I made some decent progress today research wise, although I am pretty tired from flying stand by yesterday. Although not super relevant, I figured out that the degree of Jonsson-ness I have been able to prove for the spaces so far is actually enough to guarantee that the algebraic version of the property is holding. In short, being omega-strongly Jonsson implies the algebraic property even without choice. I also fleshed out more of the Rowbottom properties for some of the combinations, and finished writing up some proofs for the combinations of length 3. I still don't feel like I'm hitting the right level of generality with my statements though.
I also got informed about the UCI summer school event by Cummings, and I went ahead and submitted for that. It would be a cool opportunity to see more of the axiom of choice side of the combinatorics I have been facing, and who knows, some of the pcf theory may prove useful. On the other hand, cofinalities are very complicated in L(R), and the best machinery I have seen to deal with them is the Jackson's theory of descriptions.
Tuesday, May 17, 2016
Saturday, May 14, 2016
5/13/2016
Day 2 of the CUNY Graduate Students in Set Theory Conference
Kameryn Williams started the day out with a talk on minimal models of second order theories. He discussed some second order alternates of ZFC: GBC and KM. While GBC, the weaker of the two extensions has least models, KM does not. This involved an intriguing exploitation of inaccessible cardinals and technique that allows one to build ill-founded models over a fixed well-founded part. There are number of good open questions surrounding this result in the form of slight weakenings on "least" or the consideration of specific nice families of models.
After this, James Walsh talked about extracting ordinal notations from theories. There are countable ordinals that can be associated with different theories of arithmetic, but in the past this has always been ad hoc and relied on seemingly inessential syntactic features. The work Walsh presented is exploring a way to more canonically and uniformly find these ordinals. You have to love math, for perfectly acceptable reasons that Walsh explained to me later, there are objects in this theory called worms. One theorem was proved by "riding the worm." Amazing. In all seriousness though, this work is pretty cool; I always found epsilon_0 interesting and this work is a mega extension of that. Moreover, once he has obtained an ordinal notation system in the manner he described, he has a computable way of generating the cantor normal form of the ordinal.
After the coffee break, Miha Habic gave a talk on grounded Martin's axiom. Even though this was a forcing talk about a fairly technical (though natural) generalization of MA, Habic made a thoroughly understandable 20 minute talk. There was brief mention of a hierarchy of Martin type axioms. It looks like it may eventually be as messy as the choice hierarchy. grMA doesn't do quite as much as MA, but it can still do a lot, and it seems to be quite a bit less restrictive than MA. The talk ended on an intriguing note about the grounded PFA and the possibility of it having weaker consistency strength than PFA.
Finishing out the morning, we had a philosophy student from Italy. Maria Foglioni gave a defense of logic's basic rules of inference. While Foglioni was a fine speaker, I have a hard time getting excited about this material. Basically, some people quasi-formalized (in that vague way only possible in philosophy) the idea that you can't convince someone that logic makes sense. The only way to do that would depend on them accepting logic in the first place. On the other hand, this guy McGee wrote up what he thought was an empirical counter example to modus ponens. Except the second line of the MP isn't strictly speaking true, only probable, and on close inspection the third line doesn't actually line up with the consequent in the first. Great. Again, Foglioni spoke well on the material, but It was still just odd to me.
We broke for lunch. More Asian food. Mathematicians always want Asian food. (Unless you are with Greek mathematicians in which case they want Greek food)
After lunch, Shehzad Ahmed talked about Jonsson cardinals and PCF theory. PCF theory is very hard to get into a 20 minute talk. So much notation, so many ideals. Still, Ahmed did an admirable job motivating what was happening and why we should be interested in the old question of whether or not successors of singular cardinals can be Jonsson. It is strange to see a talk on Jonsson cardinals and have basically no familiarity with the techniques discussed. Choice based combinatorics and determinacy based combinatorics are very different animals. We did find some common ground though, and I think some of the constructions from the choice context will have relevance on the consistency strength questions I have about Jonsson cardinals without choice.
Francis Adams spoke next about definable graphs and dominating reals. Essentially, he wants to know when a graph can be countably colored, allowing for this to happen in a generic extension. He obtains a nice sufficient condition that is essentially combinatoric. The interactions between this question and the question of definable colorings are not straightforward however, if they exist at all. So this may represent an effectively new kind of question about graphs.
After this we broke for an "open problem" session. We all submitted our favorite problem and then people voted and we took the top three or four. I continued to evangelize for recursive tic-tac-toe and that got picked. So we spent a good bit of time playing with recursive tic-tac-toe, which was great. Some joker submitted the PFA conjecture, and some other jokers voted for it. Wisely, no one attempted it.
Hossein Ramandi gave a chalk talk on the minimality of non sigma-scattered linear orders. This is a large class of linear orders which extends the ordinals. Surprisingly, the talk veered immediately into supercompact cardinals and some intense large cardinal forcing constructions. Its always surprising which quesions are connected to each other. Ramansi ended up focusing on a particular kind of linear order and is able to get some results if these are of size at most omega_1.
Ian Smythe closed out the conference with "Towards a Local Gowers Dichotomy." Yay Ramsey theory. Having finally learned why happy families are called happy, I'm not sure if I'm amazed or dissapointed. Happy is not the opposite of m.a.d. These local versions of the Ramsey property that Smythe discusses are interesting, and I think they are true for the combinatorial results I have been proving about R (as a space. The older results are on the integers and are coded into reals). Smythe introduced a theorem of Rosendal which is able to prove a Ramsey type theorem in a functional analysis context. This is pretty exciting as there is not even a straight forward pigeon-hole principle in this context. Smythe got to briefly his local version of this problem, but ran out of time. I wish he had been able to talk more about his part, it sounds very interesting.
Kameryn Williams started the day out with a talk on minimal models of second order theories. He discussed some second order alternates of ZFC: GBC and KM. While GBC, the weaker of the two extensions has least models, KM does not. This involved an intriguing exploitation of inaccessible cardinals and technique that allows one to build ill-founded models over a fixed well-founded part. There are number of good open questions surrounding this result in the form of slight weakenings on "least" or the consideration of specific nice families of models.
After this, James Walsh talked about extracting ordinal notations from theories. There are countable ordinals that can be associated with different theories of arithmetic, but in the past this has always been ad hoc and relied on seemingly inessential syntactic features. The work Walsh presented is exploring a way to more canonically and uniformly find these ordinals. You have to love math, for perfectly acceptable reasons that Walsh explained to me later, there are objects in this theory called worms. One theorem was proved by "riding the worm." Amazing. In all seriousness though, this work is pretty cool; I always found epsilon_0 interesting and this work is a mega extension of that. Moreover, once he has obtained an ordinal notation system in the manner he described, he has a computable way of generating the cantor normal form of the ordinal.
After the coffee break, Miha Habic gave a talk on grounded Martin's axiom. Even though this was a forcing talk about a fairly technical (though natural) generalization of MA, Habic made a thoroughly understandable 20 minute talk. There was brief mention of a hierarchy of Martin type axioms. It looks like it may eventually be as messy as the choice hierarchy. grMA doesn't do quite as much as MA, but it can still do a lot, and it seems to be quite a bit less restrictive than MA. The talk ended on an intriguing note about the grounded PFA and the possibility of it having weaker consistency strength than PFA.
Finishing out the morning, we had a philosophy student from Italy. Maria Foglioni gave a defense of logic's basic rules of inference. While Foglioni was a fine speaker, I have a hard time getting excited about this material. Basically, some people quasi-formalized (in that vague way only possible in philosophy) the idea that you can't convince someone that logic makes sense. The only way to do that would depend on them accepting logic in the first place. On the other hand, this guy McGee wrote up what he thought was an empirical counter example to modus ponens. Except the second line of the MP isn't strictly speaking true, only probable, and on close inspection the third line doesn't actually line up with the consequent in the first. Great. Again, Foglioni spoke well on the material, but It was still just odd to me.
We broke for lunch. More Asian food. Mathematicians always want Asian food. (Unless you are with Greek mathematicians in which case they want Greek food)
After lunch, Shehzad Ahmed talked about Jonsson cardinals and PCF theory. PCF theory is very hard to get into a 20 minute talk. So much notation, so many ideals. Still, Ahmed did an admirable job motivating what was happening and why we should be interested in the old question of whether or not successors of singular cardinals can be Jonsson. It is strange to see a talk on Jonsson cardinals and have basically no familiarity with the techniques discussed. Choice based combinatorics and determinacy based combinatorics are very different animals. We did find some common ground though, and I think some of the constructions from the choice context will have relevance on the consistency strength questions I have about Jonsson cardinals without choice.
Francis Adams spoke next about definable graphs and dominating reals. Essentially, he wants to know when a graph can be countably colored, allowing for this to happen in a generic extension. He obtains a nice sufficient condition that is essentially combinatoric. The interactions between this question and the question of definable colorings are not straightforward however, if they exist at all. So this may represent an effectively new kind of question about graphs.
After this we broke for an "open problem" session. We all submitted our favorite problem and then people voted and we took the top three or four. I continued to evangelize for recursive tic-tac-toe and that got picked. So we spent a good bit of time playing with recursive tic-tac-toe, which was great. Some joker submitted the PFA conjecture, and some other jokers voted for it. Wisely, no one attempted it.
Hossein Ramandi gave a chalk talk on the minimality of non sigma-scattered linear orders. This is a large class of linear orders which extends the ordinals. Surprisingly, the talk veered immediately into supercompact cardinals and some intense large cardinal forcing constructions. Its always surprising which quesions are connected to each other. Ramansi ended up focusing on a particular kind of linear order and is able to get some results if these are of size at most omega_1.
Ian Smythe closed out the conference with "Towards a Local Gowers Dichotomy." Yay Ramsey theory. Having finally learned why happy families are called happy, I'm not sure if I'm amazed or dissapointed. Happy is not the opposite of m.a.d. These local versions of the Ramsey property that Smythe discusses are interesting, and I think they are true for the combinatorial results I have been proving about R (as a space. The older results are on the integers and are coded into reals). Smythe introduced a theorem of Rosendal which is able to prove a Ramsey type theorem in a functional analysis context. This is pretty exciting as there is not even a straight forward pigeon-hole principle in this context. Smythe got to briefly his local version of this problem, but ran out of time. I wish he had been able to talk more about his part, it sounds very interesting.
Thursday, May 12, 2016
5/12/2016
Day 1 of the CUNY Graduate Students in Set Theory conference:
Embarrassingly, I did not know the math lounge was on the fourth floor. So I sat in the presentation room (which was a nice room) like a square until someone showed up there. I perpetually feel out of sync with the rhythm of the group at math conferences. Everyone is very nice though, so its not a big deal.
Kaethe Minden opened the day up talking about subcomplete forcing and Trees. She gave a nice, intuitive introduction to subcomplete forcing and then showed an interesting application of the concept. Subcomplete forcings do not add branches to omega_1 trees. This proof had a fun flavor of large cardinal embedding/forcing proofs and a bit of diagonalization to it.
Next, I went. I feel the talk went well, but not as well as in Salt Lake City. Still, people had questions, and I didn't make any major errors. I thought I would feel more in the loop with the other combinatorial set theorists, but combinatorics under determinacy is just too different from combinatorics under choice.
We took a break here, The lounge (now that I knew where it was) is nice. Lots of chalkboards. The Berkeley lounge has a better view.
Jeffrey Bergfalk then gave a talk on homological characterizations of small cardinals. Essentially, the finite alephs can be characterized by what kind of simplicial complexes they admit. Particularly omega_n is least with no n-dimensional simplicial complex of a particular form. These complexes also have connections to other combinatoric properties. He provided some nice intuitive explanations of the relevant homology, but there is more choice occuring in these proofs than I can apply to my work.
Maxwell Levine closed the morning session with a talk on Weak squares and very good scales. Scales and good may be the most overused terms in set theory. Although this talk was quite technical, I came out of it with a better understanding of square principles and supercompactness than I had before. (The word compact is used because properties occurring on a large set below the cardinal imply properties about the cardinal) Other than that, I mostly gathered that there are some very complicated relationships between these two notions. They essentially contradict each other, but are also compatible in a lot of ways. Also, some very easily stated question about square are quite open it seems.
We had lunch at an Asian buffet. NYC is very crowded, and NYC cheap is Texas moderate. The food was good though.
After lunch, Matt Foreman gave essentially a plenary talk. He gave a broad overview of the interactions of set theory with dynamics, and introduced a particular problem Von Neumann asked. For a general audience, this talk did a good job of being self-contained, and the final result that the isomorphism problem for ergodic measure preserving diffeomorphisms is analytic is quite interesting. Two lines were quite entertaining from this talk:
"Set theory follows behind the rest of math by 50 years"
"The 2-Torus is a smiley face picture"
We took another break.
To start the afternoon session, Alexander Block spoke on the modal logic of the set theoretic multiverse. This is of course an idea every set theory generally gets after studying some forcing, but probably doesn't formalize. The real formalization is quite nice, and there are some nice subtle problems here about the different kind of multiverses one can form and what kind of modal logic they create. A surprising is known, although for more specific types of forcing it is still open.
David Nichols finished the day with some reverse mathematics. In particular he was interested in variations of the 2-dimensions ramsey problem and how they relate to each other. It was previously that over the minimal second order arithmetic they are equivalent. David, however, was able to show that under a strong notion of computable, they are not computably equivalent, and in fact form a strict hierarchy. I always enjoy the results of reverse math.
I'm glad to have my talk out of the way, and look forward to just sitting back and enjoying the talks tomorrow.
Embarrassingly, I did not know the math lounge was on the fourth floor. So I sat in the presentation room (which was a nice room) like a square until someone showed up there. I perpetually feel out of sync with the rhythm of the group at math conferences. Everyone is very nice though, so its not a big deal.
Kaethe Minden opened the day up talking about subcomplete forcing and Trees. She gave a nice, intuitive introduction to subcomplete forcing and then showed an interesting application of the concept. Subcomplete forcings do not add branches to omega_1 trees. This proof had a fun flavor of large cardinal embedding/forcing proofs and a bit of diagonalization to it.
Next, I went. I feel the talk went well, but not as well as in Salt Lake City. Still, people had questions, and I didn't make any major errors. I thought I would feel more in the loop with the other combinatorial set theorists, but combinatorics under determinacy is just too different from combinatorics under choice.
We took a break here, The lounge (now that I knew where it was) is nice. Lots of chalkboards. The Berkeley lounge has a better view.
Jeffrey Bergfalk then gave a talk on homological characterizations of small cardinals. Essentially, the finite alephs can be characterized by what kind of simplicial complexes they admit. Particularly omega_n is least with no n-dimensional simplicial complex of a particular form. These complexes also have connections to other combinatoric properties. He provided some nice intuitive explanations of the relevant homology, but there is more choice occuring in these proofs than I can apply to my work.
Maxwell Levine closed the morning session with a talk on Weak squares and very good scales. Scales and good may be the most overused terms in set theory. Although this talk was quite technical, I came out of it with a better understanding of square principles and supercompactness than I had before. (The word compact is used because properties occurring on a large set below the cardinal imply properties about the cardinal) Other than that, I mostly gathered that there are some very complicated relationships between these two notions. They essentially contradict each other, but are also compatible in a lot of ways. Also, some very easily stated question about square are quite open it seems.
We had lunch at an Asian buffet. NYC is very crowded, and NYC cheap is Texas moderate. The food was good though.
After lunch, Matt Foreman gave essentially a plenary talk. He gave a broad overview of the interactions of set theory with dynamics, and introduced a particular problem Von Neumann asked. For a general audience, this talk did a good job of being self-contained, and the final result that the isomorphism problem for ergodic measure preserving diffeomorphisms is analytic is quite interesting. Two lines were quite entertaining from this talk:
"Set theory follows behind the rest of math by 50 years"
"The 2-Torus is a smiley face picture"
We took another break.
To start the afternoon session, Alexander Block spoke on the modal logic of the set theoretic multiverse. This is of course an idea every set theory generally gets after studying some forcing, but probably doesn't formalize. The real formalization is quite nice, and there are some nice subtle problems here about the different kind of multiverses one can form and what kind of modal logic they create. A surprising is known, although for more specific types of forcing it is still open.
David Nichols finished the day with some reverse mathematics. In particular he was interested in variations of the 2-dimensions ramsey problem and how they relate to each other. It was previously that over the minimal second order arithmetic they are equivalent. David, however, was able to show that under a strong notion of computable, they are not computably equivalent, and in fact form a strict hierarchy. I always enjoy the results of reverse math.
I'm glad to have my talk out of the way, and look forward to just sitting back and enjoying the talks tomorrow.
Wednesday, May 11, 2016
5/11/2016
Day 0 of the CUNY graduate students in set theory conference.
I flew in super early on stand by, so I pretty much had the whole day to myself in NYC. The traffic is comical. If it were shown in a scene in a movie I would say its unrealistically cliched. So much honking. I ended up getting a good amount of work done. I still have a bit a to write up to finish out these frustrating finite combinations, but I think I have the right way of looking at the combos of length 3 now. On a whim I decided to try and visit the MOMA, but it was basically closing. I should have planned that out a bit better. Last but not least I put come finishing touches on my presentation for tomorrow. I'm excited and nervous, but less so than I was for the AMS sectional meeting.
I have some general thoughts about size structure and the axiom of choice floating around in my head right now. I will probably turn them into a post when I have some time.
I flew in super early on stand by, so I pretty much had the whole day to myself in NYC. The traffic is comical. If it were shown in a scene in a movie I would say its unrealistically cliched. So much honking. I ended up getting a good amount of work done. I still have a bit a to write up to finish out these frustrating finite combinations, but I think I have the right way of looking at the combos of length 3 now. On a whim I decided to try and visit the MOMA, but it was basically closing. I should have planned that out a bit better. Last but not least I put come finishing touches on my presentation for tomorrow. I'm excited and nervous, but less so than I was for the AMS sectional meeting.
I have some general thoughts about size structure and the axiom of choice floating around in my head right now. I will probably turn them into a post when I have some time.
Tuesday, May 10, 2016
Teaching
I was thinking about teaching math, and I think I know part of why it is hard to do well. You essentially have to operate on a variety of levels of abstraction simultaneously.
At the most abstract level you want to keep in mind the formal reasons why what you are teaching is true. You then need to decide which parts of this, if any, will help the students.
Next you have the generalizations and applications of the concept. Again you have to be careful what parts of this you share.
Then you have the intuitive picture or pictures of what you are currently teaching. The hand wavey big picture bits that make them feel like they are doing something real.
At the same time you have to keep in mind where this topic sits in relation to the other topics. What is the flow of material? Along these same lines, how do you use this topic to build problem solving skills?
At the bottom you have the explicit process/algorithm you are trying to get them to memorize. How do you chunk the steps together?
Below this, in the basement where the students live, you have pure syntax/arithmetic issues. How do you write the solution down? Which basic calculations do you show?
Outside of all of this, you want to somehow convince them to actually pay attention to the material and try to understand it. Do you use humor? Do you use outside videos?
Subscribe to:
Posts (Atom)