Category Archives: Personal

Trust me, I’m a doctor!

Finally!

Six months after I had turned in my dissertation, I have finally received the approval on the damn thing.

I would like to take this opportunity to thank my advisor, Menachem Magidor, and to my overseeing committee, Uri Abraham and Moti Gitik. Their help was indispensable, and I could literally have not done any of that without their support.

In the same breath, I would like to extend these thanks to Azriel Levy, from whom I had learned a lot in our time teaching the basic set theory course for three years straight.

Wooooo!

Moment of Zen

When one is ascending a difficult path uphill, it is a good idea to keep your eyes on the path as you move forward. However, it is not a bad idea to stop sometimes, look back, and appreciate the beauty of the ground you have already covered.

This understanding extends to any difficult task which is completed in a long series of steps.

What a long strange trip it’s been…

As some of you may have noticed, I don’t use this blog to write about my papers in the “traditional way” math bloggers summarize and explain their recent work. I think my papers are prosaic enough to do that on their own. I do use this blog as an outlet when I have to complain about the arduous toil of being a mathematician (which has an immensely bright light side, of course, so in the big picture I’m quite happy with it).

This morning I woke up to see that my paper about the Bristol model was announced on arXiv. But unbeknownst to the common arXiv follower, this also marks the end of my thesis. The Hebrew University is kind enough to allow you to just stitch a bunch of your papers (along with an added introduction) and call it a thesis. And by “stitch” I mean literally. If they were published, you’re even allowed to use the published .pdf (on the condition that no copyright infringement occurs).

My dissertation is composed of three papers, all of which are on arXiv (links in the “Papers” page of this site):

1. Iterating symmetric extensions;
2. Fodor’s lemma can fail everywhere; and
3. The Bristol model: an abyss called a Cohen real.

Of course, the ideal situation is that all three papers have been accepted for publication, but all three of them are still under review. So it puts me at this odd situation where I will have essentially four sets of referees (one for each paper, and then two additional referees for my thesis), and so the output can end up oddly different between the resulting dissertation and the published papers. But that’s fine.

In any case. Those of you who are interested in reading my thesis can find it in those three papers. I am probably going to post the final thesis online when it will be approved, but the only thing you’re currently missing out is an introduction with some minor historical background and a summary of the three papers. So if you read all three, you don’t really need that introduction anyway.

Good. So what next? I have a few things lined up. More news will follow as reality unfolds itself like a reverse origami.

Got jobs?

Good news! I’m about to finish my dissertation. Hopefully, come summer I will be Dr. Asaf Karagila.

So the next order of business is finding a position for next year. So far nothing came up. But I’m open to hearing from the few readers of my blog if they know about something, or have some offers that might be suitable for me.

Do not hesitate to contact me or leave a comment.

Thanks!

Farewell, Matti

My mentor, teacher, mathematical confidant and generally good friend, Matti Rubin passed away this morning. Many of the readers here know him for his mathematical work, many knew him as a friend as well, or as a teacher.

Matti was a kind teacher, even if sometimes over-pedantic.

Recently he was diagnosed with lung cancer. I spoke with him two weeks ago, shortly before I went to the Arctic Set Theory conference, and he sounded positive. He felt that he might not survive this battle, but he was optimistic about the short time era. I was hoping to talk to him once more and to tell him about a student of mine that he would have surely liked.

Alas, this morning Matti left this world. Apparently there had been a sudden deterioration of his condition, and this morning, tired and hurt he left us here. I take solace in the fact that I have known him. I had learned from him. I have created a minuscule copy of the original in my mind, to remind me where I’m being mathematically dishonest while writing proofs (“This is not obvious, and that is not clear”).

He will be sorely missed by all of us here.

A photo I took from his logic course back in the Spring of 2009. Probably talking about model theory or Turing machines…

Iterating Symmetric Extensions

I don’t usually like to write about new papers. I mean, it’s a paper, you can read it, you can email me and ask about it if you’d like. It’s there. And indeed, for my previous papers, I didn’t even mention them being posted on arXiv/submitted/accepted/published. This paper is a bit different; but don’t worry, this is not your typical “new paper” post.

If you don’t follow arXiv very closely, I have posted a paper titled “Iterating Symmetric Extensions“. This is going to be the first part of my dissertation. The paper is concerned with developing a general framework for iterating symmetric extensions, which oddly enough, is something that we didn’t really know how to do until now. There is a refinement of the general framework to something I call “productive iterations” which impose some additional requirements, but allow greater freedom in the choice of filters used to interpret the names. There is an example of a class-length iteration, which effectively takes everything that was done in the paper and uses it to produce a class-length iteration—and thus a class length sequence of models—where slowly, but surely, Kinna-Wagner Principles fail more and more. This means that we are forcing “diagonally” away from the ordinals. So the models produced there will not be defined by their set of ordinals, and sets of sets of ordinals, and so on.

One seemingly unrelated theorem extends a theorem of Grigorieff, and shows that if you take an iteration of symmetric extensions, as defined in the paper, then the full generic extension is one homogeneous forcing away. This is interesting, as it has applications for ground model definability for models obtained via symmetric extensions and iterations thereof.

But again, all that is in the paper. We’re not here to discuss these results. We’re not here to read some funny comic with a T-Rex and a befuddled audience either. We’re here to talk about how the work came into fruition. Well, parts of that process. Because I feel that often we don’t talk about these things. We present the world with a completed work, or some partial work, and we move on. We don’t stop to dwell on the hardship we’ve endured. We assume, and probably correctly, that most people have endured similar difficulties one time or another. So there is no need to explain, or expose any of the background details. Well. Screw that. This is my blog, and I can write about it if I want to. And I do.

So, the idea of iterating symmetric extensions came to me when I was finishing my masters, I was thinking about a way to extend symmetric extensions, because it seemed to me that we ran this tool pretty much into the ground, and I was looking for a tool that will enable us to dig deeper into the world of non-AC models. It was a good timing, too. Menachem [Magidor] had told me about this interesting model that they constructed in Bristol at some workshop, and it seemed like a good test subject (dubbed “The Bristol Model” from that point onward). When I settled into this idea, and Menachem explained to me the vague details of the construction, it immediately seemed to me as an iteration of symmetric extensions. So I set on to develop a method that will enable me to formalize and reconstruct this model (I did that, and while I have a set of notes with a written account, I will soon start transforming those into a proper paper, so I hope that by the end of July I will have something to show for).

The first idea came to me when I was in Vienna in September of 2013. I was sure it’s going to work easy peasy, and so I left it to focus on other issues of the hour. When I came back to this a few months later, Menachem and I talked about it and identified a few possible weak spots. Somehow we managed to convince ourselves that this is not a real issue, and I started working the details. Headstrong and cocksure, I was certain there just a few small technical details which will be solved in a couple of days worth of work. But math had other plans, and I spent about a year and a half before things worked out.

Specifically because I kept running into small problems. Whenever I wrote about some statement that it’s “clear” or “obvious”, there were troubles with that later. Whenever I was sure that something has to be true, it turned out to be false. And I had to rewrite my notes many times over. Usually more or less from scratch. Luckily for me, Martin Goldstern was visiting Jerusalem for a few months during the spring semester of 2015, and he was kind enough to hear my ideas and point a lot of these problems. “Oh, just make sure that such and such is true” he would say, and the next day I’d find him and say something along the lines “Yeah, it turned out that it’s false, so I had to do this and that to circumvent the problem, but now it simplified these proofs”. And the process repeated itself. This long process is one of the great sources for this blog post of mine, and this post and also that post.

Closing in on the summer, Yair [Hayut] was listening to whatever variant I had at the time, and at some point he disagreed with one of the things I had to say. “Surely you can’t disagree with this theorem, it only relies on the lemma that I showed you as the first lemma, and you’ve agreed to that”. He pondered a little bit, and said “No, I actually disagree with the lemma”. We paused, we thought about it, and we came up with one or two counterexamples to that lemma. It was exactly the issue Menachem and I identified, and suddenly all the problems that were plaguing me because obvious consequences of that very problem.

I had worked very hard over the course of the next two months, and I managed to salvage the idea from oblivion. It was a good thing, too, because shortly after I’d visit the Newton Institute, and I had the chance to present this over the course of 8 hours to whomever was interested. And a few people were. But the definition was just terrible. I was happy it’s working, though, so I left it aside to cool down for a bit, while I worked on other projects of my thesis.

And now, I sat down to write this paper. And as I was writing it, I realized how to simplify some of the horrible details, which is great. This caused some of the proofs to be clearer, better and more of what you’d expect of these proofs. And that’s all I ever wanted, really. It took me two years, but it feels good to be over with, I hope. Now we wait for the referee report… and a year from now, when I’ve forgotten all about this, I’ll probably grunt, groan, and revise the damn thing, when the report will show up. Or maybe sooner.

Well… I’m done venting. Next stop, writing up The Bristol Model paper.

Addendum:

Okay, maybe this sounds like I’m treating this as a rare process. And to some extent, it is. This is my first big piece of research. You can only have one first of those. Yes, mathematical research is a process. A long and slow process. I’m not here to complain about this, or argue otherwise. I’m here to point out the obvious, and complain that I never heard people talking about these sort of slow processes. Only about the “So he hopped on a plane, came over here, and we banged this thing together in a couple of weeks time”, which is really awesome and sort of exciting. But someone has to stand up and say “No, this was a slow and torturous process that drained the life out of me for the better part of two years”.

Quick update from Norwich

It’s been a while, quite a while, since I last posted anything. Even a blurb.

I’m visiting David Asperó in Norwich at the moment, and on Sunday, the 12th, I will return home. It seems that the pattern is that you work most of the day, then head for a few drinks and dinner. Mathematics is eligible for the first two beers, philosophy of mathematics for the next two, and mathematical education for the fifth beer. Then it’s probably a good idea to stop. Also it is usually last call, so you kinda have to stop.

If luck is with us tomorrow, there might be some great news in the near future. If not, then there might be some other, good, or at least interesting, news in the near future.

And in other unrelated news, there are some updates coming in the next couple of weeks. I hope.

Goodbye, Oren.

I recently heard the news that Oren Kolman passed away a couple of weeks ago.

Some of you may have known him through MathOverflow as “Avshalom” where he often appeared in the comments with generally useful references, and some of you may have known him in real life as a teacher or a colleague, or a student. Some of you may have even knew him as Eoin Coleman.

I met him on the outskirts of Pavilion C in Cambridge when I visited there a couple of months ago, and I immediately recognized him from his old MathOverflow avatar. I told him who I was and he immediately recognized my name. We had a lively chat for the better part of ten minutes, then we had to depart to different directions. That was pretty much my last interaction with him, even online.

Goodbye Oren, you’ll be missed.

See also on the ESTS website.

Blurbs!

I don’t like social media very much. I never really subscribed to the whole Friendster, MySpace, Facebook, Twitter, Google Wave, Google+, and what have you social network sort of approach that you need to have “friends” and “followers” and “follow” other people.

I always preferred to be the master of my domain. The king of my castle. But literally, not the Seinfeld euphemisms sense. In any case. I’ve been thinking about a page where I can post short thoughts about math, life and otherwise. The blog is not suitable, since I’m not going to add a post each time I have a new thought. So instead I’ve started a blurbs page. Each blurb has a number, and an anchored link that you can use in case you want to share it.

You can find the first blurb here. And it is, in fact, set theoretic in nature. It is something that I realized one time when I was talking to students on the first class of the semester in the introduction to set theory course.