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.