# Young Set Theory Workshop 2011

It’s been a while. In case you were wondering where the heck I have been these last two months, check out mathblogging.org and our blog at wordpress.com.

The last week I was able to spend at the Young Set Theory Workshop thanks to the generous DFG. This was my first time at the YST although I have followed its success story via friends since it began three years ago. The workshop is, well, a workshop so the focus was on tutorials and so-called discussion sessions which were open to all topics and were decided upon spontaneously.

## The tutorials

The tutorials were all very good. Personally, I was particularly amazed by Joel David Hamkins and Juris Steprans. Joel gave the very first tutorial (each tutorial had 2 parts) and talked about Boolean ultrapowers which (at least when presented by him) seems an amazingly simple point of view to understand both forcing and large cardinals. I keep wondering if this non-technical approach to those set theoretically heavy-weight topics might be a way to finally bring these tools to a more general mathematical audience. I had never met Joel before but I knew him from afar as THE mathoverflow user; suffice it to say that he is just as friendly and helpful in meat space as he is on MO. The second tutorial on Monday and Tuesday was given by Slawomir Solecki who gave an introductory course on Borel equivalence relations. Though I enjoyed most of his tutorial, I did have trouble to follow the beginning of the second part. The problem for me was that he continued a technical proof that he had started at the end of the first part of his tutorial and I just couldn’t remember enough of the details and definitions from part 1 to follow. Since I have experienced this often when there’s a two part proof, it made me wonder what kind of strange tradition this is in mathematics. Why not just skip the rest of the proof? If you cannot expect your audience to actually follow, it should be better to skip it since there’s nothing to be gained (except the apparent tradition that every talk of sufficient length must contain a complicated proof). Of course, realistically Slawomir Solecki did everything right, I was just not fit enough to follow that proof — I didn’t have the impression that many shared my trouble. In any case, the rest of the lecture was luckily self-contained in the sense that the actual techniques in the proof were not important. It also left me with the goal to one day give lectures (and courses!) that are always perfectly self contained (one can dream, right?).

On Wednesday and Thursday Ali Enayat’s tutorial offered insight into models of arithmetic and models of set theory and their interactions. After some initial confusion on my part, I did enjoy this even though it was not a blackboard talk as the first two tutorials had been (and I’m always in favor of those if time permits). I can’t really say a lot else because the topic was rather far away from anything I know. Thursday and Friday also saw Juris Steprans’s tutorial on amenability. This was shockingly interesting for me. When I did my PhD in Berlin Sabine Koppelberg had a student who wrote a master’s thesis about shift invariant means. And even though I went to each of his many talks, I must admit that I had never understood what it was really all about. This probably only shows that not all talks I don’t understand are in vain in the long run, but on the other hand I am really annoyed with myself that I never took the time to understand this stuff back then. I’ll definitely have to look into at least one result (I think by ~~-Magidor~~- Foreman): under MA every ultrafilter on \(\omega\) allows for the construction of a subgroup of the symmetric group \(S(\mathbb{N})\) with a unique translation invariant mean. This is exciting because I am always looking for ultrafilter related invariants that are finer than the Rudin-Keisler order (mostly because algebraic properties are not really invariant under RK-equivalence). Juris Steprans’s tutorial at times was a little strange for me. Whenever he talked to the audience freely, I was amazed at the clarity he communicated, but every so often he’d turn to his slides and suddenly that great clarity was gone until he started to speak freely again. In the second part of his tutorial he also gave a lot of technical details for a proof which I couldn’t really follow. But for me it was all good since the topic was so much fun. I must admit, I always favor the applications of set theory to the “purest of the pure” kind of stuff…

[[Note: The slides of all tutorials are available online ]]

## The discussion sessions

Every day after lunch it was time for the so-called discussion sessions. These were rather informal and the ones I visited were very different from each other. On Monday, I joined a session led by Minami Hiroake about \(F_\sigma\) ideals and maximal almost disjoint families. This was in a way the best and the worst discussion session I joined. It was good because after a slow start four, five people really discussed a research problem. Unfortunately, the other ten people in the room did not seem to be able to join in. I guess people weren’t ready to walk out of the very first session just because it wasn’t perfect — the other sessions I went were much more relaxed in any case. On Tuesday, I joined a session by Luz Maria Garcia Avila (sorry for missing accents) who is at the University of Barcelona and presented a really nice proof of Ramsey’s Theorem using the Rasiowa-Sikorski Lemma and the partial order used in Mathias forcing. I can’t shake the feeling that I have seen this argument before, but this does not make her proof less beautiful. I’m all in favor for giving credit for (re)discovering known results. In my experience we mathematicians are too often obsessed with originality and should give far more credit for independent rediscoveries and high quality presentations of known results. Anyways, after the proof of Ramsey’s Theorem, Luz went on to describe a similar approach to prove Hindman’s Finite Sums Theorem which she was working on. My immediate response was “that should work straight away” because Andreas Blass often talks about Baumgartner’s combinatorial proof of Hindman’s Theorem as a forcing argument. This led to a couple of good discussions with Luz later (with a lot of confusion because we thought we were talking about the same presentation of that proof) and I hope she’ll be able to make it work.

On Thursday and Friday I headed two discussion sessions myself. The first was a short one because of the excursion later that day and I mostly presented the basics of algebra in the Stone–Čech compactification (again sorry for the missing accent, I’m too lazy right now) and on some recent problems regarding union ultrafilters. I tried to focus on the general feel of the area rather than proofs because even the elegant Galvin-Glazer proof of Hindman’s Theorem has some subtleties that I feel are difficult to communicate in a short amount of time — and a senseless copying down of a proof seemed inappropriate in such a setting (well, actually, in any setting). The session on Friday was on Teh Internetz. I had hoped to get a discussion going regarding a possible meeting point for set theorists but I think I didn’t quite do that right. I started off with mathoverflow and thankfully Joel David Hamkins was there and gave us his view of the platform. I hope I can convince him to write a little about it some other time, especially an extended FAQ describing the shape of the community and what to expect when posting there. After MO, I talked about some of the tools I use online and offline. After almost an argument about hidden vs open meeting places (I’m in favor of visibililty and openness but others raised concerns that I couldn’t quite pin down) we finally arrived at my favorite hobby — recording and broadcasting seminar and other talks. There was very positive feedback from Joel David Hamkins and Juris Steprans, so I hope to get a small project via posterous, tumblr or wordpress going to collect talks where the “locals” would offer to record or broadcast.

## The postdoc talks

The final research piece of the conference were the talks by some of the postdocs. Unfortunately, I missed Katie Thompson’s short talk on LOTS on Monday because I was caught up in a research discussion with David Chdounsky. On Tuesday, Assaf Rinot gave a great talk on “his” Ostaszewski square. Even though I have a hard time with these purest of the pure talks, Assaf is simply an awesome speaker. He relies little on slides and could really just give a talk — no blackboard or slide needed — amazing and extremely enjoyable (although with this kind of skill, he could put even less on the slides, real presentation zen). On Wednesday, there were two talks, Sam Coskey talked about applications of Borel equivalence relations to the conjugacy problem and Dilip Raghavan talked about the P-ideal dichotomy. I enjoyed both talks immensely not just because they were blackboard talks but because they both presented a well chosen amount of proof; and even better: those proofs were presented in a fashion which is suitable for talks, namely as sketches, hand-waving arguments and proof by picture with only little formal notation. On Thursday, David Schrittesser gave a very technical talk which I was unable to follow and on Friday Grigor Sargsyan gave a talk on inner model theory. Even though my own research is about as far away from inner model theory as is possible within logic, I always enjoy his talks because he really has a mouth-watering kind of way of telling you why people ended up doing what they are doing in that part of set theory.

I must nevertheless admit that I didn’t understand the motivation for this part of the workshop. Not that I think that postdoc talks are not interesting — quite the opposite, in fact. After all, a young postdoc can usually relate better to problems a grad student might have with a topic. What irritated me was the randomness. It didn’t feel that the speakers were chosen for specific reasons or talks but just for the usual kind of talk you’d give as at an arbitrary conference. I’d have preferred either discussion sessions guided by the postdocs or some kind of tutorial. This also connects to something I kept wondering about. The workshop has grown considerably. The first one had, I think, 30 participants, this year there were over 70. Even though I didn’t go to the earlier ones, I think it might be worthwhile to think about restricting access somewhat. Not by some random notion of ‘elite’ but simply by academic age, say people who are +-2 years from a PhD. Combine those with a select choice of great tutorial/discussion speakers (great speakers, not just great researchers), possibly increase discussions by hooking up some more postdocs via video broadcasting. I don’t know, just a thought.

## The social ~~-network~~- experience

Finally, all of the above experiences would be much less meaningful without the social aspect of (especially) this meeting. After all, the **young** set theorists are the focus. When I was on my way to the workshop, it dawned on me that, assuming I manage to stay in research, this would be a large part of my long-term peers. This gave the social component of the workshop a particular flavor and I often stopped to wonder how comfortable I’d be in this group. From the gender debate on quomodocumque a while ago, that one sentence stayed with me: “mathematics is often not worth the people you have to put up with”. My feelings in this respect were very mixed. I find it very difficult to introduce myself to mathematicians if I don’t have a meaningful question or comment. In a conference where there are relatively few speakers the easy way out for me (to pick something from the talk and start a conversation) wasn’t available. So I didn’t end up getting to know as many new people as I would have liked. Among those that I got to know for the first time where both very nice people and people that I could not connect with at all (or rather would prefer not to connect with), so I guess that part is absolutely normal. A few times though I was really disturbed by people, the most disgusting part probably was to overhear an extremely sexist comment. Unfortunately, I heard it from far away and couldn’t react at all, but such an open display shocked me. I guess you could say “well, quite normal in our society” but I do feel that such smart and highly educated people should be able to think beyond such discrimination (and certainly I cannot accept an excuse along the lines of “wasn’t meant that way” from somebody this educated).

I was nevertheless extremely happy to meet many old and new friends. To have a memorable conversation with Stefan Geschke, Gido Scharfenberger-Fabian (Berlin “siblings” of mine), David Chodounsky, Jonathan Verner, Marcin Sabok, Philipp Schlicht (head of the organizing comittee and master’s “sibling” ages ago in Munich), Thilo Weinert, Alexander Primavesi, Andrew Brooke-Taylor, Daniel Soukup, Sam Coskey, Wolfgang Wohowsky, Luz Avila, Dilip Raghavan — so many people, it was a pleasure.