This is Carle, and when Charlie asked me I had no idea why he chose me to write this episode. Well, asking Allen or me is fairly obvious, since we're the main characters. So I asked Charlie, "Why me, not Allen? He's the senior."
Charlie said, "I admire your ego, Carle. And it's going to take a very strong ego to tell this story and not sell Allen or yourself short."
I'm still debating whether or not that was a compliment, but Charlie insists that it was. Well, here goes.
There are really three stories to tell: of our involvement with the Higgs team; of our work on climate change modeling; and of our relationships with the Gang, especially the Higgs team. All three were exciting.
Allen and I divided our work on sixth and seventh order equations, which were important for both the search for the Higgs and for the analysis of massive amounts of weather related data. While we did bounce ideas off of each other, we felt we needed to divide the work in some logical manner. So I took the job of reviewing the proof of the Abel-Ruffini theorem of the nineteenth century which stated that fifth and higher order equations cannot be generally solved by factorization into radicals (i.e. algebraically), though they can be solved trigonometrically. Allen took on the task of searching for key subsets of such equations which could be solved algebraically; if the theorem held, but the set of equations that could be solved algebraically could be sufficiently expanded, we would've resolved the issue.
I suspect that I've already shared more mathematics than you're interested in, but I'll have to give you a little bit more. The Abel-Ruffini theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, and Niels Henrik Abel, who provided a proof in 1823. Évariste Galois independently proved the theorem in a work that was posthumously published in 1846. It was my job to tackle those proofs and see if they were, in fact, correct, and if they were, to see if they presented any clues to Allen's half of the problem.
It took very little time to realize that the proofs would be difficult to overturn. The theorem states that there is no general case algebraic solution to fifth and higher order equations. To prove that there is no general case solution, one need only find a single case with no solution and one has proved that there is no general case solution. It didn't take long to determine that single cases existed that could not be solved. However, the proof was still very complicated, and following through all of its pieces–especially the work of Galois–showed a lot of the possibilities for expanding the realm of solvable cases.
My part of the work was now done: The Abel-Ruffini proof was valid. At the same time, I'd discovered groups of equations that were solvable and so had Allen. Allen continued on that quest, while I set out to define exactly the nature of the equations that we'd have to solve to analyze the Higgs and climate data.
I won't even try to go into details, but after about a month of work I'd pretty much defined the equations that we needed to solve and Allen was able to determine that they fit into his solvable sets. Now the problem was to find the algebraic solutions that our theoretical work suggested had to exist.
That was the hard part, but with Allen's assistance, I finally developed a protocol that worked for the equations we anticipated would need to be solved as the Higgs, and subsequently climate, data was generated.
That was about two months work, and except for the grand night at The Hideout that Tim has already described and the events surrounding our joining the Gang, we didn't see much of Kay, Kevin, and Roger during that time. Then one day we walked into their office at 314 Witmer and dropped copies of a summary of our work on their desks. The key part of the document was the step-by-step protocol for solving the equations we anticipated.
Kay picked up her copy and said, "What's this?"
I said, "That, madam, is the solution to your problem. We're done. You don't need us anymore. Oh, yes, that document's close hold. It's for your work on the Higgs only. It'll become our Ph.D. dissertations, and can't be publicly released until we've defended."
Roger, who'd been listening to the interchange, said, "My God. I doubted that you could accomplish the job, much less in your timeframe of six months. But you did it in two months. I don't mean to doubt, but I have to ask, are you sure?"
Allen said, "We are, but Kay's going to have to take some real data, start the analysis, develop the equations needed to be solved, and then apply the protocol. If it works, then you can be as sure as we are."
Kevin said, "And where do we get real data, since the machine we're designing hasn't been built yet?"
Kay said, "Kevin, that's simple. We know what these data streams will look like, we can have a computer generate random streams, that are within the parameters that are possible. Data analysis of such data streams will usually lead to the conclusion that the data are invalid, but the analysis will be just the same as if we had valid data. And, from time to time, the random data will be valid, but will certainly fail to show a Higgs. It's an impossibly small possibility that a random data set could show the existence of the Higgs (think of monkeys typing Shakespeare), but we don't need that to be confident of our procedures."
She continued, "Listen, you two, can I get you to help me write the program that'll develop the random data streams? It's very clear that when mathematics is involved, you two simply must be involved."
I answered, "Sure, we're on board. But we don't want that last thought spread around campus. We want to work on climate data, not everybody else's research project. Doing this for you has been great fun, and the work is as important for our climate research as it is for your Higgs research. Other research projects on campus won't dovetail like that."
Kay said, "We'll keep mum. And we'll keep this protocol under wraps. Once an equation is solved, the protocol isn't needed to check it, so people that review our work don't need to have the protocol to establish that our work is valid."
Kevin said, "It's celebration time. Let's have steaks at the Dakota Steak House and then really celebrate at Gangland."
Kay said, "And I think we'll leave our non-professional partners at home with each other: Mitch and Grant will know how to entertain themselves, and I'm sure that Cam and Noreen will as well."
I asked, "You sure that they won't mind?"
Roger said, "Mitch's already talked about the need for the scientists (his word) to get together from time to time. Besides, that way he doesn't need to feel guilty about leaving me to my physics while he plays with Grant–sometimes painting and sometimes symbiosing (my word)."
"Dakota Steak House, it is. How about 7:30?"
Allen and I hadn't been back to Gangland since we'd been introduced to it along with the unveiling of our pictures. As we thought about it, and discussed it as we drove to the Dakota Steak House, we decided that this evening was going to be a sort of bellwether event. We were moving from sexual engagements that were planned by others to make us more comfortable with our sexuality and with sex with other than our partners to sexual engagements that we initiated because we enjoyed them. The five of us tonight were going to have a ball, and it wasn't to be a learning experience, but simple a social or recreational experience with very close friends; perhaps they would be better characterized as lovers.
At dinner Kevin asked, "Are you two familiar with the shower in Gangland?"
"No, is it similar to the one at The Hideout?"
"Yes, but more so. I think that's where we should start tonight."
Coming back from the steak house Allen rode with Kay and Kevin and Roger rode with me. Roger confided, "You know, Kay, Kevin and I have only had sex together the time we were together with you. My sex has almost exclusively been with Mitch."
"You know, Roger, Allen and I aren't much more experienced, except that we were introduced to the Circle, I think largely to give us a broader sexual experience."
"The Circle must be a pretty extraordinary group."
"Wild and wooly, but also kind and considerate. And they're not all out of one mold. They range from divers to football players, from gardeners to university administrators. But they get along amazingly well. The place seems to function like a Swiss watch."
"So what're we in for tonight?"
"Kay and Kevin are familiar with Gangland, and the ways of the Gang, of which the rest of us are simply the newest members. I think we let Kay and Kevin take the lead tonight."
As we all rode the elevator up to the fourth floor of The Carl, I think that Kay and Kevin realized that they were going to have to be the leaders. But when we got to the door, they stepped aside and let Allen and me approach the door. I'd forgotten the combination, but Allen was somehow able to pull it out of his deep memory bank. Kevin laughed and said, "Carle, you're going to be missing a lot of fun if you can't remember that combination."
We all headed through the door and down the corridor. I was intrigued with all of the pictures and started to look at each one very carefully. Kay came up beside me and said, "Gorgeous, sexy people, aren't they?"
"They sure are. Is there a rule in the Gang that fat people aren't allowed?"
"I guess it looks that way, doesn't it. I'm sure that we'd never think in those terms. But you have to remember two things: first, a lot of the Gang came together through sports; second, the group ethos is to keep active and fit. Just being a part of this group encourages those who develop paunches to work to get rid of them. Ronnie runs a mile every morning to keep his under control."
"It certainly works. He doesn't have any paunch at all."
"As a teenager he started to develop one, and just being a part of the Gang encouraged him to do something about it. He asked for help from the Gang–then eight boys–and he got it. That's when he started his mile-a-day regimen."
"Are there other examples?"
"I guess, but that's the story that's part of the 'lore' of the Gang."
"As I look at these pictures, the more recent ones seem to be more pornographic."
"Sid started out with a goal of 'erotic.' Over time, he's admitted to pornographic tendencies."
"I'd call the one with us controlling a helicopter 'playfully pornographic'."
"Sid would love that. Tell him the next time you see him."
Allen came up and asked, "So, are these pictures sort of a menu?"
"You could say that. The rule in the Gang is that it's always acceptable to ask, and it's equally acceptable to say, 'No.' So, have no qualms about asking, and don't be offended by, 'No.' But I think the noes will be rare. You and Carle are just too cute to turn down. By the way, that rule works two ways: don't be offended by invitations, don't be afraid to turn them down."
Kevin said, "Let's hit the showers," and before we all realized what he was doing he had stripped, dropped his clothes where he stood, and trotted to the shower. Carl had originally installed four shower heads, but decided that wasn't enough. So now each head had a hose and hand-held shower head, with a diverter that allowed both the fixed head and the hand-held to flow at once. He turn the fixed head on himself and stood ready to spray the genitals of each successive person to enter. Kay said, "Thank goodness you used the hand-held attached to your head, that way you can't spray us with cold water as you shower under warm."
"I didn't think of that; I should have."
By now all of us but Allen were standing under a shower head, and all of us were spraying Allen with a hand-held. He was wetter and warmer than any of us. Then Kevin motioned to Allen to take his place, and Kevin was the object of the hand-helds. He quickly knelt in front of me and took my dick in his mouth, sucking gently.
Kay said, "Don't let him finish you. Before you come, take his place and suck someone else." That led to a chain reaction, of which I was number two. I had to pick number three and get started. With three gay men, it'd be an interesting question of who would tongue Kay. I decided that I would, rather than pass that challenge on to a different gay man. So I knelt in front of Kay, and had a new experience of trying to drive my tongue into her while she and three boys were spraying my face and my genitals with strong shower sprays. She seemed to enjoy it, but soon pushed me aside and headed for Roger, who continued the chain by sucking Allen, who then sucked Kevin. And that was just a warm-up; there had been no ejaculations!
We got out and dried each other off (that was interesting). Kay then said, "OK, we have three gay men who are fairly new to this sort of thing. Kevin and I have talked and we suggest that the three of you take turns sucking him, and the one who collects his cum, will be the first of the three of you to fuck me. And I expect to be fucked by three very horny men, all of whom think they're gay, but all of whom are going to find out that straight is mighty exciting."
I won Kevin's prize, which he insisted that I swallow. Luckily the Circle had insured that wasn't a new challenge for me. While I'd played with Marge at the Circle, and had fucked Noreen's ass, I'd never had vaginal intercourse. I was ready for a new experience. Kay was a good guide, and it was a very satisfying experience. After we'd separated Kay said, "You're still gay, aren't you. I could tell."
"Yes, but that was a most delightful experience. And you were the perfect guide. Thank you."
"Believe me, you were a lot of fun. Being someone's 'first time' is always exciting. Thank you."
Roger and Allen had lost their virginity the night I ass-fucked Noreen, but they were equally enamored with Kay. When they'd finished, Kevin herded us all back into the shower for a much needed clean up. We got dressed, and Kevin announced, "OK, everyone back at Witmer at eight; we have to start working on a computer program." We all groaned. Kevin added, "Kay and I'll clean up here; Carle and Allen, will you run Roger home?" We all offered to help with the clean up, but were told that the sheets and towels would wash quickly and then be tossed into the dryer and left. They'd make the bed while the old sheets were washing, and be out in a jiffy. We were told to go, and we went.
The next morning we all met in their Witmer office, and started designing several different programs that would be necessary for the Higgs research. We learned very quickly that we could work together effectively, in pairs or larger grouping. In talking about that later, as we reviewed our efforts together, we credited two things for our ability to work so well together: First, our deep mutual respect for each other's abilities. This came from knowing each other's backgrounds, but also from being exposed to their work in progress. Second, we were all convinced that our closeness outside of the work environment led to closeness in the work environment that enhanced our effectiveness. We're well aware that there's a conventional wisdom that says that sexual relationships in the workplace undermine effectiveness, but we don't believe it. We agree that would be true for secretive or illicit relationships, but we firmly believe that our open relationships are highly conducive to our work successes.
And were they successes! You know, of course, that Kay, Kevin, and Roger were entirely successful in their search for the Higgs Boson, and ultimately shared the Nobel Prize for Physics for their work. We were included on the trip to Stockholm when they received their prize. We were totally delighted at their achievement, and we certainly appreciated the credit that they gave us for the underlying mathematics needed to design the hadron detector.
After getting the three of them on the mathematical road that would ultimately lead to the Higgs, Allen and I debated our next move. We were in an odd situation: I was just finishing my junior year as a math major, and Allen was finishing the second year of a three year program that would lead to an M.A. in Education and an M.S. in math. But we'd completed research that most certainly would qualify as Ph.D. theses, and outstanding ones at that. We decided that we needed to seek advice, and we turned to Dr. Anderer.
He listened as we described our situation, and understood immediately the issues involved. He started with, "You're telling me that you have advanced the analysis of fifth and higher order equations by, first, showing that the universe that can be solved algebraically is vastly larger that we have believed, and, second, developing a protocol for solutions to large numbers of that universe. Is that right?"
I said, "Yes. Exactly."
"And you believe that the first is worth a Ph.D. dissertation, as well as the second. Two men, two dissertations, two Ph.D.s; is that right?"
Allen said, "Yes. Exactly."
Dr. Anderer said, "I know better than to challenge your mathematics. If you say you have accomplished those two totally impossible tasks–at least, impossible by all conventional mathematical wisdom–I'll take it on faith that you have. As to your analysis of the worth of this accomplishment, Ph.D. dissertations are trifles compared to the recognition work like that'll get you.
"OK, let me ask you two other questions. First, and this is very important, to me and to Tim. Are we likely to be able to keep you two here at this university after we hand you a Ph.D.?"
I said, "The answer to that is very simple. In the last few months our lives have been totally shaken up by our relationships with Kay, Kevin, and Roger, and a number of other people in Grand Forks. We really want to continue our lives here in Grand Forks, on the faculty of this University if that can be worked out."
"Believe me, it can be. Tim and I have already talked about what it would take to keep you here."
"And what did you conclude would keep us here?"
"We were thinking along professional lines, not the personal relationships you've developed."
"And what did you conclude would keep us here?"
"The development of a well-funded institute to study climate change, with emphasis on mathematical modeling. We, well, I, envision it being a part of the Mathematics Department. I'm looking at its first two co-directors."
I said, "We accept. But you can't really appoint us to that position without the degrees we haven't yet achieved."
"Oh, yeah? Try me. But let me move on to my second question. I know you're both very smart mathematicians–honestly, the best I've ever met, and that includes all of our faculty. But that doesn't mean that you don't have a lot of mathematics, and other things, to learn here at UND. In the normal course of things you'd spend another four years as students here, at least one of which would be consumed by dissertation research. If we lopped that year off, could you use three years profitably?"
Allen said, "We'd like to take every advanced math class you offer. We also need more physics and chemistry, as well as meteorology. We can fill three years."
Dr. Anderer asked one more question. "You did this work on higher order equations on behalf of the Higgs project, didn't you?"
"Tell me how relevant is it to your climate modeling?"
"Completely relevant. We'll have to deal with the same masses of data as the Higgs people have to. It's just that ours is weather data. But digital data is digital data, and our equation protocol will be essential to the analysis of climate change data."
"Wonderful. Then the publication of your research will be the basis for my appointment of you two as co-directors of our new institute. We have to move this forward quickly. Just how soon do you think you could have publishable dissertations?"
"We'd need to be in a program, have advisors, submit proposals, all sort of things."
"That wasn't my question. I'll be your advisor. How soon can you publish and defend?"
I said, "Well, the math is done. We reviewed the nineteenth century work–that was when the proofs were developed. We just assumed that nobody had seriously challenged the state of the art since, or it would've been widely published and a lot of fuss made over it. But we'll have to do a lot more literature research. We'll have to deal with the advancement of trigonometric solutions, to show that algebraic solutions are so much better."
"So, how long?"
Allen said, "A month or two."
"My God, you guys work fast."
I said, "Actually when he said a month or two, I was thinking a week or two. It's somewhere in the middle. We're quite certain we aren't going to come up with anything that's going to get in the way of our publication. Most of the writing is done, it was in the document we gave to the Higgs people."
"Well, I have to issue one caution–to all of us. What you can do, so can some genius at M.I.T. or Harvard, or even NDSU. And if somebody publishes first, you guys are shit out of luck, if you'll pardon the expression. So let's move. I'll talk to Tim, he's the boss of this place, and tell him we need to get you admitted to a Ph.D. program, right now. Somehow I don't think that'll be a problem for him. As soon as that happens, I'll officially become your Ph.D. advisor. Bring me drafts A.S.A.P. Better yet, let me have a copy of the document you prepared for the Higgs group. Then get me dissertation drafts as soon as you have them more or less complete."
As we left Dr. Anderer's office I said to Allen, "That was quite a meeting. I wonder if he can really make all that happen."
"Clearly, he's been talking to Tim. In that case, I'm sure he can make it happen. So, just what did we get out of that?"
"Well, first, we're going to be Ph.D. candidates very soon. It isn't unusual for a Ph.D. candidate to be working on a Master's degree preliminary to the work on the Ph.D. But working on a Bachelor's is certainly unusual. And it appears we can lop a year off our expected program length. And we seem to be the chosen leaders of a soon to be created Climate Modeling Institute, or some such name. It makes my head swim."
The next day at lunch the RA for our dorm found us in the cafeteria where we usually ate and told us that President Tim was looking for us. We should get to his office right away. We ate quickly, and headed for Tim's office. His door was open, and he saw us walk into the outer office. He came out, greeted us, and invited us into his office. He got right to the point, "Damon Anderer was here early this morning, quite excited. He told me about your conversation yesterday, and now I'm excited. By the way, he was talking to me about you a year ago when you published that article in the American Journal of Mathematics. He, and I, decided right then that you two were keepers. We were right, too. Of course, Damon doesn't know my personal history with you two. He might enjoy that story, but it needs to come from you, not me."
This certainly made us feel good, but we hadn't had a chance to get a word in when Tim continued, "Now, we have to get a few things accomplished pretty quickly. I've talked to the graduate dean and while it would be quite unusual for an undergraduate to be admitted to a Ph.D. program, it is possible. So, here are application forms for admission, and I'm afraid you'll both have to fill them out. There is a request for an essay, simply put in 'Waived by Tim.' The bibliography we will need, in detail. We have all your transcripts on file, so just note that. Get those back to my secretary tomorrow, if you can. I know that you're going to be working hard on shaping your dissertations, but you need to sit down with Damon Anderer and shape the plans for a climate modeling institute, or whatever it's going to be. And don't just accept what Damon suggests, tell him what you want. And don't be frugal, be expansive. If we're going to do it, we're going to do it right."
Allen said, "Would you mind if I caught my breath, just a little. This is going pretty fast."
Tim continued, "You'll catch up. You're both smart kids. We need to think about your committees. You need separate ones, but Dr. Anderer will chair both. I think I'll be on yours, Carle, and Charlie will be on yours, Allen. We'll be the outside of the department members. Use Kay and Kevin as well. You can each pick a couple of mathematicians as well, preferably ones that can understand the mathematics, or are you two the only persons on campus that can understand what you've done?"
Allen replied, "No, creating it was difficult. Understanding it, and seeing that it works isn't that difficult for a good mathematician, and there are a number of them in the department."
"I hate to lay another burden on you, but you both have to have journal articles all ready to go to AJM just as soon as your defense is approved. We don't want this to trickle out, we want to make a loud boom."
I said, "I think that this will take the full two months that was Allen's outside time estimate. Maybe a little longer."
"Dr. Anderer will help as much as he can, but of course he can't write for you. But he's a good editor, don't feel bad about using him."
That afternoon we stopped by Dr. Anderer's office and left a copy of the paper we'd prepared for the Higgs group. He came in just as we were leaving and he brought us into his office. He said, "Look, you guys, I know you feel like you've been caught in a whirlwind, but Tim is determined that this all go forward as fast as possible. He doesn't want to be scooped on this. I don't think he's worried about somebody upstaging your research, but he doesn't want any announcement of another climate modeling program made before UND makes its announcement. He wants UND to be the leader, not a copycat. Don't worry about your current classes; I'll arrange for you to get incompletes if needed, and the work can follow. Now, I never thought I'd be saying this to a college junior, but go finish those dissertations."
"We've already started, Dr. Anderer."
"And skip the Dr. Anderer stuff. We're colleagues. Call me Damon, and I'll continue to call you Carle and Allen."
"That'll take some getting used to."
A few days later the graduate dean raised a new issue. The rules stipulated that you could not officially submit a dissertation proposal, much less a finished dissertation, until you were formally admitted to candidacy, and that meant passing the Ph.D. preliminary examinations in mathematics. Damon called us in and asked us if we thought we could pass the prelims now. Allen replied, "I'm sure that I can, and I can't believe that Carle can't. I assume that the prelims don't go into mathematical specialities that only a minority of Ph.D. candidates deal with. If the prelims are limited to a standard common core that all Ph.D. candidates need, then we're fine."
We were fine. The department approved a special session for our prelims, and we spent (wasted?) a day showing that our mathematical talents were sufficient to be considered Ph.D. material. In fact, Damon told us that we'd achieved the highest scores on record for prelims in the math department. He refused then, and since, to tell us which of the two of us had the higher score.
Meanwhile, we were transforming the little summary document that we had prepared for the Higgs folks into two dissertations. There were three parts to the research: The determination that the proof of the Abel-Ruffini theorem was, in fact, valid. This was fairly straightforward, but it involved some complex mathematics that led directly to Allen's development of sets of equations that could be solved algebraically. This logically fit into Allen's dissertation, but I'd done the work. To get around this awkwardness, it was agreed that I'd write an brief paper containing my work on the Abel-Ruffini theorem and submit it to the math department for comment, unpublished. Allen would then cite the paper, and include it as an appendix to his dissertation, as would I.
My dissertation involved the development of the protocol for the algebraic solution of fifth and higher order equations that met the requirements to be included in one of Allen's sets of algebraically solvable equations. I would like to have included a proof that the protocol had general application to all of Allen's sets of equations, but I couldn't accomplish that. Suffice it to say that it led to the solution of all of the equations we tested, and those solutions, which included each of the sets that Allen had described, were all included in my dissertation. The most difficult part was trying to describe the process of developing the protocol. How does one describe insight, good guessing, trial and error, and blind luck. Bouncing ideas off of Allen, I had developed a protocol, tweaked it every time I got an equation we couldn't solve, and eventually had a protocol that evidently could solve any of the defined equations that was thrown at it. But I couldn't prove it would always work! Allen had been part of that effort, and that was acknowledged in my thesis, just as my work was acknowledged in his. I think it was clear that this was joint work that we had divided as fairly as we could into "his work" and "my work."
We each turned our dissertation into a journal article that would share the basic mathematics with the world. All of these papers were created amazingly quickly, thanks to our intensive work, and Damon's feverish editing. He refused to compromise on the quality of either the writing or the mathematical clarity, for the sake of speed. When, after 65 days work, which included the day we had to stop to take our prelims, we had both dissertations and both journal articles ready; we were all proud and satisfied with the products.
Then Tim stepped in. You didn't think he could keep his nose out of this, did you? He got me, Allen, Damon, Kevin, Kay, Roger, Ronnie, Sharon, and Kyle in his office and asked, "OK. I want to know just how important a mathematical achievement this is."
Damon said, "It's staggering. It opens very important equations that were considered unsolvable for over a century to algebraic solutions. It's mind-blowing."
"Are we sure the mathematics is correct?"
"Yes. You provide the equation; it's fairly simple to see if it fits into one of Allen's defined sets; if it does, you apply Carle's protocol, and you solve the equation. It's not simple mathematics. You're going to have to be a well-trained mathematician to do the work. But once the work is done, the fact that the equation is solved is easily established by any second year mathematics student."
"How do we convince the world that we really have something without simply handing out the two journal articles. If we do that, then we have no further rabbits to pull out of the hat."
Kay said, "That's simple. We issue a press release that describes the work, states that the dissertations will be defended at such and such a place and time. Then you provide a list of previously unsolvable equations, along with their solutions. It won't take good mathematicians long to realize that these were solutions that they thought didn't exist and which they could not have produced. They'll be demanding more very quickly. And the 'more' will be at the dissertation defenses."
Tim asked Damon, "Do you agree with that?"
"Yes. The question is whether people will wait for the journal article or will come to watch the defense. It's a long way to North Dakota. And, if we hand out the journal articles at the defense, we undermine the American Journal of Mathematics."
Tim said, "OK, we get AJM involved, I presume they'll be eager to be associated with this."
"Why don't we invite them to publish a special issue that contains the two articles? It can be sold by them outside the room where the defense is to be held."
I said, "It can also contain my unpublished paper on the Abel-Ruffini theorem."
I'll make a long story short. The AJM editor, a Professor of Mathematics at Purdue University, was, at first, skeptical, but agreed to look at Allen's and my work. We flew to Lafayette, Indiana, the next day, ready to show off our work. However, there was a fly in the ointment. We'd been advised by Charlie and the University Legal Counsel not to show any papers until a secrecy and proprietary agreement had been signed that guaranteed that nothing would be shared or published without our consent. Professor Stamm, the AJM editor was put off by that, and at first refused to sign. Charlie had warned us of this, and suggested that we have Prof. Stamm call him, Charlie, who would try to smooth the way for us.
I don't know what Charlie told him, but he signed the document, and we gave him copies of the journal articles we'd prepared. His first comment was, "This is ridiculous. It's been proven that this stuff doesn't exist. What have you got?"
We simply said, "Read it. And then give us a few trial equations that you think can't be solved algebraically."
He read for two hours while we went looking for lunch. When we got back to his office he had four equations for us to solve. Only three of the four fit Allen's criteria, and those three took Allen and me about an hour and a half to solve. Prior to Allen's work and my protocol, the solutions would've taken days or weeks, depending on the computer capacity available. Stamm was incredulous, but I provided an additional equation and showed him how to use the protocol. In about two hours, with my guidance, he'd solved an "unsolvable" equation. He said, "OK, I'm a believer. Where do we go from here. This is mind-boggling."
I said, "The Mathematics Department of the University of North Dakota and the American Journal of Mathematics will issue a joint press release announcing this mathematical breakthrough, and saying that it'll be defended as two Ph.D. dissertations at UND and published in a special issue of AJM which will first go on sale at the time of the dissertation defense in Grand Forks, North Dakota. Then get the special issue in print, and we'll set up the defense. And nobody except trusted printers sees those papers in advance. You do all the editing personally."
Then something occurred to Dr. Stamm, and he asked me, "Aren't you pretty young to be completing a Ph.D.?"
"Oh, I was just admitted to candidacy, like Allen here. I will be a Senior math major in the fall, and Allen will be completing an M.A. We were admitted to the Ph.D. program, and to candidacy, early, so that these papers could be published now."
"You're an undergraduate?"
"Thank you. Dr. Stamm, it's been a pleasure to work with you today. We'll keep in touch. We have to set a date and get that press release out. Why don't we have the PR people at UND draft it, and then we can all comment."
It was all agreed. Six weeks later Allen's defense was set for nine in the morning and mine for one-thirty in the afternoon, Tuesday, September 22, 1998. The entire mathematical world had been invited, but we had no idea how many might come. The email grapevine was full of doubters; there was considerable weight to the argument that the equations that Allen and I were claiming to solve algebraically could only be solved trigonometrically. On the other side were those that considered the endorsement by Professor Stamm and American Journal of Mathematics to be assurance that everything was legitimate. The ongoing debate seemed to fuel an interest in going to Grand Forks among both the believers and the doubters: somebody was going to be proved a fool at these defenses, and both groups were convinced that it would be the other group.
This was before the days of Facebook, and most of this conversation was in private email communications between individuals and established groups. However, both Stamm and Anderer were part of many such groups, and they were well aware of all of the email debate. The question remained, how many people should we expect at our dissertation defenses?
Tim's response to that was surprising. "Well, there were more than 200 at both my defense and Charlie's, and I'm expecting you two to double that. We'll use the mainstage theater; it seats 650. However, there's going to have to be one difference in your defenses. Normally, the public are invited as witnesses, not participants. There are people who are convinced that this is some kind of a con job. Just why they think this university would be a party to such a thing I don't really understand, but that's the reality. So, it's like a magic show where the magician gets audience participation to show that the thing isn't 'fixed.' Of course, magic is always fixed, just not in the way the audience suspects. In your cases, the obvious fix would be using equations that have been previously solved trigonometrically, and then pretend to solve them algebraically. The only defense against such accusations is to work on at least a few equations that are produced by the audience, and I'm sure that they'll come with them."
Allen said, "We'll have to invite such equations during my defense, and I'll have to analyze them to determine if they fit into one of my solvable sets. Those that do can be given to Carle in the afternoon."
Does this sound like we were getting ready for a three-ring circus? It turned out that we were. Just after Labor Day; which, because Labor Day was late, was about two weeks before the big show, Damon started getting telephone calls and emails from math colleagues saying they were coming to see the show. He reported that they were more likely to use the word show than defense. Those that called were generally people who believed the publicity and were eager to see a significant advance in mathematical knowledge. Final count: 512 people filed into the theater for Allen's defense. This included all of the UND math department, and a good percentage of the faculties in the hard sciences, and all of the scientists from the Supercollider and the IAP. That was about 250 people; there were about 100 Gang members; over 75 out-of-town visitors; and the rest simply curiosity seekers.
For those who weren't fairly advanced mathematicians it was completely over their heads. Allen showed how working with the proof of the Abel-Ruffini theorem led to his understanding of different sets of equations and how they related to the proof. He then went to the simplest of his sets and worked through his proof that an algebraic solution was possible. He was only able to present a proof of solvability for two of the eleven sets he worked with. For the others he showed his reasoning that they could be solved, but added that only the inability to find one that couldn't be solved was the basis for his conclusion.
He then gave an analogy to the four color map theorem. For years it was accepted that any two dimensional map could be made with only four colors, with no areas of the same color touching except at a point. This was the basis for printing maps for centuries. And it worked. The theorem was only proved in the 1970's using a controversial proof by computer. Although some still dispute the proof, more recent formulations of it have convinced most mathematicians. Allen insisted that mathematicians today could use his work, along with mine, to solve equations, just as map makers could print four-color maps, regardless of the lack of proof of their respective theorems.
Then the committee gave Allen a list of about fifteen equations of orders fifth through eighth. He was to determine if they fit his various sets, and therefore could or could not be solved algebraically. He was to "show his work" and with the aid of a large whiteboard he worked through the list, twelve of which fit into solvable sets. Then Damon Anderer came to the front of the theater and said, "We realize that many of you have equations that you'd like to present as a challenge to Mr. Kramer's work. But we cannot ask him to work on 500 equations. We've selected five seats in the theater at random, and will invite the persons in those seats to submit equations, or pass the privilege to a friend or colleague. We have also invited five distinguished mathematicians from five major universities to submit equations." This was done, and very rapidly Allen determined that seven of the ten were solvable, and showed how they fit into his different sets.
The nineteen equations, as yet unsolved but claimed to be solvable, were left prominently displayed on the whiteboard. The committee then proceeded to vote to accept Allen's dissertation and approve the defense.
It was now well past noon. Everybody was told where lunch could be had quickly, and invited to return at 2:00 (rescheduled from 1:30) p.m. for my defense. I had, of course, been looking at the equations on the whiteboard, and was convinced than none of them would be impossibly difficult or time-consuming. With that Allen and I set off for Damon Anderer's office where the three of us ate sack lunches and rehashed the morning. It'd been a triumph–except that it was all meaningless unless I could show that the equations were actually solvable. It was also important for me to show that without the set screening that Allen's work allowed, enormous amounts of time could be wasted trying to use my protocol on unsolvable equations. In other words, it was important to make clear that there were, in fact, two critical pieces to this work, each of which was, in and of itself, a major contribution to mathematics.
Attendance was up a little in the afternoon, but in general I had the same audience that Allen had in the morning. In the lobby outside the theater clerks hired by the American Journal of Mathematics were selling copies of the special issue that contained our three articles. Of greatest importance, Allen's article made it clear how to determine whether an equation fit into a solvable set, and my main article provided the solution protocol. In other words, for a mere $10.00 you could buy instructions on how to solve equations which, up to now, required the work of the largest computers on earth. AJM sold out of seven hundred copies, but many weren't sold until after my demonstration that the protocol worked.
The committee moved me through the preliminaries very quickly. Clearly they, and the audience, wanted to see me solve equations. Of course, the committee had already seen such demonstrations as we had worked through the dissertation. And they'd each produced their own set of "impossible" equations for me to tackle. So they were content to let the nineteen equations on the whiteboard constitute the demonstration.
Of course, there'd been a significant attempt, especially in regard to the equations that had come from the audience, to select equations that would be difficult to solve. The problem was, since they hadn't seen the protocol (it had been very close hold), the people proposing the equations didn't really have a way of judging which equations would be difficult to solve. My view of the list suggested that they'd come up with a pretty good mix.
I spoke to the committee and the audience and said that I wasn't going to tackle them in order, but would do the simplest ones first. I explained that in that way I could better explain the use of the protocol, which was, after all, the crux of my dissertation. If it worked, I got my Ph.D. If it didn't, it was back to the drawing board. To the audience it looked like high drama: could I, or could I not, solve these nineteen equations? To me, and to the committee, it was clear: I could, because I'd solved every one they'd tossed at me for over a month. Frankly, I was getting bored with the protocol. But I'll have to admit, the present situation was far from boring. Exhilarating would be a better term.
It took me two and a half hours to solve all nineteen equations. It took up to four whiteboard to solve each one, and as the solutions were checked, those boards were photographed and erased. The photographs were posted on the World Wide Web that evening. When I finished the last equation, and without waiting for many people in the audience to check my work, there was a long, loud standing ovation from the audience. Even the doubters realized that they'd just witnessed an amazing performance that would change certain areas of advanced mathematics forever.
Tim took the stage, congratulated Allen and me, thanked Damon Anderer for all of the support he had given, and then noted that this basic mathematics was destined to support the team of physicists that were looking for the Higgs boson. He had thus made the first public announcement that the University of North Dakota was in the race to find the Higgs boson. It was a tremendous challenge to Kay, Kevin, Roger, and their entire team. He then invited the committee to formally vote to accept my dissertation and its defense, which they did with alacrity.
The next day Damon Anderer called a press conference to announce the University of North Dakota Institute for Climate Modeling, with Carleton Holmes and Allen Kramer as Co-Directors.
We'd been working with Damon designing the Institute for Climate Modeling for about six weeks. Fred had been involved, and had, in fact, been the driving force. His proposed staffing model and budget was about triple what Damon and the two of use were thinking about. Even with Allen's equation protocol to help with analysis, we needed huge computers to collect and organize the massive amounts of weather data that was available. And our stated goal was, quite simply, to synthesize all of the weather data that existed in the world. It was a massive, impossible undertaking, but as the data was collected in any quantity it improved the climate model that our computers slowly produced.
The problem with a climate model that predicts weather trends–we never claimed to be, nor attempted to be, weather forecasters–fifty years into the future is that it takes most of fifty years to prove that you were right–or wrong. There was pretty substantial scientific agreement that the earth was warming, but total disagreement on how fast and why. There was an enormous tendency for people, and especially politicians, to discount the speed at which warming would take place and the extent to which human actions were the cause. An Institute for Climate Modeling, no matter how smart, no matter how much data they crunched, no matter how clear the evidence amassed, could have very little affect on public and political opinion on the subject of global warming, as the phenomenon was coming to be called.
That didn't obviate the need for the best possible data on what the future of the world would look like. And it was the goal at the Institute to collect as much data as existed and analyze it to the best of our ability. Our scenarios for the future of the world were not pretty. A warming climate would affect the poles first, raising sea levels throughout the world. Deserts would expand. Agriculture would migrate northward or southward away from the equator. I should note that as agriculture moves away from the equator, the amount of land available for a particular kind of agriculture shrinks, because we are on a globe not a flat surface. As you move north winter days dramatically shorten which has a significant effect on people. Food production would be affected. Extreme weather events would become more frequent. The list went on and on. The Institute was called the Doomsday Institute by not a few persons.
Tim, Charlie, Damon, and Fred remained our staunchest supporters. They all agreed that the truth must be told, even if the world was determined to ignore and discredit (if not shoot) the messengers. It was our job to get the message as right as we possibly could, and publish it in any and every forum available. One way was the journal Climate Trends, which the Institute published quarterly, and which quickly became a highly respected source of high quality science regarding weather and climate.
Over the last decade we have tried to remain true to our mission.
And we've stayed true to each other as well. We've been introduced to the glories of sex within the Gang. We've been amazed at the joy that can come from intergenerational sex (who would've guessed?). We've learned that there is, indeed, a little bit of straight in us, perhaps more than a little. But most of all, we've learned that loving each other within the bosom of the loving group called the Gang, is the most wonderful thing in the world.
The night after our defenses, the end of a day of unimaginable personal and professional triumph, we decided that we'd like to spend the night in Gangland. We started in that grand shower, and moved to the huge bed. We hugged each other till we could hardly breathe. Then we wiggled around to a 69 position and made glorious love. Then we slept the sleep of true lovers, locked in each other's arms, and we awoke just in time to make it to Damon's press conference.
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