# Russian Mathematics

By mlipton on Sep 30, 2015 in Russia

Mathematics is a special discipline in that one’s culture and upbringing rarely affects its content. A historical account can be shaped by your nationality, your interpretation of a novel can be affected by your race, and even the observations you make in a scientific experiment can be affected by your cultures (albeit fields like physics and chemistry don’t suffer that much from this). But mathematics has nothing to do with the human condition, and as a result, enjoys almost unanimous agreement as to what is considered “truth.”

So when I enrolled in the Math in Moscow program, I was unsure what “Russian mathematics” was. I only participated for the opportunity to take high-level courses not offered at Willamette. And while it’s true that there isn’t some fundamentally distinct perspective on the mathematics itself when studying it in Russia, the pedagogy and the structure of the courses diverges from the typical North American style. In most American math classes, you listen to a lecture, do a few homework problems that solidify your understanding, followed by taking an exam. The reason it’s like this is because this basic structure has been tried and tested over the course of centuries.

In Russia, the math courses are much more interactive. All of my classes had a dedicated “problem solving” portion where the students would collectively work on the day’s problem set, or some question posed by the professor. But the most important part was our reliance on each other for the homework problems. After the second week or so, the problems we were assigned were so difficult, that just every student in the class had to consult each other out of necessity. It’s not that the course itself is inherently difficult; it’s easy to assign trivial topology problems, but the university felt we would get more out of the classes by working on difficult problems as a team. One of my favorite memories was working with two other students on describing how to construct a universal Turing machine. The previous lecture gave us the definition of a Turing machine and a few examples, but it was sometimes up to us to prove some of the more substantial problems. Universal Turing machines are a core concept in Computability Theory, so I suspect that if I were to take the course in America, this would just be another lecture. This mimics how mathematics research is done. Mathematicians work on difficult problems and collaborate with others to seek new insights.

Furthermore, the classes emphasized the interconnectedness between seemingly disparate branches of math. This might fall on deaf ears, but the prime example I can think of is how contour integration of a meramorphic function around a pole is strikingly similar to determining the index of a singularity of a vector field on a surface. The central idea is that the specific geometry of a loop around a pole or a singularity of “nice” functions is not as relevant as the fact that the object in question is contained in said loop. That is, “nice” functions make seemingly infinite calculations straightforward. This is a recurring theme in mathematics, much like how one could say the desire to retain power is a theme that runs throughout history.

I would highly recommend any student at Willamette interested in learning how mathematics is done beyond the confines of our small community to study abroad. Whether it’s Math in Moscow, Budapest Semesters in Math, or even the MASS Program at Penn State, there are so many opportunities available.

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