People
Fulton Gonzalez Professor
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
BromfieldPearson
Room 203
Medford, MA 02155
Email @tufts.edu:
fulton.gonzalez
Phone: 6176272368
Personal site
Expertise:
Noncommutative harmonic analysis, representations of Lie groups,
integral geometry, and Radon transforms
Research:
Principally, I study two subjects in pure mathematics: harmonic
analysis and integral geometry. The first field has a long history,
dating back to Leonhard Euler's work on Fourier series back in the
1740's. The second is almost equally old, with roots in the
geometric probability methods used to solve Buffon's needle problem
in 1777.
In harmonic analysis, we learn how to decompose phenomena, expressed
mainly by functions, into simpler constituents, as determined by the
geometry of the space in which the phenomena occur. Thus, for
instance, a human voice or the sound a violin makes can be
decomposed into "pure" tones in order to be analyzed more
effectively. Likewise, a fingerprint or any other twodimensional
picture can often be decomposed into simpler constituent functions.
One application of this kind of analysis is that it allows us to
remove the "noise" associated with measured data. Another
application, which forms the subject of my research, involves the
reconstruction of functions from indirect data, such as their
averages over curves or surfaces. This subject, integral geometry,
is a large and active area in mathematics. The archetypal problem in
this subject is as follows: how does one reconstruct a function in
the plane from its averages over lines in the plane? The solutions
to problems like these have many applications, for example in
medical imaging and sonar.
I decided to pursue math as a profession when I was in college; when
I went to graduate school I didn't really have any idea about what I
should be interested in until I took a fantastic course taught by
the professor who became my Ph.D. thesis adviser. It was an
introductory course on integral geometry, and it was taught with the
right combination of rigor and geometric intuition that was very
appealing to me. It also helped that the class had some really
bright and talented students, with whom I could discuss the
mathematics being taught, and who have now become some of the
world's top experts in integral geometry and related fields.
The thing that fascinates me about integral geometry is that one
needs to know "a little bit of everything" in order to do research
on the subject. In conjunction with this, one finds research
problems in it which requires the collaboration of mathematicians
whose expertise lies in disparate fields. Thus my main research
collaborator, a Japanese mathematician from Tsukuba University named
Tomoyuki Kakehi, is an expert in partial differential equations,
which I know very little about. Likewise, Kakehi knows very little
Lie group theory, a mathematical area I am familiar with.
