Popa cocycle superrigidity have been instrumental in works of Adams-Kechris, Thomas, and Hjorth (among others) in realizing complexity in the partial order of Borel reducibility among countable Borel equivalence relations. We introduce an elementary notion of rigidity which interacts better with Borel reducibility, allowing us to localize various complexity results to just above measure hyperfinite in the class of treeable equivalence relations. This is joint work with Ben Miller.

Tuesday, 4/16:

We give the first examples of computationally complicated residually finite finitely presented groups. This is a joint work with Olga Kharlampovich and Alexei Miasnikov.

Wednesday, 4/17:

Tuesday, 4/23:

Constructions of layered Borel complete sections with regularity properties are important for hyperfiniteness proofs and general study of countable Borel equivalence relations. A classical theorem of Slaman-Steel gives the existence of vanishing layers of Borel complete sections. In this talk I prove a boundedness property for any layered Borel complete sections for the Bernoulli shift on Z. This implies that layered Borel complete sections with certain desirable properties do not exist. This is joint work with Steve Jackson and Brandon Seward.

Wednesday, 4/24:

Tuesday, 4/30:

We find two computability theoretic properties on the models of a theory T which hold if and only if T is a counterexample to Vaught's conjecture.

Wednesday, 5/1:

Tuesday, 5/7:

Algorithmic randomness defines what it means for a single mathematical object to be random. This active area of computability theory has been particularly fruitful in the past several decades, both in terms of expanding theory and increasing interaction with other areas of math and computer science. Randomness can be equivalently understood in terms of measure theory, Kolmogorov complexity (incompressibility), and martingales.

In this context, we present a novel definition of betting strategies that uses probabilistic algorithms also studied in complexity theory. This definition leads to new characterizations of several central notions in algorithmic randomness and addresses Schnorr's critique, a longstanding philosophical question in algorithmic randomness. Moreover, these techniques have yielded new proofs of complicated separation theorems and suggest new approaches for tackling one of the biggest open questions in the field (KL = ML?). This is joint work with Sam Buss.