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Paolo Aluffi's Faculty Page

Paolo Aluffi
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Ph.D., Brown University, USA, 1987

Detailed description of research

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.