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Web Extra
Math and Peanut Butter: Duluth
July/August 2008
by Joshua Batson ’08
The popular image of
the research mathematician is a lonely, awkward man hunched over a tome of arcane
formulas, calculating integral after integral. But real mathematics is neither so
boring nor so straightforward—my research last summer required all the creativity,
patience, and peanut butter I could find. I began my work the Monday after I arrived
at the University of Minnesota-Duluth, when Joe Gallian, Beatles fan and director
of the Research Experience for Undergraduates, presented me with a recent journal
article. The article contained a conjecture, and my job, apparently, was to prove
it. Unfortunately, the language of mathematics has many dialects, and I could not
read (much less write) Additive Number Theory. I felt as if I had been handed a
packet of Pablo Neruda’s poems and instructed to learn Spanish, memorize the volume,
and then compose its sequel.
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Paul Erdős defines a mathematician as a device for turning coffee into theorems. |
After a perilous walk (I violated
my mother’s dictum not to read while crossing busy streets), I returned to the apartment
complex where I lived, worked, and ate with the program’s eight other students.
After two hours at my desk, mind full and stomach empty, I needed a break. I shouted
for my roommate, Alan, and we trundled down our apartment’s treacherous spiral staircase
into our sparsely stocked kitchen. Two peanut butter and jelly sandwiches and one
hour of commiseration later, we were both ready to reengage with our respective
problems. The next three days followed a similar pattern (read, lunch, read, snack,
read, dinner, reread, second dinner), and by Thursday night, I had internalized
the article and could reproduce it from memory. But research is not the same as
coursework, where Exercise 1.6 can be solved with Theorem 1.1. I knew that the techniques
I had just learned would not be powerful enough to prove the conjecture (rarely
is an author kind enough to leave such low-hanging fruit unpicked), so rather than
attempt nighttime brilliance, I joined Alan for peanut butter, bananas, and a game
of gin before bed.
I awoke late Friday morning
painfully conscious of Paul Erdős’s famous definition of a mathematician as a device
for turning coffee into theorems, and set out for Bixby’s Cafe in search of caffeine
and inspiration. Latte in one hand and pencil in the other, I sat down to work.
Attacking the conjecture head-on was too daunting, so I experimented with simplified
versions and small cases. I soon found a pattern, generalized it to a statement
even stronger than the original conjecture, scribbled something resembling a proof,
and ran back to the apartments to share the good news. Before gloating, I wrote
a quick computer program to empirically verify my statement. Five minutes later,
it spat out a counterexample. I soon found the hole in my argument (mathematicians
forget minus signs too), and consoled myself by spending the evening with my good
friends peanut butter and spoon.
Saturday’s efforts proved
no more fruitful, and I awoke Sunday expecting another disappointing day. As I lay
in bed, idly pondering my problem, the proverbial lightening struck: I saw a way
to rephrase the conjecture geometrically, and look at lines and distances instead
of integers and sums. In this new context, the solution was obvious. Fearing that
this proof too would evaporate under closer scrutiny, I could not bring myself to
write it out formally. Instead, I took a long shower, allowing the new ideas to
simmer in my subconscious. While I made breakfast (peanut butter on toast, this
time), my mind repeatedly returned to the delicate construction, checking to make
sure it had not collapsed, before fleeing again. As the day progressed without contradiction,
the realization slowly dawned that I had actually proven the conjecture. That afternoon,
when an adviser verified my argument, I was absolutely elated. After just six days,
I had somehow appended my own short verse to the epic poem of mathematics.  |
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