| 10:00-10:30 | Breakfast |
| 10:30-10:35 | Welcome and Announcements |
| 10:35-11:35 | Paul Baginski |
| 11:45-12:45 | Alexei Kolesnikov |
| 12:45-2:45 | Lunch |
| 2:45-3:45 | Alf Dolich |
| 3:45-4:15 | Coffee Break |
| 4:15-5:15 | Artem Chernikov |
The model-theoretic properties of stability and א0-categoricity have been studied extensively in the context of groups and rings. Individually, each property induces a rich framework upon these algebraic objects; the joint presence of these properties severely restricts the algebraic structures one may encounter. Baur, Cherlin, and Macintyre [2] and, independently, Felgner [3] showed in the 1970s that a stable, א0-categorical group is nilpotent by finite. Baur, Cherlin, and Macintyre further conjectured that a stable, א0-categorical group is abelian by finite, a conjecture which remains open.
Contemporaneously, Baldwin and Rose considered stable, א0-categorical associative rings and showed that they are nilpotent by finite. They further conjectured that such rings are null by finite, i.e. up to a finite ring extension, multiplication is trivial. This conjecture is equivalent to the conjecture on groups.
Rose generalized many of the results on stable rings and א0-categorical rings to the nonassociative context of alternative rings. However, he was unable to show that a stable, א0-categorical alternative ring is nilpotent by finite. We have succeeded in proving this conjecture and we will outline the proof. We will also discuss the obstacles in generalizing this result to other nonassociative rings.
I will discuss a recent project to improve bounds on sample complexity of differentially private learning algorithms. Here is the set up. Let X be a set and let C be a family of subsets of X. A learning algorithm inputs a finite set, called sample, of points of X labeled with 0 or 1, depending on whether they belong to some c* in C. The output of a learning algorithm is a probability distribution over C. It is known that a family of sets can be probably approximately correctly (PAC) learned with differential privacy if and only if it has finite Littlestone dimension (a property connected to model-theoretic stability). We study bounds on the size of the sample needed to achieve specified accuracy and privacy parameters. This is joint work with Vince Guingona and REU students.
Expansions of ordered groups of low dp-rank are good candidates for ``tame'' expansions of ordered groups. A basic examples of a result witnessing that small low dp-rank expansions are tame is Simon's result that an expansion of a densely ordered group which is definably complete and of dp-rank one is o-minimal, and thus by most reasonable. In this talk I will survey ongoing work with John Goodrick on what we can say about definable sets in expansions of ordered groups which are of low dp-rank (but bigger than one).
A randomization of a first-order structure M, introduced by Keisler, is a structure MR in continuous logic whose elements are the random variables on a fixed probability space taking values in M. Randomization preserves certain model-theoretic tameness properties, e.g. stability and NIP. However, it was observed by Ben Yaacov that the randomization of a simple unstable structure is never simple.
We demonstrate however that Keisler randomization of an NSOP1 theory is NSOP1, and describe Kim-independence in randomizations of simple theories. As an application, we obtain a 3-amalgamation statement for Keisler measures in simple theories, generalizing from types. This appears to be the first general positive result suggesting that Keisler measures in simple theories exhibit tameness properties, as opposed to some recent negative results.
Joint work with Itaï Ben Yaacov and Nicholas Ramsey.