September 23 Progress

Reading Courses:

Structural Consequences of AD by Stephen Jackson
HOD as a Core Model by John Steel and Hugh Woodin

Other Reading:

The Core Model Iterability Program by John Steel
Non-Standard Analysis for the Working Mathematician

AD and Jonsonn Cardinals in L(R)

I finally got my website up and live, http://math.unt.edu/~jkh0150/. I'm proud that I made it all my self in html, although its really not that hard of a thing to do.  I finished a few things up since last time: The counting equivalence classes paper should be ready for a GLG talk anytime now, and I presented on Lindstrom's characterization of first order logic. Core Model progess is still slow, but its starting to really come together. I need to read some more out of the Cabal, but I'm not sure exactly what I should be reading. It probably wouldn't hurt to look at scales in L(R) again. Non-standard analysis continues to be interesting, the measure theory results in particular. I don't have much else to say, other than teaching twice in the morning seems to wipe me out for a good couple hours afterwards. I'm not sure what to do about that, if anything.

August 28 Progress

Reading Courses:

• Structural Consequences of AD by Stephen Jackson
• HOD as a Core Model by John Steel and Hugh Woodin

Other Reading:

• Invariant Descriptive Set Theory by Su Gao
• The Core Model Iterability Program by John Steel
• Non-Standard Analysis for the Working Mathematician
• Counting Equivalence Classes for Co-\kappa-Souslin Equivalence Relations by Leo Harrington and Sarahon Shelah
• Infinitesimal by Amir Alexander

The past week and a half has been a bit crazy, this is the first of class and there was a lot of preparing to do.But the structure of having to be on campus for meals and teaching seems to be helping my productivity. Dr.Gao's chapter on analytic equivalence relations has me excited about possible interactions between determinacy theory and the theory of equivalence classes. I talked to Dr.Jackson about it and he pointed me to the paper by Harrington and Shelah. If I can tame it, I will present it to the graduate logic group. The first chapter of "Non-Standard Analysis for the Working Mathematician" has me annoyed, I think their half-job of describing first order logic is more complicated to think about than the real thing, and it seems they only do it to be able to Prove Los's theorem, which they do in full generality later anyways. I had a good meeting with Dr.Jackson on Tuesday, and I am reminded to always always always try to draw a picture first. The core model iterablity is as hard as advertised, but I am pushing through. Finally, Infinitesimal has taught me that Thomas Hobbes thought he had squared the circle, despite all of his proofs being debunked. That should be filed under great historical blunders.

August 17 Progress

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Structural Consequences of AD by Stephen Jackson
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: The Core Model Iterability Program by John Steel
Pop Math: Infinitesimal by Amir Alexander

It was a good week of progress, despite having to deal with class schedules and financial aid. Competing administrative requests make for an unhappy grad student. I feel like I'm developing a cool library of equivalence relations from Gao, and on that note it looks like we will be working together on a short project sometime this year. I would like, eventually, to have work done in descriptive inner model theory, invariant descritive set theory, and non-standard analysis. CMIP is slow going, but it takes time to work through sucha barrage of detail. I'm definitely a little caught off guard by how much it relies on FSIT. Between Jackson's qork and Gao's, I am struck by how we simulatenously know so much about complexity in the AD context, and how we little we know in general.

August 11 Progress

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Structural Consequences of AD by Stephen Jackson
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: The Core Model Iterability Program by John Steel
Pop Math: Infinitesimal by Amir Alexander

Two more weeks until class starts. I hit the first big iterability proof in "The Core Model Iterability Program," so I will be spending a good amount of time on that tomorrow. I think I can be completely done with "HOD as a Core Model" by the time the semester starts, and finished with prelimaries section of "Structural Consequences of AD" as well.

Stray Thoughts:

• Iteration Tree proofs are complicated. Really complicated, and these are the simple ones. 
• I was not expecting hypersmooth countable equivalence relations to be hyperfinite, that was cool.
• I'm a little burnt out on the HOD stuff, but its good to see the more general presentation.
• I think Amir overstates the importance of this battle over infinitesimals. It is interesting that the Jesuat order was destroyed, but until I look into it further myself, I'm going to assume there was at least one other reason.

Agust 8 2014 Progress

Current Reading

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Structural Consequences of AD by Stephen Jackson
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: The Core Model Iterability Program by John Steel
Pop Math: Infinitesimal by Amir Alexander

It is hard to remember to post on this at the end of the day. So I finished the papers on tracks 2 and 4, and have started new papers. Even on track 1 I am now solely on the L(R) part. So there isn't too much to report. Infinitesimal continues to be interesting, the book really excels when it focuses on the personal rivalries and politics of the time.

August 4 2014 Progress

Current Reading

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander

Today seemed like it would be a very unproductive day at first; I wasn't feeling very good and I overslept. With lots of breaks, however, I was able to make progress on all but track 1. I am now done with the introduction to Core Model theory, and will be moving on to the CMIP. Either tomorrow or the next day I will be done with "Large Cardinals from Determinacy", and I will be moving on to "Structural Consequences of AD." My most recent reading in "Infinitesimal" has made me very glad that I am not living in a time where I have to worry about being in the pope's favor. Galileo has now crossed the story a few times, so that's been interesting. I suppose instead of popes I have to worry about being in the favor of the NSF.

August 1 Progress

Current Reading:

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander

More of the same on tracks 1,2, and 4. For track 3, the chapter on countable equivalence relations started out with a bang. Countable Borel equivalence relations are generated by countable Borel actions, and that leads to the fascinating marker theorem. I am excited to see what comes next. After reading "Infinitesimal" a bit more, I am afraid I would have been on the side of people who thought that the proto calculus being done was nonsense. Its clearly more powerful, but Toricelli has lines being composed of different infinitesimal width, which just sounds like a terrible way to go about things. It is fascinating, however, to see more of the background that Newton and Leibniz were working in.

Again, there was a lack of posting due to travel, though this time was for pleasure and highly necessary. Its looking like I will have to travel again in August as my father is moving to Korea at the end of October. Maybe I can get a good three day weekend. Because it is the start of my fifth year I no longer am receiving my initial tuition scholarship, but fortunately my department offered me a new tuition scholarship. I have one more month until the school year starts, so its looking like I will be able to completely finish almost all of my projects.

July 25 2014 Progress

Current Reading:

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander

Not too much of interest to report today. HOD is going well, everything is quite familiar and intuitive at the moment. The generation theorem in "Large Cardinals from Detemrinacy is going to be my weekend project. I finished the Glimm-Effros dichotomy chapter in IDST, and I am excited to move onto the general theory of countable Borel equivalence relations. The Introduction to Core Model Theory is, just like the HOD paper, quite familiar and intuitive at this point. It is nice to see the exposition still. Infinitesimal finally got a bit into the math, and now I'm hooked. Amir has elucidated some interesting differences between the Euclidean approach to geometry and the proto-calculus approach. It seems to be the start of rigor vs. inuition. I do, however, find it difficult to follow geometric proofs when they are presented in text. I feel if I am having to struggle with it, the average reader would probably just get lost. Maybe instead of completed geometric diagrams, it would be better to draw the diagram and add the labels in steps. The text brings up an interesting paradox: starting from axioms and chasing deductions is highly stable and very checkable, but the physical intuition seems capable of "justifying" the axioms and simplifying the work, whereas the one thing the axioms can never justify is themselves. Which approach provides deeper knowledge, or better, what's the proper combination of approaches?

July 24 2014 Progress

Current Reading:

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander

I am finally back to where I stopped in "HOD as a Core Model" before the slew of family drama started. I stopped at the wrong place when I went to deal with that. So I am almost done with HOD in L[x,G]. My advisor wants us to start reading the HOD in L(R) part starting next week, so the timing is almost perfect. I am at the main proof of "Large Cardinals from Determinacy," and its quite long. I am debating breaking it up into parts, as it is one proof. I think tomorrow I will just make a go of the whole thing. Tracks 3 and 4 are coming along nicely. I have dropped the bonus track for now, as I am not sure if I want to keep looking at Chang and Keisler. If I drop that I will go back to "Iterations of Rational Functions."

I decided to post this before I began reading Inifinitesimal today as I figured I would not be willing to make a post afterwards. Regularly updating this is harder than regularly doing research.

An Update on Updates

There have been a lack of updates lately. This started because there were a few days where I couldn't get work done because of family drama. Then I was getting work done but was too busy with friends in the evening to post on the blog.

In the meantime I have met with my advisor twice, and we finished reading HOD in L(R). We will be moving on to more current HOD analysis and then start on a research problem. I did ditch the mouse sets paper for now in favor of the introduction to core model theory. At this point its probably a little silly to read it, but I still think I'm better off with it under my belt before I read CMIP. I finished the section in Chang and Keisler on non-standard analysis. I would reccomend it, as the exposition of the material on superstructures there is better than I've seen it presented elsewhere. I have plans to read "Non-standard Anaysis for the Working Mathematician" in the fall. Anyways there should be more posts coming.

July 15 2014 Progress

Current Reading:

Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander

I finally bothered to look up the correct name for track 1, I should have done it earlier but I was lazy. It's more correct as technically the paper talks about L(R) and L[x,G]. Anyways, I covered pseudo-normal-iterates today in detail. My gut feeling is that this definition, like many others in this subfield is a highly refined, exactly what works definition. But the idea of describing the models that show up when following a strategy L[x,G] can't see in a way that L[x,G] can see is very much the same as the approach to HOD in L(R) and I find it intuitive.

Track 2 was not very exciting today, but I am building up important groundwork. It's just a generalization of the Turing cone measure. I say "just," but the generalization is intuitive and the results are strong. I now remember that I did start to cover strategic determinacy. I don't have a good feeling for this definition yet, I think it will require a good amount of meditation and use before it becomes clear, the excellent exposition in the paper not withstanding.

For track 3, I learned about the Effros dichotomy. I know this is an important result, and I am excited to see the applications of it to come. For whatever reason this material feels intuitive. Its probably because I have a good amount of DST experience.

Again I couldn't bring myself to look at track 4 today. I'm seriously considering leaving it for now and coming back to it when I can better understand its relevance. Its annoying to leave the paper so nearly finished, but I may have to follow my gut. If I do leave it I will start reading on the core model induction.

On the bonus track I waded halfway through the section on nonstandard universes. Chang has a comment about not liking the term nonstandard analysis, which he wants to replace with Robinsonian analysis. I get the historical context he's coming from, but the reality is that the method of analysis is nonstandard, and I don't see the name changing any time soon. So far the material has been review, but a second look never hurts.

"Infinitesimal" moved on to start telling the story of how math became more of a focus for the Jesuits. The story of the inception of the Gregorian calendar is fascinating, but I find this historical background to maybe be a bit too long. I hope the book starts getting to the controversy soon.

July 14 2014 Progress

Current Reading:

Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander

Today was good work day, not perfect, but good. "HOD in L(R)" continues to be interesting, but I find it difficult to keep the myriad versions of iterability straight in my head. I'm trying to form some intuition for the various forms, but I suspect that will come later. As for "Large Cardinals from Determinacy," I find the level of generality breathtaking, and I haven't even reached the most general point yet. What's nice is how this and "HOD in L(R)" are overlapping right now. My studies for a while now have been centered on L(R), its nice to be able to spend time in HOD. "Invariant Descriptive Set Theory" has finally started to get into gear, today I covered E_0 and there are interesting Dichotomies to come. Track 4 got no love today unfortunately.

I finally reached the section in "Model Theory" where they cover regular filters. As much as I love my complete filters, studying interesting filters that more naturally exist has its own appeal. Tomorrow I cover the section on nonstandard universes, I hope the more complete model theory background will help make these wonderful objects more accessible. Finally I got through the first chapter of "Infinitesimal." The book certainly starts off tediously enough, I felt as though I were watching a discovery channel documentary for a while there. Finally though he starts telling the story instead of trying to convince me the story is worth telling, and the first chapter is a fascinating, if brief, history of the fracturing of the Catholic church and consequent formation of the Jesuit order. No math has been discussed yet, but I do feel like I understand the sociological motivations of the anti-infinitesimal side a bit better. As far as learning what kind of writing allows a math book to be popular, I fear I am so far only able to reach the sad conclusion that excluding the math and emphasizing the people and the history is the answer.

First Post

Current Reading:

Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander

I didn't get too much reading today since I took my girlfriend to have her wisdom teeth removed. I'm currently feeling a little frustrated as I am at the start of a new section in all of my readings. The theorem of Woodin on mouse sets paper should be finished next week, and then I will start reading on the core model induction. "Infinitesimal" has already surprised me: I had no idea that there was important infinitesimal work done before Newton and Liebniz.

The weekend is ahead, and I plan to get a lot of recreational reading done, and maybe make an opinion post. We'll see.