This summer some model theorists around Berkeley have been meeting to discuss things around groupoids and amalgamation as well as galois cohomology and variants. Basically, Alex Kruckman discussed some of the ideas relating amalgamation problems to groupoids which was developed in a series of papers. Alex has given me his notes which we discussed over a series of lectures.
In a slightly different direction, but with similar ideas in mind, Adam Topaz gave some informal lectures on simplicial sets. I don’t have a reference, but a lot of the pictures that Adam drew looked like this.
The other things we talked about this summer had more to do with model-theoretic versions of galois cohomology, along the lines laid down by Pillay. I outlined Pillay’s paper and some recent work of Sustretov on elimination of imaginaries and Galois cohomology. In particular, we discussed how the non-definability of various Zariski geometries relates to elimination of certain generalized imaginaries.
This post might be edited as various people send me notes (or if I write some down).
Recently, we had a special day-long seminar at MSRI on difference and differential algebra; my notes are linked at the end of the post. Of course, I took this as a perfect time to give a talk not directly about differential algebra… I gave a talk about superstable groups, of which differential algebraic groups are a specific example. Normally when one wants to specialize results from stability theory to differential algebra, you do things directly because fits into whatever stability class you are considering. Sometimes the specialization does not work, especially when you assume finite rank. When you try to generalize results from differential algebra, similarly, you just see what sort of assumptions are being used and which you can get around.
I can’t promise, but I think this is the only post I will make which involves a French prime minister. The sort of differential equations which Ronnie talked about last week are named after Painlevé; they were originally isolated in studying the Painlevé property: the only movable singularities are poles. In particular, the Painlevé equations are families of order two differential equations which take parameters. It turns out that all of the solutions of second order equations with this property can be expressed in terms of solutions to one of six families of equations. Showing that these six equations (for generic values of the parameters) are actually irreducible (I don’t mean irreducible in the Kolchin topology) was a longstanding open problem, which was solved by Nishioka and Umemura. Essentially developing the right notion of irreducible was part of the issue; you can read the definitions and a nice back story of the problem in Ronnie’s first paper with Anand Pillay, where they imported the technology of Nishioka and Umemura and used model theoretic techniques to prove more. They also give a very detailed account of the story I just sketched above.
A few weeks back in the Berkeley model theory seminar, Will Johnson gave a talk on elimination of imaginaries in algebraically closed valued fields. In case you missed the talk and aren’t familiar with the subject, I will make a few comments (and give a few links) before linking to Will’s article.
You probably know about elimination of imaginaries if you are a model theorist. If not, let me give a very short explanation. Suppose that you have a structure with at least two -definable elements (I am writing to mean definable without parameters). If you are a non-model theorist, you are probably already interested in structures which have at least two -definable elements, since fields in the language of rings are an example. Now suppose that you have a -definable equivalence relation on . Then has elimination of imaginaries if there is an -definable function for some such that . Again, since you are reading this paragraph, you are likely a non-model theorist, so it might not be clear why this is useful. The most obvious answer is that this allows one to view naturally occurring quotients which your structure interprets as definable sets (in the above formulation, this is the image of the function ).
Let be an algebraically closed field with a nontrivial valuation. Let denote the valuation ring and the maximal ideal. denotes the residue field. Here is the basic problem: in ACVF there are certain naturally interpretable sets which can not be viewed as definable sets (if you are a model theorist or you just read the previous paragraph, then you know what I mean). The first example of this situation in ACVF consists of the which are isomorphic to . Such objects are called lattices. denotes the set of lattices in . This set can be identified with . One also has to add sorts which consists of .
The main theorem which Will talked about is due to Haskell, Hrushovski and McPhereson who write that they first got the idea for their approach from Holly. The theorem says that ACVF with additional sorts and eliminates imaginaries. The theorem has a certain tightness as well; ACVF does not eliminate imaginaries in any finite set of sorts. Will has produced a shortened and perhaps more conceptual version of the proof of the theorem.
Now that the semester has started again at Berkeley, there will be occasional posts with seminar notes from the model theory seminar appearing here. This week John Baldwin is talking about Vaught’s conjecture, which you have probably heard about before.
In case you are wondering, the red herrings are:
That descriptive set theory plays a central role in finding models with absolute indiscernibles.
That the existence of a model in
rather than the embeddability relation in
That complexity of individual models without embeddability conditions is a sufﬁcient tool.
After John’s talk, slides and perhaps some more notes will appear here. For now, here is a link to the paper on which this talk is based. Update: Here is a link to the updated paper. John writes, “This links to the submitted version of the paper with Laskowski and Koerwien that I presented in Berkeley in January. After some alarums and excursions the results are actually stronger than before.”
Also, for those curious about notes from last week’s talk from Will Johnson, those may eventually appear, but not this week. Also, since the seminar audience has grown quite a bit this semester, there may be new participants looking at this site and these notes. Please feel free to leave comments.
Tomorrow I will give the last seminar talk of the semester, and it is generally a part of a biweekly series of talks about NIP. Here is a link to my notes, which are handwritten for the same reason that there have been no posts lately (I have been traveling and busier).
Part of the seminar time tomorrow will be a planning meeting for the Spring semester, when there will be a lot of model theory in Berkeley. So, the natural thing to do when you only have part of the time for the seminar is to pick multiple things to talk about. In this case, I don’t have a lot to say about either of the topics. You may recall that we generally worked through the paper On nip and invariant measures (we actually got through most of it). This week I am going to talk about two unrelated papers, both of which might be related to various things going in the paper we are following. Continue reading
This week in the model theory seminar (during our NIP thematic series), I am giving the talk. I am speaking about properties of NIP groups, all of which are consequences of the group having finitely satisfiable generics (FSG). This is a property which has proven useful in exporting techniques from stable group theory to the NIP context. For instance, Hrushovski, Pillay and Simon prove the equivalence of two types of generic types (both of which we study this week). Also, Ealy, Krupinski, and Pillay prove some analogues of results from groups of finite Morley rank (which had been previously generalized to various settings, which we will not mention here) under the assumption of FSG and superrosiness.