Schubert's theory, convex geometry, and the journey to solving complex problems. An interview with Valentina Kirichenko.

— If you look at the feed of scientific news, you can always understand what is currently trending, so to speak. What attracts attention, what is easier to get grants for, and so on. However, mathematics is different. It remains a closed system that can only be understood from the inside. Is there fashion in mathematics? And what is currently fashionable in your field – algebraic geometry? Who is considered a star today?

— I work on topics related to convex geometry — with polyhedra and everything that can be touched. The field itself grew out of toric geometry, but now it has expanded significantly. One of the titans of modernity in this area is Andrey Okunkov. There is even a term called “Newton-Okunkov convex body.”

— Quite an impressive co-authorship.

— In fact, Newton had polyhedra, and Okunkov has developed this area in modern times. Today, it is indeed a fairly fashionable science. What I like about it is that you can reformulate quite abstract results from algebraic geometry in terms of convex geometry. For instance, how many integer points are there in a polygon? It turns out to be related to some algebraic curve, which is not easy to explain. Yet, at the same time, integer points in a polygon and Pick's formulas can be explained even to school students.

— What unsolved problems interest you?

— I am not so much engaged in classical algebraic geometry as in its intersection with representation theory. There are algebraic objects with significant internal symmetry. This is called “Schubert calculus,” which is also a very fashionable topic due to its multifaceted nature. It includes combinatorics, which can be tackled by people completely distant from algebraic geometry and its representations. There are many interesting open problems in this area. I really enjoy taking all these problems and trying to reinterpret them through convex geometry, searching for more explicit and beautiful solutions.

— Which specific problems are you interested in, and what is the main difficulty in finding solutions?

— There was a Hermann Schubert — and he was not a composer. “Berlioz is not a composer,” as it was said in Bulgakov's works. Hermann Schubert lived in the 19th century; he was a school teacher and also a renowned mathematician. He loved Hamburg, which at that time did not have a university, and he taught mathematics in a gymnasium while already being a world-famous scientist. Nowadays, it is hard to imagine a school teacher publishing scientific articles known worldwide. He even had a very famous monograph titled “Calculus of Enumerative Geometry” (“Kalkül der abzählenden Geometrie”). In essence, he solved various problems almost through black magic. For example, one problem is: given four pairwise intersecting lines in three-dimensional space, how many lines intersect all four? It is hard to even imagine whether such a configuration is possible — some four lines, and we need one that passes through all of them. Interestingly, physicists have recently used this problem. They actually needed the equation of a line that intersects all four.

— What was that experiment?

— There is a machine called HADES (High Acceptance Di-Electron Spectrometer), which is almost akin to Hades from ancient Greek mythology. This spectrometer is located in Darmstadt. German physicists, in collaboration with our scientists from Dubna, were analyzing results brought by this machine. Physically, they had a particle suspended in their setup, and sensors were trying to determine its trajectory under the assumption that it was moving in a straight line. They took two sensors, drew the equation of a line, and checked whether the particle intersected it or not. This process is quite lengthy; computing all trajectories would take ten years on a modern computer. They realized they needed to consider four sensors, taking into account that these are not points but segments of lines. Thus, based on those four lines, they needed to write an equation for a line that intersects them. They immediately created some sort of installation! They mentally took the example of saw horses, on which logs are sawed. You can always place a fifth log on four logs, and it will definitely touch all four and can be sawed. They discovered all this themselves, and then saw my article on Schubert calculus and approached me as the main Russian-speaking expert on this topic. Schubert had a method that allowed him to accurately determine how many such lines would exist.

— What was his black magic in this case?

— Well, this is the simplest problem; there were more complex ones. For instance, suppose three circles are drawn on a plane; how many circles will touch all three? Schubert found an answer to this problem! Among other things, Schubert solved a problem whose answer was a nine-digit number! And it was completely correct! As modern algebraic geometers write, it is as if a person was blindfolded and landed a huge airplane perfectly on the runway. In other words, Schubert developed a method, but from a modern perspective, it had no justification. The Schubert calculus method is truly black magic without any possibility of justification.

— This is very interesting because, if we return to physicists, it was previously believed that theoretical physics gives many ideas to mathematics. Now it seems less obvious.

— Yes! But interestingly, mathematicians usually interact with theoretical physicists, while here we have experimental physicists. It is always interesting to see how applicable your problems are to our Universe. Perhaps they are some abstract theories without applications. So it’s great when physicists take this and apply it, and they have processed a lot of calculations using this method. Otherwise, they simply wouldn’t have managed!

— If we continue the connection between the real Universe and algebraic geometry, GPS and similar systems come to mind. But is there something less obvious?

— The Schubert calculus became popular in the 19th century, but then interest somehow waned. Hilbert had one of his problems numbered 15, which was precisely about justifying Schubert calculus. Somehow, it was established, more or less, but not everything was justified. There are some problems that remain unproven by modern methods. Well, they decided that they had generally justified it and that was it; they set it aside and forgot about it. Then, a hundred years later, it turned out that all these abstract calculations were needed for theoretical physics. Interestingly, mathematicians tried to calculate something and couldn’t. They reached some limit, and then our science said it could go no further. But physicists calculated further. String theory helped here. You can believe in it or not, but it is very interesting for mathematics as it solves purely mathematical problems using completely non-mathematical methods.

— Today, there are several universities in Russia where students are lining up to study mathematics. I thought that the top place has always been the Faculty of Mechanics and Mathematics at Moscow State University, but in the 90s, after a robust mathematical school, you did not go there. Why?

— In the turbulent 1990s, it became possible to study in non-state educational institutions. It seemed a bit scary, but it was possible. Interestingly, at that time, there was absolutely no competition at the Faculty of Mechanics and Mathematics at Moscow State University, meaning you could study as much as you wanted. People were heading towards economics, psychology, and law faculties. It was just impossible to get into those. But everyone who wanted could get into the Faculty of Mechanics and Mathematics.

— Why did you ultimately choose the Independent Moscow University?

— Well, I didn’t need a deferment from the army, so I decided to give it a try. There was always a lot of bureaucracy at the Faculty of Mechanics and Mathematics; you had to attend physical education and similar subjects, which, I thought, distracted from mathematics. However, many students studied both there and at the Independent University because the Independent University held classes in the evening — specifically for additional education.

— Were names not a factor back then? It seems that everyone left, but someone must have been teaching?

— Yes, in the 1990s many left, but some maintained connections with their homeland through the Independent University. For example, Askold Georgievich Khovansky worked both in Toronto and in Russia. Eventually, it turned out that the Independent University was a place of strength where prominent specialists came to teach Moscow students. The Faculty of Mechanics and Mathematics was not such a place at that time. So overall, I don’t regret graduating only from the Independent University. Although now I see that almost no one followed my example. Especially since later the Independent University lost its license for full education, and it became clear that private higher education didn’t quite thrive in Russia.

— How did students choose what to study in the 90s? What seemed promising?

— Economics became very popular. During Soviet times, everyone hated economics because it was Marxist, but in the 1990s, a