

A Simply Fascinating ISearch Paper
Maciek Sakrejda English Period 3 December 8, 1995 Mr. Anderson
First off, I might want to explain how and why I chose my topic, Mathematics
Research. Used to be just mathematics, but now it's mathematics research.
Despite the expected Oh's I got when I answered "What're you doing for your
ISearch?", I stuck to it. "Lose your dreams and you will lose your mind," as
the Stones put it in "Ruby Tuesday". A few months ago, I somehow decided  I
don't know exactly when or how  that Mathematics is what I want to do when I
grow up. I've always been good at math, and have always liked it. I some people
see in it a certain quality  an interesting term I've heard recently used to
describe it is beauty  and I guess I'm one of them. I haven't had time to
consider it yet, but personally, I think it's more like elegance. Then again,
they are the ones who are Mathematicians. Mathematicians, though, are too broad
a topic. I decided to narrow it to research after looking around a bit. My mom
is a Nuclear Physicist, and I considered that topic for a while, but then when I
gave up on it, I missed the "rocket scientist" aspect of it. It's just the idea
of being a Nuclear Physicist, or Rocket Scientist, or whatever; the first
impression people have of you when you tell what you do for a living; but it's
also so much deeper than that. Just being a researcher for yourself, especially
if you enjoy whatever you're researching. Looking in the mirror and saying to
yourself: "I'm a Researcher!" Now you probably think I'm quite mentally ill,
but if you don't, you understand exactly what I'm trying to say. That's not why
I want to be a mathematician, but it certainly is an irresistible perk.
So after narrowing my topic, I decided to look on America OnLine for
Mathematicians through their profiles. AOL provides a fairly easy access to its
member directory through the use of key words, so I did just that. I found
dozens of people, but they were mostly math teachers. I didn't want math
teachers, I wanted researchers. If I had wanted math teachers, I would've
started with math teachers at school. AOL, as usual, messed with me and
considered the key word "mathematician" the same thing as "math" for some reason.
I found some people whose profiles looked like they might be the work of
mathematicians, so I decided to send out some email. I included these and a
couple of college professors  PBS specials about things always seem to have
experts from universities, and it seemed reasonable that math might be researched
there. Some days later, I checked to see what responses I had received. I had
about ten, of roughly twice that many letters sent out, but considering that most
AOLers seem to be comatose  just go into a chat room and see for yourself 
you're lucky if you can find five of the room capacity of twenty three talking 
I wasn't disappointed. Not, that is, until I looked at the responses. They all
were positive, but only one said the writer was anything close enough for me to
be interested in. I wrote back to Tatejb, Professor J.B. Tate in the Illinois
Mathematics And Science Academy asking him to tell me more about what his profile
called "Presenter at Math Conventions". He sort of did: he said he presents on
current topics in mathematics and sent me a letter containing some of his old
notes from such presentations. Unfortunately, they were nothing more than
applications for graphing calculators  not exactly what I was looking for. And
even more unfortunately, when I asked Mr. Tate for an interview, he didn't, for
whatever reason, answer.
I was disappointed, but continued searching. We
were coming up with many interesting and seemingly useful interview tips in
class, so all I really needed was an interviewee. But that wasn't easy. I had
given up on finding anyone on AOL, I imagined that "real" mathematicians have
internet access at work  much like my mom does at LBL  so they wouldn't need
the "internetfordummies" which AOL really is. It's not bad, it's just too
simple, too shallow, and gives too much of what you don't need for someone who
knows what he's doing. I did try to look for some info on math research in the
vast archives of AOL. It has electronic versions of dozens of magazines  none,
of course, about math  so I looked through some of the science ones and found a
couple of scattered articles, like ones about Zeno's paradoxes, which state that
motion is impossible because an object traveling even the smallest distance has
to take some amount of time to cover half the distance, half that time to cover
half the remaining distance, half that time to cover half the remaining
distance, and so on. Since the distance can shrink to be as small as you want,
the object never even starts moving. These, surprisingly, were unrefuted till
recently. This is another of the reasons I love this field  how would you like
to be the guy proving motion possible? The proof is a little complicated,
involving almost infinite  "unlimited"  numbers and almost zero ones 
infinitesimals. You can understand it if you think about it, though. While
interesting, this told me nothing about the researchers who actually did this
stuff. Doing an ISearch on veterinarians and interviewing a cured puppy  the
puppy and the problem are just very specific instances of your subject's
work.
I decided that if I wanted to find something really meaningful on
AOL, I'd have to look for hours, and given that that's not reconcilable with the
online fees, I gave up. I had to find another way, and I did. My mom somehow
found an electronic flyer for an MSRI (pronounced emissary  hey, they came up
with it, not me. Look funny at them) presentation at UC Berkeley. I saw it and
thought that this is exactly what my ISearch is about. MSRI stands for
Mathematical Sciences Research Institute. I awaited the presentation eagerly.
In the meantime, my mom took me digging around the World Wide Web at LBL, kicking
her off onto a comparatively measly Pentium and using the $40,000, 64 Megs RAM,
Netscapeequipped Silicon Graphics workstation. If you know anything about
computers, that's about fifty times better than the computer the average bear
needs. They pretend to collide heavy ions, amongst other things. That was the
first time I had used the WWW, and it was great. I found CECM  the Centre for
Experimental & Constructive Mathematics at the Simon Fraser University in
Vancouver, Canada  and it was really the sort of place I was looking for. I
emailed some people there and a few I found on the math postings page CECM had a
link to, asking for recommendations about books or magazines about math research
and for einterviews. After I found what I needed for my ISearch, my mom still
had to do some work, so I looked around and found a web page for Napoleon XIV,
the guy who sang "They're Coming to Take Me Away, Ha ha".
The next day I
checked for responses (some of which were written to my mom at work, even though
I said I wasn't my mom about ten times  hers is the eaddress they were sent
from, but I asked them to respond to Octahedron@aol.com. I just love that even
at the intellectual top they still make stupid mistakes.) and I found that almost
everyone was nice  one (Not from CECM, of course) was unpleasant and rather
arrogant, but recommended some good books  and two  both from CECM  offered to
help. Luis Goddyn who gave me an interview via email, and Loki Jorgenson, from
CECM, who told me: "Well normally I don't answer Email from cranks posing as
high school students doing school projects but anyone who quotes Die Fledermaus
can't be all bad." after I sent him a request ending with one of my favorite
quotes: "If he tries to kick you in the stomach, it's probably not the real
Santa!" He said that they were really busy, but that he'd pass around a short
list of questions with "NO GUARANTEE" that I'll ever see anything back. I did.
Loki sent back three responses. He was helpful from the beginning, but I knew
when I asked him about his name (I thought I heard of a demon named Loki
somewhere) and he told me: "Loki was Thor's halfbrother from Scandinavian
mythology. Depending on which version you read, Loki was the trickster god, who
was either evil or insane. Either way works for me :o)" I knew he was a neat
guy. I found all the researchers that answered see the future of "pure math" 
just research  as shaky, but applied math will be very much used. As one of the
mathematicians put it, "We are beginning to see that many 'pure' mathematicians
didn't do a good enough PR job to let people know that the results they needed to
make their computers work, their robots put bolts on correctly, the predictions
of chemical reactions come out right, were mathematics." Most seemed to agree
with him. But they didn't see math research as gone, just going; I don't think
they actually believe pure research in this field will just die. Some said that
even "useless" math should be studied; it is beautiful, like art, music, or
poetry  though maybe not as marketable  , and they are a normal part of society
while being "useless". Still others told me that the applications for math are
often so unpredictable that it's hard to tell what's useful and what isn't. Then
I asked about their actual research. Most vaguely explained what they were
doing, and I wasn't surprised. Luis Goddyn from CECM told me that only the best
mathematicians know more than about 1% of Math, and these people were doing real
Math. I doubt I would've understood a much more detailed explanation.
Flowinduced Vibration, Circuit Covers of Graphs, etc. Luis said he'd send me
one of his papers, but I haven't gotten it yet. Most researchers do things other
than just researching, the main one being teaching. Mathematicians work with
graduate students, write talks for presentations, talk to their colleagues to see
if that sheds new light on a problem, do administration stuff (if they're at that
level), answer email, and go back to old problems to see if they can succeed
with a new try. Only about a quarter of their time  depending, of course, on
the person  is spent doing actual research. Once every five to seven years, on
the average, if they work at a university, they go on sabbatical. All this means
is that the university gives them half pay for about a year and they can leave
and do whatever they want, like going to CECM to work on stuff out in the Real
World, as opposed to the confines of the university. The idea is to keep the
professors from getting moldy and outdated. They need to see what's being done
now instead of teaching the stuff that's five or ten years old. They can use
what they learn on sabbatical in their teaching, and even if they can't, they can
certainly benefit from researching and use that in "unrelated" areas. Many
mathematicians, though, work at places like CECM, and they need a way to keep
informed. The answer is business trips. There are generally two kinds:
conferences, meetings of often several hundred mathematicians, to which one comes
using one's grant money and invitations to smaller meetings, where the host pays
the way. The first are, of course, more plentiful, but both are pretty much the
same. The small ones are just more specific. Then I asked how they got started
in this field, and the responses were almost unanimous  all found that "special
something" in math in high school. The one exception was Luis Goddyn; he flunked
7th grade math because he couldn't do times tables, but now he's a mathematician,
so, as he says: "...this is NOT the skill needed for mathematics." Can't argue
with facts. Well, you can, but that's another paper... Then I asked about the
hierarchy of the field, and I learned that there really is none. It's pure
chaos; "An academic is like a cowboy of the wild west." You need more than
brains to go up. You need speed and PR skills. At a certain level, the moving
up costs you research time as you have more and more administrative tasks. But
the pay goes up too. Some move to fulltime administration and have to abandon
their research. After that, Loki the "Lexically Placable" answered a couple of
quick questions for me  namely, where does the money come from and how much
money do you get. I know that can be a touchy question, so I said I understand
if they don't want to answer, but the way I see it, it's how much they pay you,
not how much you earn. Loki the "Rhythmically Touched" told me that CECM is
funded by the federal government and "...also smaller provincial funding
agencies. Small amounts are provided by the university [Simon Fraser University]
but it primarily provides just the lab, supplies, faculty, students, etc. We
also get some funding from private/industry sources but this is new for
mathematics." The second question was answered by both Loki and Mr. Goddyn, but
they seemed to disagree a little. Both said salaries start at $2530K and the
average being around $45K, but Goddyn said that these climb to as high as $120K
at Princeton and Yale, but Loki said that they almost never go sixfigure and
seldom top $60K. Goddyn gave me his answer in US$ and was talking about the US;
Loki talked about Canadian salaries and I converted them to $US. I guess that
might have something to do with it, I mean, the researchers getting paid less in
Canada.
Finally, the time for the MSRI presentation at UC came. My dad dropped
me off after school, and I found the Valley Life Sciences building in which the
presentation was taking place. When I got there I wasn't disappointed. A whole
lecture room  a small auditorium, in fact  was filled with all sorts of people,
ranging from high school students to MSRI researchers. After a brief
introduction, the presentations started. The first was Hendrik Lenstra, a Dutch
guy working at UC who looked a lot like John Inman from "Are You Being Served?",
and he was presenting Fermat's Little Theorem. If the name Fermat sounds
familiar, don't be alarmed  you haven't been taught things subconsciously in
your math class. Fermat's Last Theorem has recently, after centuries of trying,
been proven by Andrew J. Wiles. It was very frustrating, because Fermat left the
theorem on the margin of a book he was reading and said: "I have discovered a
truly remarkable proof of this theorem which this margin is too small to
contain." The theorem itself states that there is no integer n>2 for which x_{n} +
y_{n} = z_{n}. X, y, and z are all positive whole numbers. Mr. Lenstra was more than
a little loony  he showed us a portrait of Fermat and then said that Fermat's
Little Theorem was a theorem he came up with when he was little. Then he showed
us a picture of Fermat when he was little  the original picture but about half
the size. Fermat's little theorem is about prime numbers and exponents and has
useful applications in cryptography. Mr. Lenstra's talk was hilarious, and his
confident manner and accent made it even more so. Next came Ellen Gethner, a
woman working at MSRI. She presented what she was currently trying to prove, that
there are larger and larger gaps between prime numbers as they get bigger and
bigger. This had already been proven in one dimension; she was doing it in two,
with numbers known as Gaussian integers. She seemed to be a Trekkie  she used
Mr. Spock in her presentation  so I decided to ask her for an interview after
the presentation. Trekkies are friendly  they have to be. I don't know why
yet, but just try to picture a mean Trekkie... It doesn't happen. Except for
the Klingons. And the Romulans. And the Ferangi. And the Cardassians. And the
Borg. ...never mind. She was nice, OK? And furthermore, she didn't show any
possible application for her work  something I find oddly attractive in math. It
loses, for some reason, a certain quality if it's useful. The third presenter
was John Conway, a Briton from Princeton, and looked a lot like Ringo Starr does
now. He talked a lot about knot theory. Basically, he showed us two ropes, got
two pairs of volunteers from the audience, and counted by twisting the ropes
around. If, for example:
So if you start with a) and perform b) three times and then c) and then b twice
more and c) again, you'll get: 0+1+1+1 = 3, then 1/3, then +2 = 5/3, and c)
again for a result of 3/5. Fun stuff, plus, it can get you practically any
number. And it has some applications. His lecture was fascinating, but I think
he was cheating. He was more a magician than a mathematician. His grand finale
was getting a number from the audience, tying it together in a sack, and having
volunteers from the audience untie the number without seeing it by following the
rules he had set up (a, b, and c, above). They did it, and everyone was stunned.
Then the presentation ended. People started walking out, but many, including me,
stayed to ask the presenters some questions. Ms. Gethner was busy talking to
someone, so I decided to listen to Knot Man explain some things to half a dozen
loyal fans. One even asked him to autograph a math book. Makes me wish I had
mine... The room emptied, it got late, and Ms. Gethner left before I could get
myself to ask for an interview. I wasn't worried, though, because the program
showed the MSRI web page, so I thought I could just find her name there and ask
for an interview through email, and I did. Complicated, but, at least for me,
it works. She set up an interview for Tuesday at 4:00 at MSRI. Other than that,
I didn't find much useful to me on their web page, except the neat mutant
doughnut on the home page (the Eightfold Way, a really complicated symmetrical
thing: "This sculpture is a rendition of the Klein quartic, discovered by Felix
Klein more than a century ago. Mathematically, the surface is a 3hole torus
with 336fold symmetry, the maximum number possible" says the MSRI propaganda)
and an application form for research. It had an eaddress on it, so I emailed
asking if I could get an application form. I was sent one in TeX (pronounced tek
for some reason), but I replied that I didn't have a TeX reader, so I was sent an
application and a whole lot of MSRI "propaganda" by what internetters call
snailmail. I got the application Tuesday, but I had apparently forgotten to
give alisa, who did this for me, my real name, as the application was addressed
to Octahedron The Unwise, the name I use online.
Before the interview, I decided
to go to the library and check out some of the books the unpleasant mathematician
had recommended and some math magazines another guy  an economist actually; I
emailed him by mistake  recommended. I, of course, found neither, nor did I
find much else on math. I finally ended up checking out two books: "e: the Story
of a Number" and "Recent Revolutions in Mathematics". The second one was really
interesting, with many neat problems and theorems, and the first wasn't bad, but
it sounded a lot like a math or history book. Unfortunately, neither told me much
about modern math research.
But I still had the interview. After school my mom
drove me to MSRI, sitting perched above the Lawrence Hall of Science, a new yet
quaintlooking building, with "cedar chip lap siding that weathers to a saddle
brown color." I had looked at the MSRI "propaganda" that afternoon, but this
looked grand. The pamphlets showed a lot of interesting things, including an
impressive track record for MSRI. Many Fields Medal winners worked there. The
Fields Medal is the closest thing to a Nobel Prize in mathematics, which doesn't
exist. Mathematicians and others give many different stories about "why": some
say a great mathematician who antagonized Nobel brought this about while others
believe that it is because another mathematician ran off with Nobel's wife, but
there really is no official story. When we finally got up on the MSRI hill, we
couldn't find it. The Space Science Institute was there  that was right next to
it  but we couldn't see MSRI. We finally figured out that we have to, fairly
literally, drive off a cliff. You can't see the road beyond the Space Science
Institute. There's a small bump and then a steep slope. Optical illusion, math.
Maybe even on purpose. Figures. The inside looked somewhat like a hospital,
maybe a mental one, except for the black carpeting. A catwalk hugged the
secondstory walls. I went upstairs and found Ms. Gethner in her office, which
faced, of course, the wrong way. The people on the other side got the Berkeley
view. My kind of luck. My original questions were a little weak, so I came up
with some new ones and used some of the old ones, and I got an OK interview. As
I suspected, she was a Trekkie, and she told me she saw a Star Trek episode which
mentioned Fermat's Last Theorem, "unsolved for 750 years," shortly before it was
proven. She had a tape of it, but it was in Philadelphia. She didn't live here;
she just came to MSRI to do research. She told me it's nearly impossible to be
able to do only research; that most often it's research and teaching. That's
what sabbatical is for, getting away from teaching to concentrate on research.
You just find someone who's interested in what you're doing  this year
Combinatorics and LowDimensional Topology for MSRI  and you're off  if they
accept you. She showed me her work computer and one of the two programs she was
using, Mathematica. She took out a 1,000page manual and we looked for things
dealing with primes, which is what she was researching. One function showed the
number of primes lesser than a certain number, another gave you the biggest prime
lesser a certain number. It did a 10digit number in a fraction of a second.
Think it's easy? Try it on paper. Or even with your calculator. It was a
really powerful program, but she said she used the computer only about half the
time  the rest was spent thinking. But MSRI has a different idea  they want to
spend up to $300K/year on computerrelated things. I got the same response from
the CECM guys, but they're biased since a large part of CECM is using computers
in mathematics research. But she too said that computers are very important in
this field. Most math researchers use a UNIX platform like a Sun or Silicon
Graphics workstation. This is what my mom uses at work, and my dad is almost a
sort of UNIXsupremacy extremist.
When I asked how she got into this field, she
said what most others did. She was in high school, taking geometry, and then
she'd stay up all night with a problem until she solved it. You get a lot of
satisfaction from solving a hard problem. I agree with her 100%. The longer you
work on a problem, the better you feel when you break it. On the future of "pure
math", she agreed with the others. She believed that "pure math" would decline,
but that applied math would always be useful. As the young professor from CECM
told me, "But if you call it engineering, it pays better." About funding, she
said that MSRI gets most of its money from the National Science Foundation.
Someone figures out how much they need, and they just ask NSF. There is some
convincing to be done before NSF approves each time, but MSRI doesn't seem on the
verge of certain doom. They're doing OK. I then asked how all these conjectures
come up for them to prove. She answered that for every theorem you prove, you
come up with half a dozen interesting conjectures. Conjectures are basically
unproven theorems. Some are useful, some aren't. But you always get these so
there's always new stuff to work with. Infinite recursion. Mathematical, isn't
it? Another thing supplying perpetual work for mathematicians is the work of
Kurt Gödel, who proved that all mathematical systems contained elements which
could be proven neither true nor false. It was interesting that he proved this
using the same logic he was proving flawed. An example of such a blow to logic,
and therefore mathematics, is Russell's Paradox. It states that if a barber
shaves all the men in the village who do not shave themselves, he will shave
himself only if he does not shave himself. There are always holes in the system,
whatever the system is, no matter how much you try to fill them. Recent
Revolutions in Mathematics says that this also proves the human brain to be
superior to any computer facsimile, as the former will be able to handle Gödel's
freak logic defiers. This also explains a lot about humans. As one
mathematician put it, "We can no longer hope to be logical; the best we can hope
for is not to be illogical." Lastly, I asked her what she thought about the
books recommended to me, and if she had any others to recommend, and she showed
me some of the magazines I couldn't find at the library  the "American
Mathematical Monthly"  but it was far more "technical" than I had expected. She
also recommended a book she read while in high school, called How to solve it,
which was pretty much about math research. What I thought much more useful to me
was "Math Horizons", a quarterly published by the Mathematical Association of
America. As we were going to find some back issues of the "American Mathematical
Monthly", she was invited to a party by her friends. She had to leave in a few
minutes anyway, and so did I, so she accepted. As my dad drove me home I
wondered whether this was a party or one of those "social gatherings" that happen
to Nuclear Physicists. A few days later, Luis's paper still hadn't arrived and
the ISearch was almost due, so I went to LBL with my mom again  this time I got
the Pentium  and looked for research papers in electronic journals. That's how
they get published  they're submitted, they're checked for bugs, they're
accepted, and they're printed. Or uploaded, in the nineties... So anyway, I
found a couple of papers about a problem called The Problem Of The Kings  the
fact that on a 2m x 2n chess board, mn nonattacking kings can be placed. The
paper stated:
"For each m=1,2,3, ... there are constants c_{m}>0, d_{m}, and 0 < é_{m}<= m+1
such that f_{m}(n) = (c_{m}n + d_{m})(m + 1)^{n} + O(é^{n}_{m}) ( n => ì )"
Are we having fun yet? The general idea of the proof, oddly enough, is not that
complicated and I got a few basic things from it. The other paper  which was
about a more complicated aspect of the same problem  I didn't understand one bit
of. Then again, I've still got a few years.
I've found a lot about
mathematicians and mathematics in this ISearch. I learned many new things,
though my desire to be a mathematical researcher somewhat weakened. I mean, I
still like it, but I doubt I've got what it takes to do math research for a
living. I think if I am ever cursed with another ISearch, I will do better
because I'll start earlier. An ISearch is simply not something you pull off a
decent job on in two weeks. But it's a little late for that now. Most
importantly, I think that I saw that you can't do just research, which sort of
disappoints me. I don't think I could teach, which is often a key part of the
nonresearch part of a researcher's job, among things like giving talks,
attending talks, collaborating, and reading email from cranks posing as high
school students doing an English paper...
Sources
1. e: The Story of a Number by Eli Maor  I used this only a little for
background information.
2. Recent Revolutions in Mathematics by Albert Stwertka, used more heavily for
background info.
3. Luis Goddyn, researcher at the Centre for Experimental and Constructive
Mathematics (CECM) at Simon Fraser University in Burnaby, BC, Canada, interviewed
about mathematics research via email in first half of December 1995.
4. Loki Jorgenson, Research Manager at CECM, who agreed to circulate a list of
questions for me there. He also answered a couple of questions himself in the
first half of December 1995. The three sources below are his work.
5. Anonymous postdoctoral researcher at CECM, via Loki Jorgenson (Source 4). On
12/7/95, he answered some of the questions Loki circulated for me.
6. Anonymous young professor at CECM  "...record holder (with Andrew Granville
and others) for the first case of the FLT [Fermat's Last Theorem]." On 12/7/95,
he answered some of the questions Loki circulated for me.
7. Anonymous young professor at CECM. On 12/7/95, he too answered some of the
questions Loki circulated for me. In fact, "This one is practically an essay
on... [its] own," as Loki put it.
8. Professor J.B. Tate at the Illinois Mathematics And Science Academy, who
answered a few questions for me and sent me some of his presentation notes on
November 24, 1995.
9. Luis Goddyn's bookmarks to math sites on the world wide web, and, of course,
the sites themselves. The full list will be in the Journal of Documentation.
10. MSRI "propaganda"  addressed to Octahedron The Unwise, my online name 
from MSRI, sent on 12/8/95. It includes a newsletter, a couple of flyers, an
application to work at MSRI for the 9697 term, and a little "instruction
booklet" to the institute.
11. "Math Horizons", a magazine about math published by the Mathematical
Association of America. I got some background information out of it.
12. "Berkeley Mathematics Newsletter". I got a little info out of it, but it was
mostly PR stuff.
13. David Feldman, via email. I found his name on mathematics posts on the web
and he recommended some good books.
14. "The Problem of the Kings"  a paper I downloaded from the World Wide Web. It
showed me a lot about how things like this are proven, and I understood a
surprising amount of it.
15. "The Problem of the Kings"  another aspect of the same problem downloaded
from the same WWW site. To my relief, I understood very little of it.
16. Ellen Gethner, a researcher at MSRI. My live interview. Interviewed on
12/12/95 at MSRI.
17. "Numbers In Action"  MSRI organized lecture about mathematics and math
research at UC Berkeley.
