**Algebraic geometry**
(for the non-initiated)
Until a few decades ago, algebraic geometry was quite simply the study
of geometric objects defined by polynomial (that is, `algebraic')
equations. As is often the case, the field retains its original name
although in the process of its growth it has greatly enlarged its
scope and objectives. For example, the 1950's and 60's saw the
surfacing of completely unexpected connections of algebraic geometry
with other fields of pure mathematics such as commutative algebra or
number theory---this caused a complete overhaul of the techniques and
tools of the field, and opened the door to the solution of
long-standing problems and conjectures. To name the most famous
example of this sort, the 300-year-old conjecture known as
"Fermat's last theorem" was settled by Andrew Wiles a few years ago
precisely by applying the new sophisticated language and results of
algebraic geometry and number theory developed during that revolution.
Work in algebraic geometry in the past few decades has often centered
on applying the new techniques to old problems. My first contributions
to the field were in `enumerative geometry', a subfield aiming to
establish the number of solutions of certain kinds of equations,
answering questions which are usually posed in terms of the number of
geometric objects satisfying given conditions. My Ph.D. thesis
consisted of the verification of some results dating back to1872 and
that had since not been recovered rigorously. This work brought me to
develop techniques that I have later been able to apply with success
to more questions in enumerative geometry, at times pushing them
beyond their original scope. While most of the old results in the
field are now recovered, the conceptual tools developed in the process
allow us to attack and solve similar problems that were beyond any
reasonable expectation of mathematicians working a few decades
ago. This program is one of my long term projects, which I have
pursued both by myself and in collaboration with Prof. Carel Faber
(now at the Royal Institute of Technology, Stockholm).
Studying these problems led me to a parallel direction of my research.
I found that in order to sharpen my tools I had to develop new
techniques in `intersection theory'; in time, these proved to be
useful in attacking problems that were far removed from the original
motivation, and the quest for these techniques became an independent
line of work, in the theory of `singularities'. Although a precise
mathematical definition of `singularity' is highly technical, the
concept it formalizes is quite natural: creases, corners, or sharp
asperities are examples of singularities. The `black holes' of modern
physics, as well as the postulated `big bang', are thought to be
singularities of the metric underlying space-time. As it turns out, a
given geometric object cannot be forced to be singular in an arbitrary
way---for example, a flat piece of paper cannot be folded precisely
along a circle without creasing it in some way. This elementary
observation is the simplest example of a number of techniques which
can be developed to relate the internal structure of a geometric
object with the kind of singularities it can acquire. From studying
its singularities, one can construct quantities which encode the most
essential features of a space---that is, one can construct
`invariants': quantities which remain the same if the space is changed
in non-essential ways. The construction and study of these invariants is the program to which
I have devoted most of my energies in the past five years.
A further line of research in algebraic geometry is the result of
another revolution in the field, started in the last few years. This
new revolution is redefining what it means for mathematics to be
`pure'---indeed, it is blurring the edge separating pure mathematics
and theoretical physics, a distinction which has been in effect for
most of this century. The catchword of the new revolution is `quantum
cohomology', a tool originally introduced by the physicist Ed Witten
in the context of `quantum gravity'. The power of this idea was soon
understood within the algebraic geometry community, and a parallel
theory was developed in a context very close to my main field of
expertise. I intend to maintain a high level of involvement with these new
developments. They are representative of an impulse now pervasive in
mathematics---the surfacing of unsuspected interactions between the
ivory tower of pure research in this basic science and other, more
immediately applicable, fields. Like all the rest of science,
mathematics does not live in a vacuum.
The mathematics department at Florida State University is currently
very active in algebraic geometry. Algebraic geometry is relevant to
the interests of many faculty members at FSU: professors Ettore Aldrovandi, Eriko
Hironaka, Eric Klassen, Craig Nolder, Mika Seppala, Mark van Hoeij and
others. We regularly run courses and seminars on the subject, both at the
elementary and at a more advanced level: recently (in 2000) we have had
semester-long activities on the moduli space of curves, and on Grothendieck's
"Dessins d'Enfants". Several graduate students are now studying in fields
closely related to algebraic geometry. The department offers a
lively environment where to begin work in this exciting field. |