Penn Today interviewed the math department’s incoming chair to learn about his longtime passion for geometry and his hopes for the future of contemporary math research.
Tony Pantev, the Department of Mathematic’s next Chair, shares his hopes for the future of contemporary math research and his longtime interest in geometry and topology.
As one one of the oldest branches of science, mathematics has seen a lot of changes. From being taught under the broad umbrella of “natural philosophy,” which combined math, physics, and chemistry into a single subject, to the modern era of specialized subtopic research, math is no longer a solitary field but one that requires active collaborations with other fields in order to answer complicated questions.
One of math’s major and current open problems is the homological mirror symmetry conjecture. It grew out of a striking observation made by physicists working on string theory, who noticed that methods from complex geometry, or the use of polynomial equations to describe shapes and spaces, and from symplectic geometry, the use of energy-driven dynamics that describe general physical systems, produced the same answers. This insight surprised many researchers in geometry and led to the formation of the program known as homological mirror symmetry.
How did you become interested in math, and how did you end up here at Penn?
Math was something that I was always curious about and something that I tried to explore in any way I could. I went to a special high school [in Sofia, Bulgaria] for math and science. You would choose a specialty, and I chose math; we had university faculty teaching all the specialties. It was a great experience.
In Bulgaria, you apply to a given major, so I applied to math; clearly that’s what I was going to do. But then I had the feeling that I’d been doing that all my life, so I should do something different. All males after high school go to the army for two years, and I thought of army service as a break from math. I was seriously considering switching to philosophy or creative writing, but by the end of my first year in the army I decided to stick with math.
The moment I got to university, my perception of the subject totally changed. I got exposed to really deep things that I hadn’t known about before, and things got really exciting.
When I finished college, it was 1989, the year the Berlin Wall fell, so we got access to materials from Western universities where they announced their graduate programs. I applied, came here to do a Ph.D., and that was that.
How has Penn’s math department changed in the past 30 years?
Somehow the atmosphere hasn’t undergone any drastic changes. I think it’s great that, in terms of scientific content and energy, the department hasn’t changed that much.
One thing that’s new is our contact with physics. When I was a student, we had some faculty who had joint appointments and were working with physicists, but it wasn’t something that was really visible. With physics, we are very lucky because we share a building, so it’s very easy to work together.
Many math and physics departments try to do this but not many succeed. I think it’s because we have big groups, not just one or two people working for math working with one or two people from physics. We really intermingle. We have common seminars, we supervise students together, and we come up with projects. It’s been productive and exciting.
How did you become interested in homological mirror symmetry research?
I got interested in high dimensional geometry and topology in my last year of college. I started going to physics and geometry seminars. The subject was fairly new and not that well-developed, and it was even more so back in Bulgaria because the information flow was very slow. Even innocuous things, like current research publications in abstract math, weren’t freely available.
How has this field of research evolved?
The homological mirror symmetry conjecture was formulated in 1994, and back then it was a completely wild thing. Two years before, physicists formulated the mirror duality of conformal field theories that mathematicians just didn’t believe. It was so bizarre, and mathematicians couldn’t parse it in any reasonable scientific way.
The physicist’s mirror duality had implications for an established subject called enumerative geometry. Physicists love computation, so they took those implications and started doing calculations and produced invariants. The invariants come in hierarchies, and what the mathematicians were able to do at the time was basically the first floor, but it took a lot of time and effort.
The physicists could give answers at any level of hierarchy. They produced answers at level 1, where mathematicians did have answers, but when we compared the answers they were different. Then the mathematicians double-checked their computer code and they found a bug. They fixed the bug, reran the calculations, and got exactly the physicist’s answers.
Then Maxim Kontsevich, one of the most original minds in mathematics in the second half of the century, proposed this homological mirror symmetry conjecture, a mathematical explanation based on the ideas of the physicists. He succeeded in proving parts of it, and that’s what started this whole thing.
What inspires you the most about this area of research?
It’s amazing because it bridges two completely different geometric universes that, in principal, don’t interact with each other, but through this conjecture they are truly a reflection of each other. You can ask questions in one universe and answer it in the other, and questions that are easy in one are difficult in the other and vice versa.
It’s hard to picture how having two different mathematical “universes” produce the same answer is so incredible. At first glance, it almost seems to make sense. Can you explain why mathematicians were so blown away by this conjecture when it was first proposed?
These two geometries are completely different, and on the surface have no interaction with each other. It turns out that there is this dictionary that allows you to take a phenomenon in one and recast it as a phenomenon in the other, a dictionary that tells you that they are the same.
It’s like you have two different movies that are telling the same story. They contain the same information, the same emotional message, the same intellectual content, but all of those things are packaged differently. It’s like the color in this movie is completely equivalent to the sound of this one. You can listen to the words here and you can paint the picture there.
Do you think this conjecture will have an answer in the near future?
I think we’ll have a reasonable proof of the conjecture pretty soon—not the full generality, but big enough so we’ll have what we need.
The interesting thing is that the conjecture is actually growing. There are really exciting new questions and proposals coming out, and as a collaboration we are getting easily distracted because they have implications that go far beyond the immediate goal.
Where do you hope to see the field of mathematics go in the future?
The math we teach in a standard undergraduate curriculum is probably two to three centuries old. It’s not modern math. But that is also changing in the past 10, 15 years: Modern math has come into other sciences and applications, and students get exposed to those ideas. That’s a trend that I would like to see expand.
Math is a peculiar subject. It is science in its own right, but it’s also a universal language that provides a canvas for other sciences. Math has been very successful in being a tool in scientific modeling, but it also has been revolutionized by being applied to other subjects. That’s one of the most exciting and productive things that has happened in math in this century, and it’s a trend that that’s really pushing the subject to its boundary.
The Simons Collaboration on Homological Mirror Symmetry involves Columbia University; the, University of California, Berkeley; the Simons Center for Geometry and Physic;, the University of Miami; the University of Vienna; and the Institute Des Hautes Études Scientifiques. Penn is responsible for organizing and leading the activities of researchers across eight scientific nodes, including Tony Pantev as director and Ron Donagi as deputy director.
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