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Jeff Danciger begins to explain his original
theorem.
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Jeffrey Danciger, ‘06
One Student’s Path to Math Happiness
Questions, challenges, problems, and puzzlement are an
inseparable part of Jeffrey Danciger’s life. He wouldn’t
have it any other way—he’s a mathematician.
Of course, some of Jeff’s friends had questions, too—people
who, unlike him, weren’t earning dual degrees in physics
and mathematics at the College of Creative Studies (CCS). “They’d
say, ‘Oh, you do math research? What could that be? Calculus—isn’t
that pretty much all there is?’ And I’d say, ‘No!
There’s so much more!’”
The commotion began with a freshman course called Algebra, Analysis,
Number Theory, Combinatorics, and Probability. “It showed
me that tons of interesting mathematical stuff was going on that
I had no idea even existed,” Jeff says. “Day after
day, the professor told us things that just blew my mind.”
| Jeff’s
Best Tips
Mostly,
Jeff studies alone. “I think
you have to spend a lot of time thinking by yourself to develop
your mind and your abilities.
What has been most useful to me is just to sit in front of a blackboard
and work on something until all hours of the night.”
Many
people have no idea how much thought is involved in problem-solving,
Jeff says. “Some of my friends would say, ‘I can’t
do this problem.’ Then I’d ask them how much time they
spent on it. They’d tell me, ‘About three minutes.’ No!
You have to sit there and think and think about it. If it doesn’t
click in about three days, then maybe you should get some help
with it.”
Finally, he says: “Get involved with as many things as you
can. Be enthusiastic. Work really hard. Take advantage of summer research programs
designed for undergraduates, where you’ll meet great people and learn
an insane amount!”
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The son of biologists, Jeff had cruised
through advanced-placement calculus at Calabasas High School, north of
LA—“but in this freshman course I got ideas about what math
really is.”
Jeff concluded that “to really understand these things,” he needed
a math degree—in addition to his future UCSB physics degree, for which
he had given up admission to UC Berkeley. (A 2006 graduate, Jeff is now pursuing
a PhD in mathematics at Stanford, with full financial support. “I had to
pick one field for graduate school,” he says. “It doesn’t rule
Navigating the unknown
Jeff used the immersion method to learn higher mathematics. The first-year graduate
course in modern algebra he waded into when he was a sophomore is a good example.
(Customized programs of study are a CCS hallmark.) “I had no business taking
graduate courses,” Jeff says happily. “The first day, the professor
covered more than everything I knew about algebra. The second day, I was completely
lost.”
His solution: “In any situation like that, you just push yourself and push
yourself and keep working.
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Eventually it will settle in, and you can understand.I was so
determined,” he continues. “I think I wanted to prove to myself
that I could do it.”
Jeff developed rapidly as a mathematician, says his faculty mentor
Professor Mihai Putinar, who invited him to a graduate seminar on functional
analysis to
accelerate his learning
Eventually it will settle in, and you can understand.I was so
determined,” he continues. “I think I wanted to prove to myself
that I could do it.”
Jeff developed rapidly as a mathematician, says his faculty mentor Professor
Mihai Putinar, who invited him to a graduate seminar on functional analysis to
accelerate his learning
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. “During the group discussions at the blackboard,” Professor
Putinar recalls, “we remarked on the lack and the need of a min-max principle
for complex symmetric matrices [tables of numbers]. And after an interval, Jeff
produced the desired theorem! It is very natural, has an ingenious proof, and
is totally new.”
After-hours insight
Jeff had searched for a theorem for most of that fall quarter. During winter
break, he began “deep consideration” of the problem. All the thinking
over the months began to converge. “Suddenly,” Jeff says, “in
one of my proudest moments yet, the key step to the proof dawned on me while
I was hanging out with some friends one night!
“ Coming up with an original idea and seeing that idea have an effect on
other parts of mathematics were very exciting to me,” Jeff continues. “During
my career I want to have more experiences exactly like that.”
Jeff’s paper describing the work was accepted for publication
in the journal Linear Algebra and Its Applications (v. 412, 1,
22–29). Then he, Professor Putinar, and then Visiting Assistant
Professor Stephen Ramon Garcia did equal parts of research applying
Jeff’s min-max principle to function theory. The resulting
article, “Variational Principles for Symmetric Bilinear Forms,” will
be published in Mathematische Nachrichten.
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