The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites.

About Dr. Smith:

After earning her Ph.D in Mathematics (Thesis: Tight Closure of Parameter Ideals and F-rationality) from the University of Michigan in 1993, Dr. Smith worked as a post-doc at Purdue University before joining the faculty at MIT. Dr. Smith taught for several years at MIT and had been promoted to associate professor before deciding to accept her current position at her alma mater, the University of Michigan, where she continues her research and teaching. She authored An Invitation to Algebraic Geometry, see more info about it in the sidebar.

In 2001, Dr. Smith was awarded the Ruth Lyttle Satter Prize in Mathematics for "her outstanding work in commutative algebra, which has established her as a world leader in the study of tight closure, an important tool in the subject introduced by Hochster and Huneke. It was also awarded for her more recent work, which builds new bridges between commutative algebra and algebraic geometry via the concept of tight closure." (See article quoted)

Q & A with Dr. Smith

How important do you think algebraic geometry is as a tool for solving some of the major mathematical problems today, like the Millennium Prize Problems, like it has been of solving Fermat's Last Theorem in the past. 

Algebraic geometry is the study of geometric shapes (of any dimension) that can be defined by polynomial equations. Such shapes are called algebraic varieties. Right at the center of modern mathematics, algebraic geometry influences and interacts with many other fields. In fact, often is it hard to say where algebraic geometry ends some other area of mathematics begins. This is because polynomials provide a sufficiently rich collection of equations to describe  all kinds of wildly interesting geometric shapes, but on the other hand they  are also quite concrete and easy to work with---even computers can be programmed to manipulate or graph polynomials quite easily. Even shapes and functions which are not polynomial can often be approximated arbitrarily well by polynomials.

As you point out, algebraic geometry provided the foundation for the solution for Fermat's last theorem, but it is also arguably the field of at least two of the seven millennium  problems -- worth one million dollars each if you can solve them! These are the Hodge Conjecture, and the Birch-Swinnerton-Dyer Conjecture. Descriptions of these problems can be found on the Clay website (http://www.claymath.org/ millennium/), but it is interesting to note that these two problems come from the opposite ends of the algebraic geometry spectrum. The Hodge conjecture has to do with the topology -- or general shape -- of  algebraic varieties. The Birch-Swinnerton-Dyer Conjecture is concerned with the arithmetic (or number-theoretic) nature of particularly simple varieties. Thus, while one could classify the Hodge Conjecture as topology and the Birch-Swinnerton-Dyer Conjecture as number theory, both are conjectures about  algebraic geometry.  But even beyond these two millenium problems which are directly impacted by algebraic geometry, I would not be surprised at all if algebraic geometry played an important role  in the eventual solution of some other millenium problems as well!

What drew you to algebraic geometry - which is notoriously technical nowadays?

I was drawn to algebraic geometry by the multitude of beautiful and simple examples, such as the conic sections, and the elegant algebraic arguments that backed up my geometric intuition.  In high school, I essentially discovered for myself the projective plane,  a two-dimensional geometry not unlike the geometry one studies in high school but in which parallel lines meet  "at infinity". At the time, I felt certain that one could develop an axiomatic system like we were doing for the traditional Euclidean plane in class, but I was unable to convince my teacher. How thrilled I was in graduate school when the projective plane was formally defined, along with its higher dimensional analogs, and I  was taught to do geometry in this wonderful space I'd dreamed of in high school! 

I wrote my PhD, however, not in core algebraic geometry, but in the closely related field of commutative algebra.  This is because in writing a thesis, I felt it more important to choose an inspiring and supportive teacher and advisor, than a particular topic. To this day, I feel extremely lucky that I chose Mel Hochster to be my advisor, even though I always felt pulled  in a slightly different direction, scientifically, than his own expertise.

I am not sure I agree the algebraic geometry is 'notoriously technical.' Like every field of mathematics, it does definitely have problems  which are quite technical, but the fundamental objects---algebraic varieties, or the zero sets of polynomials----remain quite concrete. Many basic open questions are quite easy to state: is every curve in projective three-space the intersection of two surfaces?  No one knows!  

What's your favorite part of algebraic geometry?

My favorite part of algebraic geometry is  the existence in many cases of {\it moduli spaces.}  These are spaces  which parametrize entire collections of interesting geometric objects, but which themselves turn out to be also defined by polynomials! So for example, we can envision the set of all "curves of genus g" to be itself some algebraic variety -- each point on  this parametrizing variety corresponds to exactly one such curve. Why should  a collection of algebraic varieties itself have the structure of an algebraic variety? This is the beautiful miracle of algebraic geometry. 

What are you working on now?

I am trying to prove that certain objects that arose independently in two different areas of math are, in fact, closely related. For example, on the one hand, we have "Fano varieties" which are certain "positively curved" geometric shapes arising naturally in the classification of algebraic varieties. On the other hand, algebraists have studied an operation called "tight closure" over the past two decades which has to do with the behavior of polynomials after raising them to some prime power p and the "reducing modulo p".  There are deep and mysterious interactions between these two areas, and many of my scientific publications have been on this connection. There is much more to discover and prove, and I am working on this with my students and post-docs.

Do you ever get to see applications of your work in other fields of study?

Not directly if you mean outside mathematics, but I do see applications of my field, and I do get involved with training students who work on applications. On the other hand, there are direct applications of my work to other fields of mathematics, perhaps most notably to non-commutative ring theory and discrete math.

What problems have you faced and how did you overcome them, being female in a male-dominated profession?

As an undergraduate, I faced some explicit sexism from a professor at Princeton, and then a lot of more subtle sexism when I tried to drop his class.  As a graduate student at Michigan, I also found my stipend suddenly suspended when I got married, because I presumably no longer met the criterion for financial need, although no one thought to inquire whether my male cohorts were married to doctors or what their financial arrangements were.

What worked for me in handling these problems was to keep complaining about it until I finally found someone (a dean, in both cases) who understood and could work with me to find a solution. Nowadays, such explicit sexism is more rare, partially because this kind of behavior is no longer socially acceptable and so perpetrators have been driven underground. I do occasionally encounter people who I know or suspect think my gender renders me a less competent mathematician. Mainly, I have been able to avoid and ignore them because they are far outnumbered by fair-minded and supportive people.
I have also found strong support  from female scientists at my university outside my own department.

If I want to pursue a math major in college, what sort of classes are best to take in high school?

Ideally, you should take college-prep classes at your own level.  I don't recommend rushing ahead too fast, as I've seen too many talented students burn out with the pressure that way. I took geometry, algebra, precalculus, and then Calculus (AB), basically the standard college prep courses. If you are interested and can find a good teacher to supervise you, or there are also nice possibilities for extra or reading courses. I was lucky my high school teacher, Mr. Drinfeld, was willing to run an extra class for a few top seniors. One of the books we read was Underwood Dudley's Elementary Number theory. I loved it! It definitely made me a mathematician. But, I think following your own interests (not mine!) is best.

What type of math questions intrigued you most when you were in high school?

One cool thing I remember was learning how to "cast out nines" to check my multi-digit multiplication in elementary school. Given a (multi-digit) number, we cross out all the nines, and sum the remaining digits, then repeat until we arrive at a single digit number. For example, the number 123599 becomes 2 after this 'casting out nines' process, since 1 + 2 + 3 + 5 = 11, and then repeating 1+ 1 = 2. Now to check a  multiplication calculation, we cast out nines for each of the multiplicands, then multiply the resulting single digits together. If the multiplication is correct, this result should be the same as what is obtained by casting out nines for the product. For example, to check that 566 times 59 is 33, 394, we cast out nines for both 566 and 59, arriving at 8 and 5, respectively. Their product is 40, which yields 4 after casting out nines. On the other hand, casting out nines on the purported product 33, 394, we also get 4. If we had gotten something other than 4, we would know there must be an error.

This trick intrigued me already in elementary school, but later in middle school, when we learned about arithmetic in bases other than are standard base 10,  I tried to apply this trick to check my work in base 8, and realized it no longer worked. I was very excited to realize that in base 8, one needs to cast out sevens instead!  Finally, in high school, when I learned a little modular arithmetic (a.k.a. clock arithmetic), I was thrilled when I finally figured out the complete generalization to any base, including a formal proof. I finally fully understood something that has intrigued me for years! A great feeling.
[For the experts: The secret is that casting out nines simply computes the number in mod 9. This is easy to prove by writing the number in its base 10 expansion.]

How do you spend a typical day? Do you teach classes, and if so, what's your favorite class to teach?

My typical day usually involves an hour or two of class preparation, then a lecture (usually advanced undergraduate or graduate level). After that, depending on the day, I may attend a research seminar for an hour or two, and almost certainly will spend several hours in deep mathematical discussion with one of PhD students (currently I have six),  a post-doc, or possibly some other collaborator or colleague. On quieter days, I may spend time working by myself on calculations or research, or perhaps writing up results for publication. 

My favorite class to teach is intro to algebraic geometry. This fall,  I am looking forward to teaching a first abstract algebra course to juniors and seniors. 

Did you participate in math competitions when you were in middle and high school? Do you think they're useful in general?

No. I am not sure whether they were offered or not at my school, but probably I would not have been attracted to them if they were. The first time I heard of such things was the Putnam in college, and I never tried it. That kind of competition never appealed to me.

I am not sure how math competitions could be useful, though I guess that they are a fun and confidence-boosting activity for the right kind of kids. Probably they offer some good experience with problem solving, though one can get such experience in a less competitive and more collaborative way, such as participating in math camps or summer research projects,  which might better approximate the way actual mathematics (and science) get done. 
My closest encounter with math competitions is through one of my former PhD students, who was sort of traumatized by his experience in the math olympiads. Too much pressure. 

Mathematical talent takes many forms, and the type of quick-witted cleverness necessary to win math competitions is just one type, not necessarily the type of talent most essential for success as a research mathematician. More important, in my observation, are creativity, curiosity, persistence, a willingness to think deeply and ask good questions. Still, no harm in developing one's cleverness through competitions, if the student enjoys!