Asaf Karagila
I don't have much choice...

Looking Back at the 2010s

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The decade is drawing to a close, and while it is entirely arbitrary, it's a great excuse to look back at this decade.

At the end of 2009 I started my senior year as an undergraduate. I both read the first part of "Introduction ot Cardinal Arithmetic" to get a hold on the basics of set theory, and also took my first course on set theory (I'm omitting the introductory course from my freshman year since that one covered very very basic set theory). I've studied with the wonderful Matti Rubin, and it was a fantastic course. Too bad that it focused almost solely on the axioms (i.e. how the axioms are not provable from others, etc.) and that we only spent a short time dealing with actual set theoretic topics (e.g. Solovay's theorem on partitions of stationary sets, etc.)

Nevertheless, I remember cramming before that exam. I've spent a week of 19 hours per day of proving all the theorems in Matti's notes, writing it into my own notes, then on the next day reproving the notes, ad nauseum. I would email him with questions at odd hours of the day (well, 4am counts as night I guess), and he would answer shortly thereafter. It was great. The reason for that was that I did not want to take the resit exam as many of my friends preferred (we had a big measure theory exam three days before the set theory one and we all focused on that one), so I wanted to make sure that I get a good grade. It worked.

At this point I already knew two things: I wanted to be a set theorist, and I wanted to know more about the Axiom of Choice. Matti had spent hours durings breaks, also in the preceding year, telling me how there was magnificent mathematical arguments and many open problems that were left there, in waiting, since the 1980s. At his advice, I approached Uri Abraham to be my advisor for my masters. The next few months were spent reading and learning the basics of forcing, although to be fair, it took another several years before I fully grokked the ideas of the method. But I do excel at using techniques that I do not fully understand, or so I'd like to believe (reading my masters thesis or my first paper is likely to make me claim the opposite, though). So that was more or less fine.

I was very naive when I started my masters, thinking that nobody had noticed some obvious questions around the construction of permutation models (with atoms), and I was certain that these conjectures were true and would be solved quickly. As I learned the forcing-based approach of symmetric extensions, I began translating my knowledge to that method. And that was fun too.

When I finished my masters I've decided to continue my studies with Menachem Magidor in Jerusalem. It was important to me that I do my Ph.D. at a place where set theory is very active, since Ben-Gurion was a terrible wasteland, and while I did go to Jerusalem about once a week, it's not the same when you're not actively there.

My summer project was spent thinking about the following thing: Symmetric extensions are similar to permutation models in the sense that they are submodels of a larger universe where we have choice and we define this intermediate model. But in the symmetric extensions case we have to use a group (and filter) from the ground model, whereas in the permutation model case we do not. For most uses this is irrelevant since the group and filter are somehow "very simple" anyway. But why should that be a thing? Why not have a method that allows us to use "new groups" and "new filters"?

I remember talking with Matti over the phone on this, and he suggested this might be difficult since generic objects tend to code generic information, or be copies of ground model objects. But nevertheless, I continued to try. And then I realised that actually what I would like to do is to iterate symmetric extensions, since there was no framework to do just that.

Speaking with Menachem at some point, he suggested that perhaps the Bristol model could be constructed by some method which extends symmetric extensions (and indeed that was one of the major parts of my dissertation, it is an iteration of symmetric extensions). Later that summer, when he agreed to take me on as a student and we sat down before he left for his sabbatical in Toronto (there was a research semester at the Fields Institute), we drafted a list of problems I might be interested in, and I added one at the end: iterating symmetric extensions.

Three months of trying to read more about Borel equivalence relations and other stuff, I felt that my heart is drawn to the symmetric extensions. So I wrote Menachem that I'm changing my direction and going that way. He was fine with it, which was great news for me. I was free to chase my heart's desire. Matti wrote one of my reference letters to the Hebrew University asked me to type it into the computer (and send him the file so he can approve it), it was his way of letting me know what he really thinks of me. He wrote there that the most important thing about me is that I am independent and I have many ideas of my own, so there's no need to tell me what to think about. I could not possibly disappint him by doing something someone else told me to think about.

And so I set out on a journey to understand how to iterate symmetric extensions, and how to construct the Bristol model by doing just that. Roughly a year later and I had some basic idea on what needs to be done, at least in principle. I've made a terrible arrogant mistake, though. Mohammad Golshani asked on MathOverflow if someone has a reference for the Bristol model. I wrote an answer and made a remark that I have high hopes of finishing the basic building blocks on that very day...

It took another year (and then some), of hard work, of a subset of \(\{\text{blood, sweat, tears}\}\), before Yair Hayut noticed some terrible mistake in my work that sent everything crashing down, crushing all my hopes of finishing my thesis on my schedule. It took a few long weeks with many long walks in the scorching summer afternoon through the ugly concrete neighbourhoods of Beersheba. But I figured out a way to resolve the problem Yair had pointed to. It took a bit longer, but I also managed to find a way around the roadblock that now stood between me and the Bristol model (this, to the initated, was the birth of "productive iterations"). So it took another year, but I finally reached a point where I could say that I have this under wraps.

Phew. We are already in the summer of 2016, that's halfway through the decade. Menachem, however, refused to let me finish my doctorate without adding at least one more theorem. Something small, perhaps, but something that he can tell others in a single sentence. Something that any other set theorist can understand without exposition. I had set my eyes on some problems, but at that point, I was tired of all of them, and I found a small and minor applications of my method: the failure of Fodor's lemma on all cardinals (in a nontrivial way, that is, so successors are regular). This was done over the summer, and I was ready to finalise my thesis. Thank goodness.

In the meantime, other side projects were taking place. I asked Yair once what he was working on, we had a nice chat, and that led us to our first collaboration. It started with "proper forcing cannot change cofinalities" and moved all the way to the paper that got published later at the Archive for Mathematical Logic. Some time after that I was talking with Yair about large cardinals without choice, and how there's no investigations towards critical points, but only towards combinatorial definitions of large cardinals (the recents papers about Berkeley cardinals were not yet public knowledge at the time). This lead to our projects on critical cardinals and a Silver criterion for symmetric extensions (both ended up in the paper "Critical cardinals" and I wrote on that project in some other post). I met David Asperó in Cambridge, and talking about properness in ZF led us to our project that culminated in a recent paper (still under review, though). And with Philipp Schlicht we worked on some idea that I had which began as an attempt to understand the basics of iterations of symmetric extensions, and ended up with a nice theorem that was finally written down just a couple of months ago in our joint paper.

Well. I've applied to a bunch of postdocs and got rejected. I've asked Martin Goldstern if he had some funding to hire me, and he said that's fine, so we did just that. As soon as I started in Vienna, I've learned that I was awarded the Newton International Fellowship, and so shortly after I moved to my current job in Norwich.

This one ends in two months time. What will be after that? I still don't know. But I did learn a few things since this decade started.

I learned how to write a paper.

  1. I've learned what counts as a proof. Always ask yourself: is Matti going to be satisfied with this? If the answer is no, then it's not a proof.
  2. I've learned how to write a paper. I've learned that one should always present results in research seminars before submitting the paper to a journal.
  3. I've learned that showmanship is important for teaching and speaking. When I began my masters I went to ask the best teacher I've ever had, Uri Onn, for advice. He said that you are telling a story, and you need to keep your audience interested in the story. And I add to that, you need to make sure that you know how to captivate with your story.
  4. I've learned that when flying to a conference, pack as little as possible. Travel lightly. If possible: buy whatever you cannot take with you and is cheap.
  5. I've learned that when you write a grant proposal, one should always write in confident terms of success. Never second guess yourself. The goals will be met, and the results will be great.
  6. I've learned that the world ain't all sunshine and rainbows. To quote Mr. Balboa. But also that small communities, like the set theory community, are great at helping others (in most things).
  7. I've learned that sometimes you have to fight for yourself, and that no one will fight for you. Even though they could, or might be able to do just that. Academics are programmed from day one of "adult academia" to avoid conflict, and that means that if you have to stand up to something, you're almost definitely on your own. But if you're lucky, you will at least get some consistently good advice, and some people will actually care about your situation.
  8. I've learned that you should never be afraid to ask questions or argue with someone just because they are "more senior" than you. Science, mathemtaics, philosophy, they all thrive on people asking bold questions and disagreeing with each other where appropriate. Of course, you should always remain respectful, civil, and friendly.
  9. I've learned that spending a lot of time on MathOverflow and Mathematics StackExchange can be a very good thing if you're using your time correctly. I've seen others feeling shame for asking "silly questions" under their exposed identity. But that's just silly. We all start somewhere.
  10. I've learned that at some point in your life, you just can't eat a whole pizza at 1am and expect to sleep like a normal human being.
  11. I've learned that the above also applies to steak, and many other foods.
  12. I've learned that writing long lists like that on your blog is kind of useless.
  13. I've learned that sometimes, it's just the time to stop.

So, we're standing at the edge of a new decade, one that is rife with opportunity, and with some luck, it will be a whole new era in my life! Let's see how that goes...

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