Continued this week to play with some toy examples to figure out what's going wrong with our earlier proof attempt. It's an interesting little set system we've uncovered. Packed up my life---goodbye Rutgers!
Prof. Allender introduced yet another technique this week, one which we think will allow an improvement to one of my results from a few weeks back. We're not sure yet how to make something coalesce, though. Later in the week, lacking novel things to try, I returned to an earlier, simpler technique to dig into the roadblock we hit there and flush out what's going wrong.
Worked with a new technique this week, one with a slightly more combinatorial flavor. Thought I had something worthwhile, and even had Prof. Allender provisionally convinced, but it turns out that log(a-b) is not equal to log(a) - log(b). Ah well.
There is little to say. The theorems ebb and flow, but mostly they seem to ebb. We continue to run into roadblocks with every proof technique we try. Did my second presentation to the rest of the DIMACS crew -- here are slides and notes. Reader beware: they are not quite consistent with each other.
The mathematics is behaving better this week. We finally have a couple of firm little facts well-outlined in the darkness. There are a couple more avenues to explore, and as more becomes illuminated I am writing a small document to send to a collaborator of Prof. Allender in hopes that he has further ideas.
I am still alive. We are not sure how to directly attack the conjecture, so we have set about fishing around in the dim for small truths. A fairly substantial one nearly surfaced, only to be drawn back under the current when a small but irreparable flaw was discovered in the heart of its proof. Disappointment ensues.
I'm working on a proof of the conjecture in the notes below. So far many little lemmata but no real proofs have surfaced (although there are plenty of likely-seeming candidates). I continue to meet with Prof. Allender a few times a week. In my copious free time, I'm working through a recursion theory book with an eye towards attacking another conjecture made by Prof. Allender in a recent paper.
I met with Professor Allender on Monday to discuss potential projects, and I've been reading in a few different directions in order to acquire a feel for the flavor of the mathematics in each potential area. I gave a short presentation on one potential path on Thursday at DIMACS. The notes are here.
I arrived at Rutgers on Tuesday (June 7). My first week, I finished my last final for school and settled in to my quarters. By Saturday, I finally had time to do some mathematics, so I worked through readings from Prof. Allender.
The University of Washington runs on quarters rather than semesters, so week one finds me still in Seattle studying for finals.