| |
Professor of
Mathematics
Professor Previato’s areas of interest are algebraic geometry
and partial differential equations. In 2003, she received the Distinguished
Teacher of the Year Award from the Northeastern Section of the Mathematical
Association of America, awarded annually to recognize dedication
to students and initiative beyond the classroom.
To visit Professor Previato’s home page, click
here.
|
| |
Q: Describe some of the research
projects you’re currently working on.
A: I am working with three undergraduates, mainly in algebraic geometry,
and five graduate students in very diverse areas (coding theory, mathematical
physics, number theory and hard-core algebraic geometry). Last year
I had two official Undergraduate
Research Opportunities Program (UROP) students, also funded by
a grant from the National Science Foundation; one of those projects
resulted in a dissertation. Let me describe one that I think the mind’s
eye might enjoy:
In mathematics, as well as in real-life applications,
billiard tables come in different shapes than you might see in a
conventional game of pool. When the boundary is an ellipse or a
hyperbola, the trajectories of the billiard ball are quite interesting
curves (and fun to watch), but when the boundary has cubic equation—for
example, if it’s shaped like an egg—nobody knows what
the trajectories will look like. I have an undergraduate researching
this and animating trajectories in a computer movie. She has found
some “periodic” trajectories, which was my main goal.
For example, after two bounces, the ball comes back where it started
and then repeats the same path. We are also developing a general
strategy for the equations of motion, and a solution in case the
curve has special symmetry. One feature of this problem that I love,
and the reason why I assigned it, is its hidden depth: It has to
do with the elliptic curves of Fermat’s last theorem. I am
enabling the student to discover these deeper facts little by little,
and the knowledge will serve her well in the Ph.D. program she intends
to pursue.
Q: What’s your favorite course or lesson to teach, and why?
A: There is, so far, no topic in mathematics that I don't enjoy, and
I always end up learning a lot together with, and from, my students.
However, the courses that I am most excited to teach have to do with
my research in algebraic geometry—courses on subjects such as
topology, differential topology, or algebraic geometry, as well as
algebra, complex analysis, lie groups, elementary number theory, and
non-Euclidean geometry.
Since 1989, I have also been
in charge of Boston University’s participation in the William
Lowell Putnam Math Competition, a national competition similar to
a Math Olympiad. The problems are best solved by an original combination
of several subjects (for example, algebra and geometry or calculus
and probability) coming to play in the same arena. It’s an
unusual experience; the students seem to enjoy it, and I certainly
do.
Q: How do you engage students in learning complex subjects?
A: I don’t know that I have a particular strategy to “engage”
students—I am simply so curious about, and fascinated by,
the beauty and harmony of mathematical structures and the world,
that this catches their imaginations and they ask for further directions
of study. Because I believe in students’ creativity and intellectual
potential so much, I always try to assign special projects, even
in elementary classes. Learning by doing is what people enjoy most,
because people are natural builders, and they come to know themselves
better in the process.
|
