General Information

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Project Description

As preliminary work, we are trying to decide certain conjectures about locally-embeddable-(into)-finite (LEF) groups. These conjectures are answers to questions which some researchers LEF-group theory are interested in. It seems that to decide any one of these conjectures, one must assume an axiom which is independent of ZFC. For each infinite cardinal ℵ₀ ≤ κ < 2^ℵ₀, we aim to understand whether it is possible to characterize each LEF group, G, of cardinality at most κ under the assumption of Martin's axiom at and below κ. If we show that it is consistent with ZFC (by proving it from ZFC + MA(κ), or something), we'd like see exactly how strong it is and whether it is independent of ZFC, so that we might understand the nature of these ultraproduct constructions. It seems that it might get more subtle when the cardinality of G is the continuum, and that there might be no analogous theorem for LEF groups of size continuum. Nevertheless, understanding this would allow us to understand LEF groups as a whole and how they depend on the underlying set theory.

There are going to be analogous questions for sofic groups. There has already been a characterization of LEF groups as those which are isomorphic to a certain kind of ultraproduct; indeed, our research on LEF groups has pertained to their characterization in terms of more specific kinds of nonprincipal ultraproducts and whether an LEF group is embeddable into the reduced product of all the symmetric groups. There is an analogous construction for sofic groups in terms of metric ultraproducts (which is not the standard model-theoretic ultraproduct of a continuous, or whatever, group equipped with a distance metric), and, I imagine, an analogous kind of work we'll have to do there to answer those looming questions. Regardless, it is necessary to understand the ultraproduct constructions for each LEF group before one can pose questions about the metric ultraproduct of sofic groups. More on this later.

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Weekly Log

Week 0:

Let's include all relevant information about my work on this project before the start of Week 1. Before the program officially began, I spent a bit of time making sure that I could trace all of the constructions, theorems, and definitions, in the material that Simon sent me, down to terms and methods I was already familiar with. I don't expect to get a perfect intuition on this area before the end of this program, in Simon's specific field (indeed I already have some in other fields), but with his guidance I will certainly make it further in this field, by the end of the summer, than I would have on my own in this time frame. I've found it important to realize when I am trying to verify/falsify a conjecture. One must be careful to never assume something without knowing that it is true when researching -- it will quickly metastasize. I've been sticking to Simon's advice and suggestions for diections to inquire. I'm trusting him like a black box, or an oracle, for similar reasons and of course I try to find the bigger pictures and ideas of the area with his guidance.

We have a theorem already, a special case of what one might call ''the preliminary research question''. It is a positive result which helps us characterize countable LEF groups. When I was given this conjecture, I knew that, by definition, I could not assume that it was true. So, I tried to derive what I could from the assumptions to see if I could get near the conclusion. Simon played a major role in resolving this conjecture and I've been very careful to make sure I know how he did what he did. Simon's advice is that MA(κ) will resolve this in general. I began trying to understand how one might translate Simon's complete proof into one which explicitly cites MA(ℵ₀). I had a few ideas, none of which worked-- ideas about what poset might be the ccc one, and what the family of dense subsets might be. Balancing my time with direct attacks at the conjectures, I'm studying Chang and Keisler's classic on model theory in addition to a few survey papers and technical papers that Simon has sent me.

Week 1:

I didn't read too much this week (except for Dostoevsky: I am waiting for Rogozhin to reencounter Prince Myskin after the former attempted to murder the latter)--I spent a lot of time deciding conjectures (modulo a Socratic exam on Monday the 12th). Simon and I found a very nice poset of finite approximations to a total function from G to the total product of all the finite symmetric groups which is an monomorphism of groups in all but finitely many coordinates. We spent a while discussing how one works backwards when constructing a poset, first picking what a generic filter must be, what it should mean to strengthen an element, and what the dense sets should encode (tasks), and hence how to use Martin's axiom at and below a cardinal κ. I wrote this up and I am rather sure that this is how one works when researching axiomatic logic in general, by assuming a consistent yet independent axiom to do something extra, and having a crystal clear picture of what it would take to get where you want, and a rather nice intuition about what each statement which follows from the axiom's assumption would mean or do for you in the context of your problem.

It suffices to show that certain sets are dense and that the poset is ccc, of course, so that we might apply MA(κ). I believe that I've done this--Simon will check on Monday and a more complete version of the argument will be written up. The hardest part is deciding whether the poset is ccc. My method is a modified version of Jech's Lemma 14.35. Under Martin, we've added another clause to spice up the conversation on independence. Of course, to show independence, it would suffice to find a model of ZFC for which at least one of the conditionals does not hold. Looking at the proof we've got, I was able to determine which entailments are provable in ZFC and which are candidates to falsify in an extension.

Week 2:

The meeting on Monday the 12th went well. Simon has been giving me intuition and methods for solving certain problems. Around the same time whenever one is to use the Δ-System-Lemma, one would also be interested in uniformizing all of the elements of the resulting Δ-System, just to make the comparison of elements in the system easy (when all you care about is the size of the system). When the elements of the Δ-System have only a countable amount of data specifying each, then such uniformization usually makes your life easier, as you can invoke "Wlog, each element in my Δ-System [uniformly] has this property" and then it is easy to find a common extension among any two (although, I think that this might get more subtle in other situations). We found two ways to uniformize which would make the argument for the existence of two comparable elements very obvious (in any uncountable subset, hence making it ccc). The informal proof which I showed him was on Monday was invalid (in the formal sense of an argument) because I did not make another uniformization, although I uniformized a lot.

I was then given an assignment to understand a result. This result is a negative result about a certain kind of Cohen forcing. I have some ideas about how this might be useful when trying to use forcing to build a model of ZFC where our Theorem fails. The idea is that we're to cook up a model of ZFC and a LEF group in that model for which the Theorem fails. The conditional proof where we used MA(κ) to construct a generic filter which encoded a map from G to the full direct product of the finite symmetric groups which is a group-monomorphism in all but finitely many places is the place where I am looking to break our Theorem, the first place; at least--because it seems to depend so directly on MA(κ) which is independent of ZFC. This has been making me wonder about the spirit of MA(κ) and how much our Theorem depends on it. Simon has told me that our Theorem is not equivalent to it but I have seen no proof. But, I am rather sure that there is some common extension of the two which is ... this is what I mean by spirit, although I should have said "relative" or ''mutual spirit''.

Other than that, I've been preparing ideas for independence proofs, accumulating potential-Lemmas, but also trying to work backwards --- but this has required me to think about the mutual spirit of our Theorem with lots of things, things which I know are independent of ZFC. Other than that subother, I've been reading Kunen's book, which is better than Jech's monograph for some things--although not all--and it has cleared up some beliefs, notions, feelings and intuitions (as well as crystallized and strengthened some methods) which were not obvious when Dima's course ran. Sometimes it is refreshing to see another side of a characterization theorem stated as a definition and then to hear the informal commentary about the novelty of this result. I also finished Dostoevsky's book, "The Idiot", and now I'm going to get either some Tolstoy or the rest of Dostoevsky. I also took 13 people out to Jazz on the 13th --- and was massively thanked, which was nice. The Jazz was smooth and the night was nice. I will probably bring the group out there again, although not this next week. This past week was more relaxed with respect to this work, in some sense which I don't care to make precise---although, all aspects of this (and all my other projects) will get more intense and productive from now on (for instance, Simon and I will be meeting twice each week from now on).

Week 3:

On Monday the 19th I presented to Simon what I had believed was a complete proof of the Conjecture's consistency. I had believed that during this set time I would be demonstrating my understanding of another Lemma he had assigned me last week, but he surprised me with this, which was great and probably more useful. It was important to double check that I understood all of the moves he and I had made and what we had done, that is, those conclusions which he deemed important or which he deemed as progress towards solving the problem (the larger one, now). I expect him to have great insight and intuition about what it means to make progress in this field. I came to appreciate the poset and the methods he has been introducing me much more when I was forced to do this summary of our work on the blackboard. We met again on Tuesday the 20th and I did a nice job proving the other Lemma to him.

I rewrote my summary of our work, plus my demonstration/paraphrase of this Lemma. They're basically fine, modulo one justification I haven't found yet. I know exactly where we are in solving our supreme LEF-research problem and precisely what Simon thinks may work. Simon thinks that this Lemma will be important, and that in a particular Cohen-forcing extension of ZFC we might be able to falsify the conjecture and hence show its independence. Paired with this Lemma I've been reading Brian P. Hall's 1961 paper "Wreath powers and characteristically simple groups". Simon believes that we can construct a LEF group which is a Wreath product of other group actions in that nice Cohen extension, and then assume one of the conditionals of our conjecture to get a contradiction with the Lemma. Tomorrow June 26th I will present to him the highlights of The Theory of Wreath Products and then show my progress and ideas. Hopefully we can fix the independence soon. I have other lingering questions, but I've kept them separate and organized; for instance, with nonprincipal ultraproducts. There's lots. I'll put this here although we can pretend (META) that it is prefacing this journal ... this is all VAGUE on purpose.

My mother visited this past week. I took her to Jazz on Tuesday and got breaded eggplant on Wednesday. I'm almost done reading Strevens' "The Knowledge Machine." He is too much of a constructive empiricist for me. Science benefits from ideology and he gives an incomplete account of scientific explanation and this lets him argue what he does so that he doesn't contradict himself when he says that "scientific reasoning is subjective" but that "official arguments from evidence are as objective as possible" and that "science has become humanity's supreme source of truth because of the latter". If he were to give a nice account of scientific explanation, then his thesis would fall apart. Science has gotten a lot from ideology ... that's what most revolutions have been and ideology is at the heart of explanation. If there is no metaphysics or ontology, then there is no science. He has dismissed many open problems in the philosophy of science, like theoretical equivalence. I should write "The Wealth of Philosophy" to counter him and Karl Marx's "The Poverty of Philosophy" ... .

Week 4:

I presented the Theory of Arbitrary Wreath Products fairly well on that Monday but Simon ended up teaching me some great intuition. I've come to appreciate his intuition as I write up the details of what we think will be a counterexample in this model of ZFC. When the factors of the wreath product are all isomorphic, there is a nice higher level symmetry always. There is too much symmetry for C₂ so we are working with A₅. On Thursday I was supposed to show more details, but I only showed that there was more we both needed to clarify. I've been checking all the Lemmas although I have taken a break this holiday weekend; he and I won't meet this Monday as he will be celebrating. Also, I won't be explicitly citing the "Lemma" (from last week(s)), like I was trying to. I'm finishing the analogue Lemma for that Lemma now. Mostly, I need to write things up.

I will send him the whole proof of independence tomorrow or Tuesday. We won't meet again until Thursday. I can get this finished up and then it is up to me to find the analogous conjectures (although I have two monographs of Pestov which I read modulo some details) for Sofic Groups, as well as introduce the area to him. This is our Socratic routine. Other than that, I read some books by Greg Chatin and some philosophy papers ... and inadvertently watched the entire season of "FUBAR" on Netflix with my cat (which should not have happened).

Week 5:

We did not meet on Monday June 3 because Simon was celebrating in NYC. On Thursday, I did a decent job summarizing how our work has gone and what we have done, and the new ideas I had. I was very excited because I thought I had a way to show the consistency of the statement at the continuum, using Diamond and CH. I believe in the spirit of my idea, as it is vague. It did not seem to work. Simon unearthed some parts of the argument for the conjecture failing, below the continuum, which I was still shaky on. I have now cleared those up. We then started to talk about sofic groups, although we did not get the most far there.

I had a very productive time after we met on Thursday, then on Friday, Saturday, and Sunday. I pushed our methods and results to their limit and then wrote it all up nicely and whence have sent it all this Sunday, July 9th. Then, I wrote a document stating how the work should continue for sofic groups. Finally, I included some of my ideas to help understand the area better. I am interested to see whether Simon is impressed. I found a theme in our results; and then posed some interesting questions and conjectures. I brought all of our work together and finished a few problems/answered a few questions/proved a few theorems we were working on. We, of course, will meet tomorrow to discuss the work. I expect to fix any errors in my summarizing document, and then hone some of those ideas and then work on the sofic problems this week. Unfortunately our regular meeting time is amiss because of the IBM trip. Hopefully Simon and I reschedule.

I read Klaus Schwab's "The Fourth Industrial Revolution". It was a bit listy and redundant. It could have used some more philosophy and content, and could have been four times as long. The book is a good idea, but it hasn't become anything more than that yet. It has helped me understand history little, but has motivated and inspired me more than that. I do believe that we are in the midst of another revolution. I hope to help, adapt, and take advantage of this fact.

Week 6:

We did meet on Monday. I learned that it should be impossible for my result to propogate in the way I had imagined at first. I have since been trying to understand how precisely that argument is invalid. I am beginning to see how the methods are independent of one another, like PFA (plus the poset I am currently trying to build) and the posets we built for Martin. It is very nice to see how this independent variable, cardinality of the LEF group, changes the dependent variable---the way that we can build an almost homomorphism into the direct product. On Monday, Simon gave me new assignments and introduced me to PFA. We did not meet on Thursday because of the trip to IBM. I had a great time at IBM; it was very inspiring. On top of that, I had a great time talking with Mattie about mathematics on the bus-ride there.

I sent Simon a very big email, just now, containing all my new work and ideas and arguments. I got comfortable with first order model theory for metric structures. This is a generalization (in a sense) of classical elementary model theory in terms of its semantics --- they're continuous. In another sense, the signatures are many-sorted. I got comfortable enough to do all of the sister arguments for sofic groups, to show the consistency of the new conjectures at the continuum. I also found a neat result to use our counterexample for LEF groups for sofic ones. Since I built the poset, I was able to bring all the independence and consistency metatheorems to light in a beautifully analogous way. I have more ideas now, and I am trying to finish the conversation with PFA---to show the consistency of the conjectures (for LEF and sofic), at the continuum, with not CH.

Unfortunately, I did not read many books this week. Despite the fact that I am writing this early Sunday morning, instead of really late on Sunday--like usual--I am certain that there will be no time for non-strictly-work-reading today. This is probably everything to be recorded.

Week 7:

We did indeed meet on Monday the 17th. I hopelessly rehearsed a chalk-talk for Friday's presentation. I actually ended up presenting via an electronic slideshow; I was an elegant iceberg. During that meeting, we talked about PFA, which, opposite to what I said last week, is probably going to be used to show that Martin's axiom is independent of our LEF conjecture, at the continuum. We met again on Thursday to rehearse, and I've garnered some world-class skills, for prepping for talks, which I believe will be useful for eternity. Most of the mathematical work I did this week was trying to understand PFA and testing out methods, straight from Baumgartner's Chapter, to see whether they have any analogues for embeddings of LEF groups, and also working to understand iterated forcing (in general, and in particular with countable and finite support). We cleared up some bad ideas I had about sofic groups, as well.

I read the first three chapters of Nelson Goodman's "Fact, Fiction, and Forecast." This is a beautiful book explaining his 'new riddle of induction', which is far more vicious than Hume's original. This is arguably so because Goodman takes advantage of self-reference (like Gödel's first Incompleteness Theorem). This is very important. I shall spend this last week, 'Week 8', trying to catch the last of the low-hanging fruit and writing up a final REU paper. There are lots of problems left, in most neighborhoods of this project. ...

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Presentations

• Presentation 0
• Presentation 1

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Additional Information

• My Mentor
  • Distinguished Professor Simon Thomas