During analysis, we moved on to measurable functions on an arbitrary measurable space. It was almost entirely new material, but there were some concepts, like lim sup and lim inf that I'd encountered in my full semester undergraduate, which came as a nice boost. Makes me feel like I understand what's going on.
Once we understood a bit about how functions worked, we moved on to the Lebesgue Integral. (Because now that all that work had been done, he was finally ready to integrate all the integrals that he wanted. Or at least more of them.) I recalled my real analysis professor teling me that, when in future math courses I got to the Lebesgue integral, I should remember the backdoor we'd found to it through the Generalized Riemann Integral. Which would be good advice if I remembered that course in detail, or had my notes with me. Without that, it was still interesting, just less familiar than it could have been otherwise. But still analysis, so still good.
The morning session of research seminar made sense to me. Which was disorienting, because I am used to being lost and confused by lunch. But, even though I wasn't entirely sure how the loop diagrams we were drawing related to Thompson Group F, I understood how they behaved, and hwo the discrete gradiant vector field we applied onto it worked. And, by understanding that example a bit better, I came closer to understanding how DGVFs in general work.
For the first time, I felt that I could safely say I left the research semianr less confused than I'd been when I entered. I understood how one can build a path and determine whether it's possible to have a non-stationary closed path. I could even come up with the reasoning behind a proof that the V-path of dimension 1 has no such paths (one one the requirements for a discrete gradiant vector field). After a bit more work, I think I was able to generalize it to an arbitary dimension. So that was nice.
That understanding vanished completely as the afternoon session began. Dr. Farley kind of realized that he hadn't explained homology well enough the Friday before. So he went bacck to give more definitions and work through an example or two. Whcih turned out to be an example and a half, since he gave up on the first example we'd been trying to work on something simpler. There were a lot of ideas to define, and not enough examples or theorems to make them concrete. I left more confused than I'd been when I arrived, which is at least familiar ground.
After dinner, we worked on a smaller example of putting a discrete gradiant vector field onto a simplicial complex, and then using that to calculate the homology groups. The pieces fit together a little better than they had before, but only slightly. None of the “I understand this, and know exactly what I should be doing next” that came with the DGVF on Thompson Group F. And it's not like that group is just magic, because I was as lost as ever when it came to computing the homology of it.
Still, I really understand one thing that's going on. That's more than I've been able to claim since school ended.
