2/25/2016

Modified and proofread the arguments for n larger than 3 today. I think I may be the worst small typo catcher of all time...too much auto correct in my brain. I sent out an email seeing if I could present at the ASL North American Annual Meeting, hopefully I hear back soon, though if I don't I will just have to submit an abstract tomorrow anyways. Not much else today, other than the unending internal quibbles over notation.

2/24/2016

"There will be time for visions and revisions."

Working with my advisor, we found an edge case that we had not previously considered and which poked a non-trivial hole in the argument. Fortunately, we were able to rework the algorithm to prevent this case from occurring. Even more fortunately, after formally writing up the new algorithm, the arguments are actually simpler than they were before. The fix is good. "Two steps forward one step back." My goal after this revision is to try and trace out all possible edge cases and make sure the argument is air tight. Curious things seem possible for triples of infinite strings when the only restriction is that two of the three are eventually the same.

For posterity, the edge case:
       and       and so on were all forcing the algorithm to break the desired pattern. Since this can
000       000 happen infinitely often, such families of strings have to be accounted for only once
000       000 by the algorithm.
000       000
000       000
110       110
101       101
010       000
             010

2/19/2016

First draft of a full Mycielski result for E_0 complete. If correct, this gives the Jonsson property for R/E_0 into R easily. I talked to the organizer of the RTG seminar today, and it looks like I will be able to give a talk there before the AMS sectional meeting happens. No date is set yet though.

2/18/2016

Halfway there for the full Mycielski style result. More details changed than I expected, and in particular keeping the injuries finite took a bit of a trick. I keep trying to think of a good way to "visualize" what R/E_0 looks like, and what it means to try and color it with R colors, but I don't really have a good analogy yet. It certainly doesn't help that R/E_0 is naturally topologized with the trivial topology and has no definable total ordering. I should be ready to present the Jonsson property for R/E_0 into R to Jackon on Monday.

2/17/2016

More progress on the Mycielski style result today. I have formulated a proof for arbitrary n. I also took a little bit of time to write out an elementary proof of the fact that every comeager set contains a non-smooth set (for completeness). Full Mycielski should be done tomorrow.

We got the schedule for the spring AMS sectional today. I'll be presenting at 5pm on Sunday, the last day of the conference. It is a bit disappointing that much of the audience will probably already be gone (the flight situation is intense), but I feel good being included at all. After all, the other time slots are taken by people who have more experience and have already been vetted. Its a foot in the door, and we all have to start somewhere. Looking through, I think 6 people from UNT are presenting, which is kind of a lot.

2/16/2016

I think I have finally pushed past the obstacle which has been holding me up for a couple months. We have needed to prove a Mycielski style result for non-smooth sets, and this has involved lots of fine tinkering with the argument in the Glimm-Effros Dichotomy. We have not yet obtained the full result, but I am very confident we now have it for a 3-dimension comeager set. Even better, the argument looks extendable into higher dimensions. All finite dimensions simultaneously will be another wrinkle, but I think nothing will be as hard as this step from 2-dimensions to 3.

In other news, I finally got word back from the college of arts and sciences, and they are going to fund me to go to the AMS sectional meeting in Salt Lake City. This will be my first time presenting at a seminar not being held at UNT. The saying "When it rains, it pours," can also apply to good news.

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.