29 July 2002 - 1507 CDT - Fargo, ND
Yes, that dateline is correct, I'm back home in Fargo. w00t! It was a long, rather boring drive, but I survived. Stayed in Fort Smith and Sioux City as planned and made it back to Fargo around 1330 today. Those last 66 miles from the South Dakota border to the 12th Ave N exit were the longest 66 miles of the whole trip, from my point of view. Yesterday afternoon I stopped at a rest area in Iowa and encountered something that I hadn't felt for eight weeks—a hot great plains wind. It was awful! They don't actually have wind in Louisiana, just gentle breezes. Don't get me wrong, I don't have much against North Dakota, but who ever decided that this is a good place to live with all the gosh-darned wind? Not bad here in Fargo today, but I know that I'm going to get hit with it again soon. Well, unless I can think of anything essential that I've left off this page, I'm going to just leave it as is and archive it for the memories. I'll refer you to the Math@LSU REU 2002 website to read my paper and those of my peers. I hope to have them all compiled into a single (humongous) PDF later this week, as soon as lazy Karli Smith gets her paper done. Well, thanks for reading, and if you want to say hi, give me a call or drop me an email.
25 July 2002 - 2135 CDT
Well, I'm busy not packing. I spent today doing not much; in fact, I actually spent some time tweaking my parlipro guide for the student senate. Five of us have submitted papers so far, with others close behind. One hasn't started, and she'll be around for another week trying to finish it up. I've figured out how to compile all of the papers into a single PDF file, so once they're all in I'll treat you all to that, if you're interested. I'm staying over tomorrow yet, and then I'll head towards Fargo Saturday morning. Fort Smith, AR, will be my first day's destination, and I'll stay in Sioux City, IA, the next night. Should be back to Fargo Monday afternoon, and Matt and I get to move into our apartment Wednesday morning. Well, I'm going to leave it here since I'm not feeling particularly inspired today.
22 July 2002 - 1556 CDT
Well, I'm (almost) officially done. I gave my presentation just about two hours ago, and it went moderately well. People keep telling me it went well (appeared a bit nervous and went fast at the outset...I'm chalking it up to uncomfortability with using the overhead to give a talk), but I don't have that great of a feeling about it. I guess it would have to be because Dr. Hoffman (as always) jumped on me midway through the talk and at the end of the talk. I don't understand why he insists on doing this all the time! The first time was my fault...I pointed out a lack of symmetry in a matrix, and he then proceeded to point out that things are still symmetric with respect to the scalar product I'm using or something. I should've just kept my mouth shut when I wasn't completely sure of a certain point, but I didn't. The other thing was rather uncalled for. The definitions that I've been using have been a little complicated and convoluted by the need to use the adjoint to move between the complex and its dual (and a constant habit of looking at things in terms of matrices), and he pointed out that there is a simpler thing in terms of just the inner product I've defined or something along those lines. I just sort of gave him a "I'm sure there is" glance and shrugged him off. Dr. Stoltzfus basically apologized for that after the talk, since he had forgotten to tell Dr. Hoffman that my lack of a background in abstract algebra has led us to the matrix-heavy approach, and that would've saved me that particular pain. I've always sort of had this feeling that Dr. Hoffman's had it in for me, and today only confirmed that feeling. I hope that it's not intentional, but I guess I can't really be sure.
I'm not as vehemently anti-overhead as I was before, but I'm still not a fan of using the things to present with. They seem too limiting in what I can actually display at any given time, and there's no good way to fix an error on them. Caught one on the first slide of my talk today, and that might have derailed me a bit. I was defining a graph and said that V is a set of vertices and the edge set E is a multi-set of two-element subsets of E instead of V. Oh, well, no big deal, but it's still something that I can't just grab an eraser and get rid of. I didn't really have a choice in preparing the talk anyway though, since I had to cover nine definitions, two lemmas, and one theorem in 20 minutes. I also needed to bring in many large matrices, and there was no way that I could write them all up on the board. I guess my feeling is that overheads have their place, but I'd rather use them only for things that just take too long to write and give most of my ideas at the chalkboard. Those of you back home that will be listening to me speak on this stuff in the fall, be expecting a hybrid sort of talk. Project the darn matrices (too damn big...who wants to write up an 8x8 matrix?), but then write on the board and draw to explain things. I guess having three places to look to see what I'm displaying is also awkward...do I look at the screen? How about the transparency on the projector? How about the plain piece of paper that I have laying on the table that has the same content? Which should I choose? Constantly in a dilemma about that, so it was a little more complicated than I wanted.
I told Bogdan about my experience with the overhead at tea, and he told me a great story about Paul Erdos who always gave talks with a chalkboard. He'd write a bit, and then pace in front of the board while speaking. Then he'd write some more. Well, one time he was presenting and didn't have an option of a chalkboard, so they explained to him how the overhead was just like a chalkboard. Write on the film, it shows up on the wall. Pace in front of it if you like. When you fill the sheet, take a clean one instead of erasing. He reluctantly agreed to give the talk that way, and things started off fine. He wrote on the transparency, and it was projected on the screen. He then went to pacing at the screen and noticed a mistake. Of course, he just wrote right over it on the screen! That'll teach them to make him use an overhead!
Well, I'm going to wrap it up for now. Tomorrow I'm going to try to meet with Dr. Stoltzfus in the morning, and he'll give me the comments on the most recent draft. He's also going to try to explain what Dr. Hoffman was talking about, and then maybe I'll wind up rewriting sections of my paper in terms of that. I'm not sure that I'm so keen on doing that, but it will be something to keep me busy until the end of the week, since it doesn't seem that I have that much left to do.
20 July 2002 - 1749 CDT
Whew, am I tired! We spent about 3.5 hours today hiking in the Clark Creek Natural Area, which is just across the Mississippi border. I wish somebody had mentioned the physical fitness required for hiking there before we went! It was beautiful there, and the hike in was no problem. The problem arose when we had to hike out of the creekbeds. Lots of uphill hiking that wore me out. One of my waterbottles sprung a leak, so I had only a small bottle that I drank at lunch, and I was trying to hold out as long as I could. Big mistake. Finally I broke down and asked Michelle for some of hers, and she was nice enough to share. I wound up splashing the last of it on my face while we were sitting around resting (with her permission), and that really helped. My legs were fine throughout the hike, but I was breathing really hard on the way out, and I could tell my blood pressue was elevated, since I could feel my heart beating in my head. Would I go back? Yes! It was beautiful there, and I could handle the hike if I could convince the others to take regular breaks on the way out, since that was the biggest problem—they kept hiking at far too steady of a pace.
Only one week left?! Seems impossible, but it's true. I'll be presenting Monday afternoon at 1400, and will cover most of my project (sans proofs). This weekend I need to work on my LaTeX slides, which shouldn't bee too difficult, since it'll basically be copying things from my paper. I was kind of hoping to present at the board, but I've got too many matrices, and none of them are smaller than 5x5. I guess this will save me from messing things up or leaving out something that I really wanted to cover. The paper is coming together well, and I've had to do a few proofs from module theory that sort of boggled me. However, I'm really starting to get a feel for modules and chain complexes. Interesting, since I know very little group or ring theory! Already got my invite to speak at Math Club in the fall, and Jim, if you read this before I reply, yes, I would love to speak! In retrospect, I might have been very frustrated at times, but I've learned things (very important!) and have some things to look at in the coming year. The interesting problems are still unsolved, and as time progresses I'll only be in a better position to answer them. My paper is nearly complete. I need to write an abstract (I know some of you out there are good at that...any tips?). I need to tweak on the examples section and maybe touch up a proof or two, but I'm in good shape to get a final copy out. When the others submit their papers as PDFs, I'll be linking them off of the REU 2002 website page, so those that are interested in what my peers did can check them out.
18 July 2002 - 1153 CDT
Well, I've got quite a bit put together, and I think that I can safely say that very little of it will be cut at this point. My paper is ten pages long and lacks any serious results, but it does wrap up with some questions that I'll be looking to work on in the coming months. There's a link at the top of the page that you can follow if you're interested in seeing what I've written. You should avoid section 4 (Example) right now, as it's still "under construction." Things in bold are notes to either myself or Dr. Stoltzfus. I'm not sure what of this I'll be presenting on next week, but I have learned that we'll be giving 20 minute presentations.
Outside of math, I'm getting ready to head back to Fargo. I've never really felt homesickness before in all the times that I've spent away from home, family, and friends, but this summer I've really been missing my friends in Fargo. I hope that I haven't taken you guys for granted in the past, and I'm fairly certain that I won't in the future. I'm looking forward to getting back to Fargo and working on something that is not math for three weeks before classes start. (Not that I don't love math anymore, but doing it all day, every day gets a little tiresome. Weekends haven't had that many activities, and we've needed to work on our projects then too, so there hasn't been much time to put math aside and think about other things.) I guess that my "other" activity will be hauling textbooks around, but the exercise will be good for me.
On Saturday we're going to a place just across the Mississippi border to do a little hiking. Apparently there's a place there with a bunch of waterfalls and such, so Dr. Perlis is organizing an expedition there. Hopefully everyone has made sufficient progress that they can go with. Well, it's time for lunch! (Coming in at 0800 has made my stomach growl by 1100 lately, so I really enjoy lunchtime when it arrives.)
13 July 2002 - 2125 CDT
OK, OK, so it's been way too long since I last wrote here, but I'll try to make up for it now, even though I'm really tired. This week was basically more of the same things that we've been through the past couple of weeks. Dr. Stoltzfus and I seem to have finally settled on something to focus on in my paper, but I'm still not certain that it's going to work. I might believe it if I knew as much about Goeritz congruence as he does, but I don't, so I don't really understand what we're planning on doing. Tomorrow I need to spend a lot of time working on the parts of my paper that I know how to write, since I haven't really made any progress on that front since Tuesday. It's almost impossible to believe that I only have two weeks left here. This coming week we'll be frantically working to finish up proofs, bring together papers, and prepare for our presentations, which are 22-23 July.
As far as extracurriculars go, we had PONGfest 2002 this past Thursday. This was a big ping-pong tourney for all of us that began with round-robin groups and then moved into a final tourney. Pics can be found on the REU webpage. I didn't win any games, but it was a lot of fun, and I wound up hanging around for a good deal of extra time after the tourney wrapped up just playing some extra games. Today Emma, Michelle, Andrew, and I headed into N'awlins, after we had a hearty breakfast of blueberry pancakes, fried potatoes, scrambled eggs (with tomatoes, green peppers, onions, and cheese), and coffee cake with Emma's roommate Alexus (who's here with another REU). It was fun making the breakfast, and it was very good, but it set us behind quite a bit time-wise. We spent our time in the French Quarter, just wandering around. We took in the sights there, had some beignets (a French pastry), and then decided to head back to Baton Rouge instead of fighting crowds and paying way too much for supper. Unfortunately, our late start kept us from having time to hit any of the museums, but that was OK. Now I can say that I've been there and walked down Bourbon Street at least. OK, I'm tired, so I'm quitting now.
7 July 2002 - 1721 CDT
(Note: At the top of the page I've added a link to my working draft.) Not much for progress on Friday. I gave Dr. Stoltzfus my draft to read over and then tried in vain to find some patterns. I did get some things to not work that we thought should have worked, so that's interesting, right? Had a lazy weekend. Did some reading on matroids in West's Introduction to Graph Theory, since Dr. Stoltzfus said it would easier if I knew something about them. Still don't feel that I know anything about them, but that might change tomorrow. Tonight we're headed to Dr. Perlis' house for make-our-own pizzas.
3 July 2002 - 1936 CDT
I meant to post this link with my earlier update. Slashdot had a story yesterday on a guy that's built some mathematical surfaces out of Legos. I found them interesting, and I figured some of y'all might as well, so check it out.
3 July 2002 - 1819 CDT
A week between updates, yet again. Fortuntately, this week it's been due to progress being made. Monday we returned to the drawing board again, since it was quickly becoming clear that the angle we were going after last week was not going to get anywhere soon. We decided to look at a chain complex that corresponds to embedding a digraph into an oriented S^2. We know that this has all the information that we want about the embedding, so that's a plus. I spent Monday trying to figure out how the heck this complex behaves, and then yesterday understanding how to prove how the signed Laplacian matrix comes from the complex. Well, I first had to precisely define the concept of the dual of a signed digraph embedded in an oriented S^2. I'm sure it's something that's been done before, but I wasn't about to run around and find it, so I just reworked it. This morning I set about writing out the proof (that I'd formulated in my head the day before) and typing it up in TeX. In this process, I realized that I would need to formalize a couple of definitions along the way, since I'd just thought about them via handwaving before. I was very tempted to try to use illustrations of my handwaving in my paper, but I realized after a few minutes that it wasn't a very wise idea to try that. After much muttering and scribbling at the chalkboard, I finally worked out precisely what I wanted things to be and now I'm just working out the rest of the proof before I send it on to Prof. Stoltzfus for review.
OK, so that's what's new with me mathematically. What else is new? Not much! Today we put on a little one-act moral play entitled "Fermat and the Devil" that Prof. Perlis wrote. I played the part of Gerhard Frey, a German mathematician that came up with an idea regarding elliptic curves that was integral in the proof of Fermat's Last Theorem. (Prof. Perlis knows Frey and said that Frey thought his idea was a joke destined to go nowhere initially!) Some good-natured fun was had at the expense of Marilyn vos Savant, the Parade magazine columnist (and "person with the highest IQ") that claimed Andrew Wile's proof was rubbish. Tomorrow we have a day off for Independence Day, so a group of us will be going to downtown Baton Rouge to catch the fireworks display. Friday is an optional day of work, but I'll be in at 0900 as usual, since I don't know how else I would spend the day. (I had thought about sleeping in a bit, but then I got a request for a ride to the airport at 0800.)
27 June 2002 - 1800 CDT
Ack! No updates this week?! Well, that coincides with an utter lack of progress on my project. I really don't even think that I could put down in coherent words right now what I'm looking at. (Apparently, I'm going to try anyway.) We'd like to be able to agebraically specify an embedding of a planar graph in S^2 by using its dual as well. It's not going too well, and I keep needing to eliminate characteristics of graphs such as loops and degree-one vertices. I meet with Prof. Stoltzfus tomorrow morning, and hopefully we can get somewhere. It's really frustrating to see the others getting something (even if it is just an example or something that's already been proven) written up as a start to their papers when I'm still muddling around. Hopefully next week goes better.
Yesterday we had a visitor from Milsaps College, Prof. Joseph Palen. He spoke to us on function fields (polynomials in one variable over a finite field) and analogies to traditional number theory. Today, Prof. Bogdan Oporowski (LSU) spoke on graph minors. The talk was really interesting, even if the first few minutes were just defining graphs and talking about Kuratowski's Theorem. The rest of the group hadn't really seen much graph theory, let alone Kuratowski's Theorem, so Andrew is in awe over the fact that there are exactly two graphs that characterize nonplanar graphs. Well, I'm kind of out of steam right now, so I'll just leave it here.
22 June 2002 - 2126 CDT
Last night we went out for some Cajun. The place was called Boutin's (pronounced boo-dan's), and the food was EXCELLENT. I had Pepper Jack Shrimp, which was fresh shrimp with, spicy sausage, and pepper jack cheese wrapped in bacon and then breaded and fried. It was served with a stuffed potato, jumbalya (which is rice with stuff (I believe it included bacon), for those that don't know), and corn that again, if I could tell correctly, included....BACON! (Bob would be in heaven down here.) The corn was in a little cornbread-like pie shell that was quite tasty. I was VERY full after supper, but it was definitely good. I also had a chance to try fried and grilled alligator, which was also good (and, surprise, surprise, spicy). The place also had a band in playing Cajun music, so it was a nice experience.
Today I went to buy chalk, since LSU only buys the cheap regular-sized stuff and I want the big NDSU size chalk. OfficeDepot.com showed it, so I figured I could buy it at their store, but they didn't have any on hand. I am now going to buy chalk online. Am I pathetic or what?! (I'm still searching for a place that will actually ship me the damn chalk. I may have to go back to Office Depot and ask them to get it as a special-order item.) I'm very tired of breaking chalk and not being able to write properly on the chalkboards. I then went to see Minority Report, wihch was very good (and I didn't have to deal with any lunatics in the parking lot this time).
Not a lot of progress on my project this week. I'm seeking some algebraic property of the medial graph that will let me know when a Reidemeister move can be made. I did find some papers on MathSciNet that might have some useful information, so tomorrow will probablyl involve a trip to the library. I gave my research proprosal presentation on Thursday, and it went well. I went over on time, but only because the professors interrupted me in the middle to have an argument amongst themselves and then discussed some stuff when I was done. People were impressed with my ability to simultaneously write on the board (quickly and legibly) and talk. I just told them that they need to spend some time with 27 trig students that want to see as many examples as possible before quiz time! Well, not much else to write right now.
20 June 2002 - 0806 CDT
Just watching CNN as I get ready for the day, and apparently they'd asked their viewers to email them about where the worst traffic is, and somebody actually nominated Fargo because of all the road construction. That was the only place that they mentioned on air, so I'm guessing that they're mocking us again.
19 June 2002 - 1145 CDT
Oops! I guess it's been four days since I last updated here. I wouldn't want to disappoint all of my loyal readers out there, so I guess I'd better write some more for y'all. We met Monday morning, as planned, to discuss which problems we would each like to work on. For the most part, we divided ourselves up pretty nicely with our first choices. Four wanted to work with Prof. Hoffman on pictures, so that went easily, three of us had Prof. Stoltzfus down as first choice (two for some graph theoretic work in knot theory and one for growth functions), and five listed problems posed by Prof. Perlis as their first choice. This would've been fine and dandy, except Prof. Perlis really only had four problems for people to work on. Michelle agreed to switch to her second choice, the one Perlis problem that no one had claimed, so that took the competition down a bit. There were still two people that wanted to work on quanta, but Andrew finally agreed to move over to the "topology" group and work on growth functions.
The division done, we broke up to meet with our advisors as research groups. There was a little dividing left to do in our group, as Ryan and Andrew were both working on growth functions, but they quickly settled on growth functions for different types of groups. With that out of the way, we set up meeting times for later in the day and a special lecture for Tuesday morning at 0900 to cover automatic groups, which would be helpful to all of Prof. Hoffman's group and half of our group (not the graph theoretic problems of course). I met with Prof. Stoltzfus in the afternoon, and he had two papers for me, both by William Watkins of California State University-Northbridge (a student of Russell Merris). The first was a follow-up to an earlier paper that was not available at the library but would be provided the next day. In the two "connected" papers, Watkins showed that two three-connected graphs are isomorphic if and only if their Laplacian matrices are congruent by a unimodular matrix. In the third paper, he and M. Lien used this and some ideas from knot theory to show that the Laplacians of a plane graph and its dual are Goeritz congruent. At this point, I knew that I would be working with the signed Laplacian of the medial graph of a knot in trying to find some relation under Reidemeister moves, but the exact direction was not clear.
Yesterday I spent a considerable amount of time doing very little, as I still wasn't clear on my project's direction. I did spend time reading and re-reading the three articles I had and looking up some others in the library, but not much happened until I met with Prof. Stoltzfus at 1430. Well, that's not completely accurate, as we started at 0900 with a lecture from Prof. Stoltzfus on automatic groups. I found the lecture quite interesting, as it spent a lot of time talking about regular languages and finite automata, culminating in the definition of an automatic group. Of course, I found his definitions of finite automata and regular languages quite lacking in rigor compared to Dr. Martin's approach. (He defined regular languages to be those accepted by an FA!) However, it was interesting overall. Anyway, I met with Prof. Stoltzfus at 1430, and that made my project's initial direction much, much, much clearer.
So, your probably dying to know exactly what I'm working on this
summer, and if you're not, your going to hear it anyway! The current
title of my project is "Integral Equivalence of the Discrete
Laplacian," which is a HORRIBLE name for it, but I'm still trying to
think of a snappy title that accurately describes what I'm doing. As I
mentioned earlier, I'm looking at signed plane graphs and considering
them to be the medial graph of links (because we can always construct
the link that corresponds to a signed plane graph). Once I've got a
medial graph, I'm starting by looking at its Laplacian and signed
Laplacian matrices. From there, I'm trying to find algebraic
conditions that will allow a Reidemeister move to be made on the link
(well, I'm basically thinking of the move in the graph, and the only
really interesting one is a Type III move which does a Y-Δ switch
on the medial graph). In case anyone needs a review, I'll give you the
definitions of the Laplacian and signed Laplacian of a graph so you
have an idea of what I'm working with. The Laplacian matrix of a graph
G, is L(G)=D(G)-A(G), where
and A(G) is the familiar adjacency matrix of G. The
signed Laplacian matrix of a graph G is
where the sum is taken over all edges of G
and 
is the 
matrix with exactly one nonzero
entry in position 
and
is +1 or -1,
depending on the sign of the edge in question.
Wow! I've written a lot today, and it's almost time for lunch, so I'm going to wrap it up here. Those that seek more mathematical information from me should email me, and I'll either provide it via email or by posting it here.
15 June 2002 - 2230 CDT
Well, not a long update, but I keep thinking of things that I should write about and then never get around to it. I think I've mentioned that these apartments are really, really, really, really nice. (Matt, do you think we'll be able to match it in Fargo? :-) We have a whole set of appliances, including laundry facilities that don't require quarters. The dryer (yes, it is a dryer and not a less-wet machine) even has a "More Dry-Normal Dry-Less Dry" setting so that I don't need to think about how long to dry things. the dishwasher works well, but I'm not sure about this garbage disposal thing. The other night I had some odds and ends that I wanted to wash by hand, and I put the stopper in the sink. Once I'd finished washing, I wanted to pull the stopper out and couldn't get it. It had this little handle-like tab in the middle, but it wouldn't let me pull it out. I went and recruited Andrew to help, and he couldn't get it out either. I finally took a fork and pried the darn thing out, at which point Andrew told me that I'd put it in upside-down. Well, at least I know how things work now! The other interesting aspect of the apartment has been the bed. It started off really high off the ground, as in about three feet. I finally got tired of that last night and figured out how to pull just right so that I could adjust it. Of course, now it squeaks, so I need to figure out how to make that go away.
I spent some time in the library today reading over some journal articles in the process of picking my research problem. I think I might have figured out where I want to work, but we'll see how things go on Monday. The library here is really nice, with LOTS of journals. it seems like most of the mathematics section is journals going back half a century in places. I now know what a real library is like! (Unfortunately, the QA section is on the top floor, so I have to truck up several flights of stairs before I get to the stuff I want. This afternoon we (Andrew, Michelle, and I) were going to go to The Bourne Identity, but the 1630 show sold out right as we got up to the window. We decided to buy tickets for the 1910 showing and come back later since it wasn't that far from campus. We got back to the theater about 1830 and again had a tough time finding a parking space. When I finally thought I'd found one, there was a guy that had been waiting for the space and came over and told me to back out. He was just a middle-aged white guy, but being the friendly North Dakotan that I am, I just backed out and looked for another space. (I bet he wouldn't have bothered if I were a big burly black man!) I think I found a better parking space anyway. The movie was good, and now I'm back here winding down before I go to bed.
13 June 2002 - 2130 CDT
Well, I guess it's time for an update on Wednesday and Thursday as we head into the end of week two. Yesterday morning started with a lot of people sleeping in, but I drug myself out of bed at the usual time so that I could go in and work on the website a bit. I met with Prof. Perlis at 1000, and it wound up lasting for 45 minutes instead of the allotted 30. I made the mistake of telling him that I hadn't had a course in abstract algebra, and he launched into a tutorial on some things. It was helpful, but he went over some things that I'd already seen (equivalence relations). After lunch we had an exercise session, where we just got to work on things and ask questions of each other and the graduate students. This was much needed, as it's always easier to express to each other that we don't understand a topic than asking a question of the professors, who tend to launch into very extensive explanations even on simple questions. At 1400 we had another Magma session, which failed miserably due to poor preparation of the activity and limited time. Professor Igusa gave another lecture at 1430, again talking about pictures. Today started bright and early at 0900 again. Prof. Stoltzfus, in response to a question, got into more topology than he'd planned in discussing what exactly it means when they talk about a K(pi, 1). Professor Hoffman talked about constructing the classifying space of a group. After lunch, we had a brief exercise session and then Professor Perlis went on about the quantum number of a polynomial and some of the problems in that area. Professor Igusa's final lecture was a colloquium talk on Combinatorial Miller-Morita-Mumford Classes or some such thing. It was clearly aimed more at the faculty and moved at quite a clip, so we gave up on taking notes very early on.
Tomorrow morning I'll meet with Professor Stoltzfus at 1000. Individual meetings for others will continue until 1200, and at 1300 the professors will each take 30-40 minutes to present the problems that they have for us to work on. We'll take the weekend to mull things over and will return Monday morning for the problem selection meeting. Right now it feels like we'll be able to split ourselves up fairly evenly, as there are some people that want to work with each professor (or at least would be willing to work with them). My current leanings are actually towards working on a graph theoretic problem in knot theory. Prof. Stoltzfus has mentioned looking at how Reidemeister moves affect the adjacency matrix of the medial graph or a problem involving edge cuts in the medial graph. I could also work with Prof. Perlis on some problems in algebra, possibly related to the zeta function of a graph or finding non-trivial Gassmann triples. I don't know when I'll post again. Possibly tomorrow if I get real excited about a problem, or else it will be Monday after we've picked our problems.
11 June 2002 - 2113 CDT
Well, I guess that it's been a couple of days, so it's time for an update on what's going on down here. We had another hot day, but the humidity seemed a lot less, so it was bearable. The last two days we've been privileged (?) enough to sit through four lectures each day. There were some revisions to the schedule that I believe I'd alluded to earlier, and now Profs. Stoltzfus, Hoffman, and Perlis are lecturing each day with an additional lecture after Prof. Perlis' by Prof. Igusa. Prof. Stoltzfus has been talking about more knot and braid theory, showing us the idea behind the Yamada-Vogel algorithm (he never did get to actually showing us the algorithm, per se) and talking about the fundamental group of a knot. Prof. Hoffman has been giving lectures that tie in with Prof. Igusa's. They are both talking about "pictures," which Prof. Igusa defined in his dissertation about 25 years ago. Pictures are essentially plane digraphs with some special features and can be used to represent groups (based on a specific presentation of the group). We also have seen how a knot is essentially a picture and considering the knot as a picture gives rise to the fundamental group of the knot. Prof. Perlis has still been talking about group actions. In the past two days we spent a little time with graph theory, examining how groups give rise to Cayley-Schreier graphs. We've specifically looked at a group acting on its cosets, which gave us cospectral graphs (when we constructed the graphs from disjoint unions of graphs based on a character relation). We also got the definition of the zeta function of a graph. The next topic that we started in on just before time ran out was groups acting on the roots of polynomials. He began talking about an idea that his son (a graduate student in mathematics at the U of Arizona) has developed regarding the "quantum number" of a polynomial p(x), which is the number of roots of p(x) that are contained in the field Q adjoin a, for any root a of p(x). We'll learn more about this on Thursday.
Tomorrow will be a rather light day. We start around 1000 with individual meetings with the profs. I meet with Prof. Perlis this time, which should be interesting. Right now I'm fairly confident that I'll either work with Prof. Perlis or Prof. Stoltzfus for my research. (We have individual meetings again Friday, so then I'll get to chat with Prof. Stoltzfus.) I'd been wondering if I would get a chance to work with any graph-theoretic (or closely related) concepts this summer, but now it appears that both of these professors will have some fairly interesting problems related to either algebra or knot theory. Of course, it's not pure graph theory, but it will give me a chance to apply some of the things that I know right now (as opposed to trying to figure out all the things that I do not know about abstract algebra). Anyway, once I've seen a better list of problems I'll probably yammer about them some more here. After individual meetings and lunch, we'll have a session in the computer lab on using Magma again, which will be followed by another of Prof. Igusa's lectures and our ever-present tea. Thursday will return to the typical lecture format, with Friday being more like Wednesday (and a little unclear right now).
Since I've been documenting my computer experiences here, let me add a little bit tonight before I wrap things up. After struggling with wireless in Lockett (and not getting any help from Zed or Jeff, the sysadmins, who are having similar troubles), I finally got permission to use the hardwiring in there as needed. Well, today I connected to the wired Ethernet for awhile and then decided to try wireless, which, lo and behold, worked! I couldn't figure anything out, but chalked it up to computer voodoo. I then tried things again in the afternoon, and no luck. I then plugged in for a bit, did some work, and then tried wireless, which worked! Seems like it's something with being on the network for a bit first or some silly thing. I'm going to see Zed in his office tomorrow in the hopes that we can get this thing solved once and for all. Oh, and don't forget to check out our REU web page that I've linked to above. I can't take the credit for the design (or the fun mouse-overs on the images). Prof. Oporowski did the initial design and then turned it over to me.
9 June 2002 - 2051 CDT
I wasn't really going to write anything today, but I was just sitting here watching the History Channel and surfing the web when a North Dakota Tourism ad came on. It was a long (60 sec?) ad that made ND look really nice and inviting; in fact, it made ND look just as good as any of the other states look in their tourism ads. Well, since I'm writing, I should comment on how little I did this weekend. Yesterday I slept until about noon and then dinked around a bit. Around 1500 I went to Lockett and did some math on the blackboards in our office. I tried to get my wireless working in Lockett, but I didn't have all the necessary configuration information. Hopefully Zed can get that to me tomorrow. Today I just spent some time proofreading my typed-up notes and reviewing other lecure notes. At 1700 we went to Prof. Stoltzfus' house, where his wife prepared a wonderful supper for us to eat. We played a game of croquet while we were waiting for supper, and I almost won. I would have too if I hadn't tried to finish it off and win the game too soon. Now I'm just sitting here relaxing, since I know that tomorrow begins another hectic week. Oh, yeah, it rained again today. I believe that makes five consecutive days that we've had rain.
7 June 2002 - 2151 CDT
Well, I've survived one week. Today's lectures continued along the same paths they've been on all week. Prof. Stoltzfus introduced the genus of a knot via Seifert surfaces this morning, and that took him a little longer than he'd planned. Thus the afternoon "exercise session" became another Stoltzfus lecture since he had to cover the Fox free differential calculus before Prof. Kyoshi Igusa of Brandeis University starts his guest lectures on Monday. (Apparently one of them had promised Prof. Igusa that we'd have seen it before he started, so now we have.) Prof. Hoffman worked on clarifying the idea of free resolutions of modules and gave us some specific examples of syzygies. After lunch, Prof. Perlis continued on with groups acting on sets and linear representations thereof. We got into Gassmann's criterion and a few equivalent statements. At tea Prof. Hoffman returned from the airport with Prof. Igusa, so we had introductions and the usual tea-time banter. This weekend we're too spend some time reviewing the week's lectures to make sure that they all make sense before we head into next week. I think that I'll have to go up to our office in Lockett and work out some of the exercises on the nice big chalk boards. Of course that will require chalk...hopefully there's already a bunch in there. Sunday evening we're having a nice supper together at Prof. Stoltzfus' house, but other than that we're on our own. I think some of the group were going into New Orleans tonight to enjoy some alcoholic beverages, but there's still a good-sized group of us back here that didn't head east.
I finally got my wireless card working tonight! It was going to be my weekend project, but I finally got make-kpkg to work on Debian, so I was able to follow the HOWTO that I found for setting up my specific card. Works like a charm with my router/wireless access point combo (which will periodically lose its DHCP address and take 10 minutes to get it back, I don't get that). In fact, I'm sitting in the living room of our apartment writing this right now, and then I plan on upgrading it. Tomorrow I'm going to head to Lockett and try to get on the wireless there. The only problem that I can envision is that I don't know the ESSID for the wireless stations in there, but I think there's some way to set my card so that it will talk to any WAP. I also have suspend to RAM and suspend to disk working on my laptop, so now I'll be in great shape for using my computer in Lockett instead of the Sun machines, where I have to continually remap CapsLock and Control (I've tried automating it, but it won't go) and am stuck with Netscrape 4. I think I'll have to downgrade my BIOS and lose functional battery status indicators, however, as this BIOS is finicky about waking up and forgetting to check the temperature of the CPU. It decides that it's always 85C, so both fans blast full speed and it's annoying.
One last thing before I leave off. I'd mentioned to Prof. Oporowski who is the webmaster here (and a graph theorist, yay!) that I was having some trouble with compiling my kernel to get my wireless card working, and he introduced me to the Math Dept's sysadmin, who just happens to be a serious Debian developer. In fact, I did some searching today and found out that he's the licq maintainer (among many other packages). (If any of you out there are still having licq issues, let me know and I'll talk to Zed about them.) He's the typical condescending, my-way-or-the-highway UNIX hacker type, but he was a little bit helpful with some things and seemed willing to chat a bit about things. He explained the issues with getting woody out the door and such, so I can tell that he's involved with more than just the packages for which he's formally responsible. Well, I'll leave off here for now and will post later, likely Monday.
7 June 2002 - 0804 CDT
Sorry for the lack of updates the last couple of days. I guess I've had to prioritize doing REU things over writing about them. Wednesday we broke out of the lecture format, starting the day with a LaTeX session. It was really boring and useless for most of us that have used LaTeX before, but at the same time wasn't enough time for those that didn't have any experience. After that, we had individual meetings with one of the three professors. I met with Professor Hoffman, and we just chatted about my background and interests. Strangely enough, he claims that there will be some problems involving graphs that come out of his lectures. I think I'll believe that when I see it. After lunch we had our first exercise session that actually gave us a chance to ask questions about the exercises that had been assigned. We were then supposed to go to the lab and do a Magma tutorial, but computer systems were being rearranged, so Professor Perlis presented more on group actions and linear representations of groups. Thursday we went back to lectures, with Professor Stoltzfus talking about monoids, free monoids, free groups, and finally the braid group. He says he's on his way to the Fox differential calculus. Professor Hoffman did more with cohomology and finished by introducing the term syzygy. I've been assigned to type up the notes in LaTeX form, so I might link to them here when I've got them done. In the afternoon we had another short exercise session where Professor Perlis re-explained some of the material from Professor Hoffman's morning lecture by request. We wrapped up the day by using Magma to put the matrices of a cochain complex into Smith Normal Form, simplifying the computation of the cohomologies. I spent most of last night working on typing up the notes from yesterday's lecture, and Wednesday night was a trip to the Super Walmart for groceries and some needed items. I also picked up a wireless Ethernet card for my laptop, since the math building has wireless in it! Of course, now I have to fight to get it to work with Linux, but that's my weekend project.
4 June 2002 - 1902 CDT - Baton Rouge, LA
(Unless otherwise stated, all datelines will be Baton Rouge from now on.)
Day two of the program down here, and it's been quite interesting so far. I would have written yesterday, but I couldn't get my Ethernet connection activation completed until today. Additionally, I needed to spend some quality time with Dummit and Foote's Abstract Algebra in order to catch up with everyone else. The weather is hot and humid down here. Temp right now is 84F, but was above 90F when I got back from tea this afternoon. All the REU people (12 students (6 male, 6 female), 4 grad assistants, and 3 faculty mentors) are really nice, and we make a good group. Our personalities are all really different, so that makes things interesting. I think I'll spend a few minutes describing the setup and then the lecture topics before I get back to studying.
The first two weeks have a format that seems more like school than an REU. We have one-hour lectures at 0900, 1030, and 1330 from the professors. At 1300 we meet with the grad assistants for an "exercise session" that is apparently going to resemble a recitation where we can discuss the exercises given by the profs. at 1500 we have our tea time in the math lounge. We've had a number of other grad students and profs stop in to chat and enjoy the cookies, coffee, tea, juice, fresh veggies, chips, and whatever else they have for us. After tea we're on our own basically. We've been issued keys to the math building (Lockett Hall) and an office in there. (It looks more like a seminar room with a lot of furniture than an office, but I'm not complaining!) Tomorrow will have a different structure, with one-on-one meetings with the profs in the morning. We'll also spend some time learning/reviewing LaTeX (a document typesetting language) and exploring Magma (an abstract algebra software package). After these two weeks are over, the fun begins, and I'll write more about structure then once I understand it.
(This paragraph will likely be filled with a lot of math jargon, and I'm not going to try to simplify it for my nonmathematical readers, sorry!) Prof. Stotzfus is teaching us some knot theory, but he's decided to avoid the topology that is needed to take the traditional approach, so we're using graphs to define our knots. We have to definitions, one that uses a four-regular plane graph with crossing types indicated where vertices correspond to crossings and edges to the pieces of the knot. The other is the medial graph, which results from coloring the regions of a knot projection. Prof. Hoffman is trying to do some homological algebra (specifically (co)homology of (co)chain complexes and group homology so far), and we're not so sure about it. I've never had abstract algebra, so that's why I had so much reading to do last night. Some of the others didn't get to rings in their algebra classes, and I don't know that more than one person was comfortable with the idea of a module. We're getting some ideas down, but I'm still feeling a little lost. Hopefully it will make more sense when I review the lecture notes. (We all have to take turns transcribing the notes into LaTeX and posting them for everyone.) Prof. Perlis is talking about group actions, beginning with actions on sets and proceeding to actions on groups, rings, modules, and "fuzzies" (any future algebraic or continuous object that someone might think up). Strangely enough, most of this has made sense to me even with my limited understanding of algebra. I have the feeling that my best shot of working with graphs comes with Stotzfus (I wouldn't mind doing some knot theory if we use graphs) or Perlis (a graph is an object with structure, so why not). Next week they'll have a visitor in that's going to talk about more homological algebra (yay?), so we'll be rotating through the mentors (two per day) and hearing the visitor in the afternoon.
2 June 2002 - 2245 CDT - Baton Rouge, LA
Well, I've finally reached my destination for the summer. I left Texarkana at 0830 CDT on an empty stomach this morning, due to my hotel not making clear where there continental breakfast was and lack of anywhere nearby that looked promising for breakfast. I had planned to stop in Shreveport, LA, about one hour into my day's journey. However, the bypass I took put that out of the question, so I just munched on the food I had in the car. I stopped for lunch in Lafayette, about 50 miles west of Red Stick. Called home and chatted; then I called Prof. Stoltzfus to let him know that I was on my way. Not too far outside of Lafayette I encountered a sign that told me I was on the (something) Bayou Freeway. I didn't think much of it until all of a sudden I'm on a bridge that is MILES long. They literally built the interstate through the swamp! This was an interesting part of the drive, and I had to make sure that I was paying attention to my driving and not gawking around at the waters around me. As I approached LSU I crossed the Mississippi River on a big high bridge that rivals some of the hills back home for steepness and such. I found LSU easily enough, but had very little idea where to find the East Campus Apartments. I drove around lost a for a bit and then remembered that the email had said if I reached the light for Highland Drive I'd gone too far. Well, I'd passed that intersection, so I returned and with some good luck found the apartments. Check-in went smoothly and I met a couple of the other REU guys in the parking lot. The apartments are VERY nice! Anyway, much time was spent unpacking. Andrew helped me move my stuff up, which was really nice considering that our apartment is on the THIRD FLOOR and it was humid. (The locals have told me that today was not humid, so I'm not sure what to expect in the future.) The campus is nice, but seems quite large. I'm sure a map will solve that. No Ethernet connection until tomorrow. This is a great disappointment, but I will survive. (I did find some public access terminals to check my email on for the first time since Friday.) At 1800 we all met up for a walk around campus. It wasn't too much of a tour, but rather hit the places that we'll need to know for tomorrow morning. Had supper at The Chimes just across the street from campus. After that, we just sort of hung out while people watched the end of the Lakers-Kings game and then dispersed for the night. I decided that I needed groceries, so I ventured out to find Winn-Dixie with directions provided at check-in. The directions had at least one fatal flaw; the most important was that it didn't say to head south of campus, so I went north until I hit a bad neighborhood and decided to turn around. I then had better luck finding the needed street, but didn't realize how close the store was to the intersection where I turned. Since there was no sign for the store along the street, I drove right by until I hit a residential district and decided to turn around. Then I found it, bought my groceries, and returned to campus without problems. I'm now winding down for bed by writing this summary. G'night!
1 June 2002 - 2200 CDT - Texarkana, AR
OK, so I would like to do this sequentially, but I just got back from seeing Insomnia. I don't know why, but that movie messed with my head a bit. It just put me in a thoughtful mood, but I'm not sure what I want to think about. I don't drink, but I have this feeling that this mood might lead me to want to.
Anyway, I'll try to describe day two of my travels as best as I can. Since I decided to eat continental breakfast at my motel, I hit the road at 0830 CDT this morning from St. Joseph, MO. The drive through Kansas City was interesting. I did well, but keeping in the right lane was interesting. The most surprising part was the stop lights on what otherwise would be described as a freeway. Speed limit didn't dip below 45, so traffic kept moving at 55-60, and red lights were a pain. Outside of KC it was smooth sailing until the four-lane ran out and I had to head for the two-lane. It's a little surprising when you're just cruising along and all of a sudden there's a sign that says the freeway ends and all traffic must exit immediately. Pretty soon I was in Arkansas, and it opened back up to four lanes again. However, I decided that I knew better than my driving directions (courtesy Mapquest). I decided this since, earlier, they'd had me drive through Neosho, MO, on some little US-71 Business Loop that I didn't need to take. Well, I'm glad that I did forget the directions for a bit, since I stayed with US-71 instead of taking I-540. US-71 goes right up through the Ozark National Forest, and it was beautiful. I didn't stop to take any pictures since the trees along the road were thick, but I hope to get some on the way home. Had lunch in Fort Smith and then hit the road again. I'd bought a map in some little town in the Ozarks when I filled gas, and it kept misleading me since it has a far different scale than I'm used to seeing on road maps (15 miles on the map looks like about 50 to me). The map did treat me well, and I made it to Texarkana at 1750 CDT without any troubles. I was a little surprised when US-71 widened out to four lanes again and there was a big "Welcome to Texas" sign. I hadn't realized that I'd be in Texas except for my planned trip to the movies. One general observation about Arkansas. Wait, make that two observations. First, every little town I drove through had a brand-spanking-new Post Office building. Apparently Ol' Bill managed to funnel some bucks into the backwoods. Second, little towns here are much bigger than in North Dakota. No, I don't mean that they always have more people, but they're more spread out. The city limits sign is usually about a mile out of what we'd consider city limits back home, so it was always confusing. I'd be thinking, "Hmmm, this is sure a small town," and then there'd be a speed zone and the town proper. Well, I think that's all. Only 340 miles to go and I'll be in Baton Rouge!
31 May 2002 - 2200 CDT - St. Joseph, MO
Day one of my travels to LSU went well. I finally hit the road (I-29 Southbound) with my car full of stuff at 0945 CDT. Driving went well, but I did manage to hit road construction in EVERY major city along my route. (Sioux Falls had a sign up suggesting that I consider an alternate route. I would've loved to consider an alternate route, but I don't know if I ever would have found I-29 again.) The South Dakota Highway Department needs to learn not to leave their cones lying about when they're out striping. There were on ramps where drivers had to stop and get out to move the cones out of their paths, and other cones blown into the middle of the lanes. Drove straight from Fargo to a rest area south of Sioux Falls for a pit stop, then continued on to Sioux City. I wasn't sure where to stop, so I finally hit a little town on the outskirts, Sargent Bluff. After I ate at Subway (a bit after 1400 CDT), I tried to get back on I-29 at the exit where I got off. Of course that was impossible (ramp completely destroyed), so I had to backtrack two miles to head south again. Then I hit Council Bluffs/Omaha at 1615 CDT on a Friday, so there was a bit of traffic trying to navigate all the one lane sections with ramps backed up quite a bit. After much nervousness about the State of Iowa conspiring against me, I managed to navigate through and continued onward. Arrived in St. Joseph, MO, promptly at 1800 CDT. Checked in at my motel and then headed off to the theater to see The Sum of All Fears. I had gotten directions online before I left, so I made it there just fine, and found Perkins along the way. The movie was excellent. I think I'll have to get my copy of the book from home and read it again, since it's been 8 years since I last read it. The drive back to the motel was interesting. Missouri drivers have been pretty good, but I was following some real bozos that like to stop short. No accidents, fortunately, but I missed my turn and had to turn around in a parking lot. I'm winding down now, will head to bed in a bit. Tomorrow I'm off to Texarkana, AR.
25 May 2002
Here I sit, alone in my apartment in Fargo, wondering what to do. I've realized that living alone is very boring. I haven't talked to another person face-to-face since the wee hours of Friday morning, so I'm feeling a bit isolated. It's nice not having any obligations, but at the same time, it makes me completely lack motivation. Oh well, soon it will be the end of the month, and I'll be off for my REU at LSU.
