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How I fell in Love with Math

Math I hated.

At some point in the middle of high school, I completely lost interest in my education. Just about everything seemed more interesting than school: friends, relationships, Irish dances, programming, and minor side projects on the Internet I had at that time. Everything but school.

In Russia, the final grade that goes on your high school diploma is calculated as the average of the grades you get all through high school. In particular, it means that if your grades are OK in the first half of high school, the second half can’t spoil them too much. Naturally, I calculated the average grade for every subject to find classes that I wouldn’t need to care about anymore. Next to each class, I wrote “FUCK IT” and never attended those classes again.

I hated everything about school. Particularly, math. 

Why in the world should I substitute numbers in the discriminant formula, then substitute the discriminant formula into another formula, just to get new numbers that the teacher so desperately wanted from me? I had no idea what I was doing. At that time, I didn’t even know that I was looking for the “roots” of a polynomial. No, it was an exercise in attention span —careful implementation of the algorithms without thinking too much. That’s what they wanted from me. 

I hadn’t learned a single proof by the time I finished high school. We had geometry lessons in which the teacher danced around triangles and lines and mumbled shamanic words about axioms and postulates, but if you asked me how to find the midpoint of a segment, I would point at the midpoint with my finger. If you told me that I must use a compass, I would point with my compass. 

Yes, I was That Kid who raised his hand and asked the teacher if I would ever need the discriminant formula in my life. Ironically, I am now probably the only kid in the history of my entire school who ended up using the discriminant formula in any meaningful way. 

The only school math I could do was execute algorithms. I could find the roots of a quadratic equation, but I couldn’t understand Vieta's formulas. I could find the angle using the dot product, but I couldn’t understand why some angles in a circle are twice some other angles. I was pretty good at logarithms, which probably says more about logarithm problems than it does about me. 

And, of course, I never participated in math olympiads. No math circles, no math summer camps, no challenging problems. I never wrote “Q.E.D” at the end of my proofs. I had no idea that some kids could participate in Olympiads seriously, attend the All-Russian Olympiad, the International Mathematical Olympiad, and then enter the best universities in the world. 

Math I loved.

I thought that if I could only choose, I would never do math in my life. But little did I know that I was doing math all the time! I always tried to understand things, but I didn’t know that a proper understanding is, in a way, math. 

When I was a kid, I played cards with my friends. And I was always losing. I desperately wanted to win, so when I was at home, I would sit on my sofa with my foot in the crack and lay out the cards for myself and my imaginary opponent, trying to come up with the optimal strategy. At first, I knew all of their cards. Later, I was concealing one card after another trying to get closer to the actual playing situation and understand the probability of them having a certain card and the optimal strategy in each case. In a way, I was toying with probabilistic combinatorial game theory!

There are many more examples coming from games: when I was playing World of Warcraft, I was calculating the ultimate armor for my hero; when I was playing with friends in board games, I was looking for the best strategy by writing out options next to each other. I even used math when I was writing my own little games, like when I had to implement an elementary physics engine or find when a line intersects a rectangle (it is way harder than you think). 

Another time, I was working on a small website that showed a new random word with its definition every day. I tried to find a way to fetch a random row from my database that would update every day. I somehow managed to find my post on a PHP forum where I asked about it back in May of 2012 when I was fifteen years old. Here’s the translation of my question.

I need to get a random row from my MySQL database every day, but in such a way that it is not possible to learn what is going to be the next day’s row. I wanted to create an entry in my MySQL database and keep yesterday’s date and then compare today’s date with the recorded one, and if the current one is more recent, then retract a row, and replace the date in the database. But this solution is not beautiful enough, how can I make it better?               

Granted, the question is not well-written. But there was something more important going on here: I was not satisfied with an ugly solution. I wanted a beautiful solution. It makes me smile to read the thread where user artoodetoo suggested that I use a random seed for the pseudorandom generator, and I agreed only when he confirmed that it is “a f*cking beautiful solution. it’s ingenious and it works.” 

Despite the solution being more abstract—the random seed and the generator of pseudorandom numbers are more complicated than storing a date in a database—I was ready to learn about it. I preferred a more conceptually sophisticated solution just because it was more beautiful. Well, that’s how math works. 

Finally, I should mention that I loved magic tricks in my childhood. And many striking tricks fall in the category of mentalism, specifically, so-called “mathemagic”. I read about mathemagic in one of the magic tricks books and was inspired to learn it myself. I wanted to perform computations faster than a calculator as part of my magic routine, so I began studying the basics of mental arithmetic. Later, I published a post about squaring numbers from 1 to 100. It was my first experience writing an article and publishing it on the Internet. 

All this happened before I was sixteen—a couple of years before I would have the courage to say that I liked math. At that time, I couldn’t tell that many of these things I was doing could be considered math: not only the content itself but the approach I had towards these questions. Because math for me was the school math. Follow the Algorithm and Get Your Grade!  

More than that, I thought that I would never go to college. I thought that I would create websites and make money from the Internet. Or do freelance work. Or, in the worst case, find a programming job in my hometown. I thought I would save money, move out of my parents’ home, find some illegal way to avoid the draft (the army in Russia is mandatory unless you go to college or have health issues), and live happily without any further years of meaningless schooling. 

I couldn’t imagine spending any more time studying things I couldn’t care less about. I had already spent eleven (!) years in this prison. Thanks, but no thanks.

Falling in Love.

My mother insisted that I should apply to college. I eventually agreed after I found out that I could get a scholarship in the first year of my studies even if I don’t attend any classes and will be expelled after the first semester. That was my plan: easy money. 

I picked the closest college to my house (Polytechnic University) and the program my grades guaranteed I’d get accepted to (engineering). I didn’t apply to any other colleges. Instead, I spent the summer hitchhiking across Russia. I found out that I got accepted to college when I was staying a night in Samara with hosts I found on Couchsurfing. The next morning, I had already forgotten about the acceptance to college. How to get from Samara at 6 am in the morning and get a car—that was what was on my mind.  

So I ended up at Polytechnic University within Siberian Federal University in Krasnoyarsk studying “computer-aided design systems in mechanical engineering”-- on paper, at least. In my head, I would get that scholarship money before getting expelled the next semester.

I showed up at the university at the beginning of the first semester to fill out the documents and get a university debit card where my scholarship would be sent. After all, I picked a university 10-minute walk from my home. I could as well visit it once. And since I was already at the building, I decided to go to the lectures to see what college education is about—just out of curiosity. 

And there it was—Mathematics.

It was hidden under a ridiculous line in the schedule: "History of algebra and geometry" (lecture) by Rybkov M.V. The course with this idiotic name was led by a young mathematician Mikhail Rybkov, who somewhat accidentally ended up teaching math at the Polytechnic University. From the very beginning, he said that he would not be going through the program material in a boring way with a droning, monotonous voice, but rather try to explain the statements and their proofs. 

This was my first lecture in mathematics. We started with complex numbers. We defined complex numbers, then we learned how to add and multiply them. I wasn’t impressed. Then we represented them in a trigonometric form. It was weird, but okay. And then we wrote down de Moivre's formula for raising complex numbers to any power. Ah, another algorithm… 

And then the teacher said: and now we will prove this formula.

I heard my heartbeat bumping.  

Prove it? It’s not another algorithm that I am going to follow? 

Prove it? I don’t need to just believe that it is true? 

Prove it? I can check every step of the proof? Understand that it is true? To be the measure of my confidence in the truth of this statement? 

We proceeded with the proof. I was blown away by a wave of new sensations, new experiences, philosophical feelings. I seemed to have touched the truth. It was an otherworldly experience! 

I came to the library and got books on philosophy and the history of mathematics: Klein, Russell, Stillwell—I read one after another. All this time there was a whole dimension around me that I didn't even suspect. Accessible and unapproachable, comprehensible and unfathomable, sacred and trivial. 

This changed everything.

Full of enthusiasm, I started to talk to the teacher every day after class. I asked what it was like to do research and be a mathematician. No one in my family had a college education. And I’d never talked to anyone from academia before. What did people in academia even do? 

It turned out that being a mathematician means that you think a lot about math, talk to other people who like math, read math papers, teach math, and occasionally (I naively believed!) grade homework. I was sold!

In one semester, instead of getting expelled as I expected (and wanted), I passed all my exams with excellence and transferred to the Institute of Mathematics, still within the Siberian Federal University in Krasnoyarsk. That was the beginning of my math journey. Thank you, Mikhail!

First days in math.

When I first came to the math department, they were not impressed with my credentials. I didn’t pass the final math test at school with flying colors, and I had zero prior math experience. How was I supposed to transfer to the math department and learn the missed material of the first semester all on my own? 

I don’t know how, but I convinced them that I could do it. 

It was my first experience making up missed material while still taking new classes. It would prove to be a useful skill, but only later. I started to attend math classes while studying the math program of the first semester on my own in my spare time. This would also prove useful in the future. 

I started to study analysis and abstract algebra, discrete mathematics, and differential equations. I learned what the cardinality of a set is and how infinities can be of different “sizes”. I learned that a polynomial with real coefficients and a degree greater than one always has a complex root. I learned the fundamental theorem of arithmetic. I was extremely happy. 

But it wasn’t enough for me. On the very first day when I came to the math department, I decided that I  would do research in math. I hardly knew any math, but I knew that I wanted to do research. So I did the best thing I could think to do. I started to go around the math department, knock on every professor’s door,  and ask them to give me a research problem. For them, I was some random guy who just transferred to the math department with zero math education. Unsurprisingly, all of them said no.

All but one. The only person who agreed to work with me was the dean of the department, Alexander Kytmanov. He gave me some problems about symmetric functions (how to express power sums in terms of elementary symmetric functions in the closed determinant form), which I eventually managed to solve. I guess it was evidence that I wasn’t entirely hopeless, as we started to work together.

Our research was about the zeros of entire functions and the extension of the resultant to the entire functions. It was quite algebraic and didn’t use much complex analysis, but I learned about residues and how the logarithmic derivative was useful for getting information about the zeros of a function. All of it would be useful later on. 

Eventually, I got my first results. I expressed power sums of an entire function in terms of its series coefficients and cooked up a notion of a resultant for entire functions. I wrote a paper and submitted it to a journal. It got accepted to the journal Complex variables and elliptic equations

Conferences.

While all this was happening, I started to go to student conferences and present my work. Student conferences are somewhat different from real conferences—for example, students get evaluated, and there is an award given for the best talk at the end. I was happy to win the best talk in my category in Novosibirsk. However, I was not familiar with the style and customs of conferences. While everyone was dressed in suits, looking their best, I gave my talk and received my first-prize diploma in a T-shirt saying “Trust me, I’m a Dr. (Dr. Dre)”. 

A little over a year after I transferred to the math department, I got completely involved in academic life. I studied math, wrote a paper, attended conferences, and was even included in research grants and got a salary! I also visited many cities in Russia when traveling for conferences. Eventually, I got enough experience to start attending real conferences (not the ones for students). 

The next crucial step in my life was attending one of these conferences in Koryazhma. This conference was different from any other I visited before. It seemed that surprisingly many good mathematicians visited this conference, and when I talked to them I realized just how little math I knew. 

At this conference, I met professors from Moscow. And I started to talk and go to the conference events with Professor A and Professor E. When I was talking to professor A, he suggested I transfer to Moscow, to the department of mathematics at the Higher School of Economics, one of the best math schools in Russia! 

The only problem was that I didn’t have enough money to live in Moscow. My parents couldn't even afford to buy me the ticket to Moscow, let alone cover my living expenses. Alexander told me that it would be possible to get a job as a teaching assistant, work as a tutor, and probably participate in grants by doing research. But at that very moment, I didn’t have any money or any job offers. 

Also, I had to pass the transfer exam. The professors told me that they could come up with a mock exam to test whether I would be able to pass the real transfer exam. Alexander wrote a complex analysis exam and professor E wrote an algebra exam. The plan was to take the exam during the conference the next day. I had less than a day to prepare. 

The exam.

Despite it not being a real exam, I wanted to take it with all seriousness. I asked them what kind of topics the exam was supposed to cover. The topics of complex analysis sounded fairly familiar because I was doing complex analysis for my research. But the topics in algebra… It was hopeless. It was one of the most humbling experiences of my life. I asked professor E what topics in algebra were going to be covered, and he answered easily, “why, groups, rings, linear algebra, all the standard material”. All the standard material. Except I didn’t know what groups and rings were and I had only the very general idea about linear algebra. I was embarrassed to say anything so I just nodded and pretended that all of it made sense. 

When I got back to the hotel where I stayed, I started to prepare for the exam. I read what groups and rings are, what their standard properties are, but a single night clearly wasn’t enough to learn new concepts and get used to working with them. I wasn’t ready for algebra and I expected the worst. Even though it wasn’t a real exam, it was supposed to show whether I should even try it or not. I was desperate. 

The next day I took the complex analysis exam. I knew approximately every topic and I had a lot of practice with the residue problems, and there were two problems on the exam that were related to the residues, so I managed to write them well enough. I felt that there was a chance. 

The next day I took the algebra exam. And it was catastrophic. 

The first question asked me to write down all abelian groups of a given order. I didn’t know anything about abelian groups. I wrote some nonsense. 

The second problem was about raising a matrix to the 100th power. The problem was aimed to test linear algebra: either eigenvectors and change of basis, or the operators on matrices and Hamilton-Cayley theorem. But at that time I didn’t know what eigenvalues, eigenvectors, or operators are. But it was the 100th power. So I couldn’t come up with anything better, but to take my matrix and square it to get A^2, then square it to get A^4, then square to get A^8, and so on till I got A^64. Then I wrote A^100 = A^64 * A^32 * A^4 and found the final expression. It was a dirty hack that didn’t use any linear algebra at all. But at least I solved the problem. 

The third problem was a blessing. It was about expressing the power sums in terms of elementary symmetric polynomials. In fact, it was the very question my advisor had asked me to do on the first day when I came to the math department! I solved it in just ten seconds because I remembered all of the procedures and even formulas themselves by heart. 

The fourth problem was a disaster. It asked whether some order (say, 56) was solvable. I didn’t know what a solvable group was. I couldn’t write anything meaningful about the problem. Nothing. But the previous night I happened to read a random sentence on a random math forum that all groups of order less than 60 are solvable. I didn’t know what it meant, but I remembered the sentence. So I just bluntly wrote the sentence verbatim with no explanation or understanding. It wasn’t even a solution, but I had nothing else to offer. 

Now I don’t remember what the last problem was, but at the time when I was taking the exam, I thought that it already didn’t matter. I didn’t know how to solve the basic problems, and the problems that I could solve were either by luck or in spite of the fact that I didn’t understand the right way to do them.

To pass the exams, I had to get a grade of at least 6/10 in both complex analysis and algebra. The next day I got my grade for the complex analysis exam: it was 9/10. I felt happy but I knew that algebra was going to be a disaster. Later the same day I got the algebra score: it was 6/10. 

This is probably the right place to reveal two facts that I only found out later. First, this mock exam turned out to be the real transfer exam I had to take to get to Moscow. My solutions were scanned and attached to my transfer application. Second, I met people who tried to transfer to the same math department but only got 5/10 in the algebra exam. And because of that, they were rejected. So it was not an illusory threat to my transfer. 

It is one of the instances of my unbelievable luck that I got 6/10 on the exam when I knew nothing about the material. Just because I managed to hack solutions and use memorized sentences from the Internet, I managed to scrape together the 6/10 that I needed. It wasn’t my merit. It was pure luck. But luck was on my side.

Moscow.

I returned from the conference in mid-August while the academic year in Moscow would begin on September 1st. So I told my parents that in two weeks I would move to the capital of Russia to study at another university. They freaked out, understandably. 

I had never lived on my own before. And I had no money. And my parents had no money. And I had no plan for what I was going to do, where I would live, or how I would make money. Forget life expenses—I couldn’t even afford the flight to Moscow! I had to borrow money from a conference professor just to buy a ticket. 

Everything was uncertain, but it was also adventurous. I loved it. Somehow I convinced my parents that everything would be fine. I wonder if I believed so myself. I moved to Moscow on August 31st. 

Long story short: I worked as a teaching assistant at several universities, tutored, and won scholarships, grants, and awards to survive financially in Moscow. Occasionally, there were accidents that left me completely broke, but this story is not about my financial struggles. 

I found myself in the capital of Russia, at HSE Faculty of Mathematics, one of the best math schools in the world. My classmates were graduates from the top schools of Moscow,  winners of the international olympiads, participants of the math camps where they did research with the leading Russian mathematicians. On the office doors I saw names from textbooks and math history books I read. It was the best possible place to flourish.

Except that I had a little problem: I was ridiculously unprepared to study there. It’s not your typical impostor syndrome: I was in the third year of the math program and I had little idea what groups, rings, or fields were. Imagine a physics student in the third year who hasn’t studied mechanics. That was me. 

On top of the third-year program that already had me in over my head, I had to cover the curriculum difference between universities—all the classes in the first two years of the program that I missed. There were a ton of classes. Because of bureaucracy, I had to take not only math classes, but also history, English, and a weird course called Personal and Social Safety. My schedule was absolutely packed. I used every single academic unit that I could legally use. To exaggerate, I had to cover the four-year program just in two. And don’t forget about research! I started writing a paper on random matrices, gave talks at conferences, and participated in grants. 

All at once, I had to get used to an independent life away from my parents, work at several universities to make money, cover the curriculum difference, study the third-year program that was beyond my comprehension, do research, participate in grants, and prepare for applications to the PhD programs. Bonus points if I could make any friends or enjoy the capital of Russia.

Many people helped me enormously. Every day after classes I met with mathematicians and asked them questions about everything that I was supposed to know. Hours and hours every day. People helped me, and I’m immensely grateful for that. Among everyone else, Alexey Klimenko alone spent months answering my questions for multiple hours several days a week. It’s fair to say that without Alexey’s immeasurable help, I wouldn’t have been able to catch up with the program. Thank you, Alexey! 

Eventually, I fulfilled the curriculum difference, passed all the courses, published a preprint on random matrices, and graduated with honors. It was the craziest two years of my life. And I totally loved it. 

Applying to Ph.D. programs.

Of course, I planned to stay in math for a little bit longer. I decided to apply to Ph.D. programs to keep doing research and stay in academia. 

But my application situation was weird. On one hand, I had huge gaps in my education. I didn’t take as many advanced classes as my classmates did. And I scored really poorly on the GRE math test—some would say it was an inadequate score for top math schools. On the other hand, I managed to do some research in my undergrad which probably could serve as evidence that at least I would not be hopeless in producing new mathematics. 

Another problem was, again, money. The applications are expensive! Each one costs about $75, plus another $45 to send the test results to each school. I needed the GRE General, the GRE Subject, and the TOEFL exams, which cost about $600 in total. I just didn’t have that kind of money. In fact, never in my life had I possessed that much money at once. 

Because of these problems, applying to PhD programs in math wasn’t the best idea for me. A safer plan would be to apply for a master's degree in Moscow, learn more math, become more mature in research, and then apply to PhD programs after the master's degree. In fact, my advisor recommended that I stay in Moscow. It was a reasonable recommendation. 

But that’s just not how I do things. 

The application process was another little adventure, but in short, I prepared my application as best I could and found people who agreed to write recommendation letters for me. Again, generous people in academia lent me money which I agreed to return from my PhD salary. 

And I finally applied. 

Stanford and Quals.

It’s probably impossible to adequately describe the feeling that you get when you find your first offer. For me, at that time, it was the long-awaited reward for all the struggles I experienced while transferring from university to university, covering curriculum difference after curriculum difference, struggling to survive in expensive Moscow, borrowing money, again and again, working in multiple places to support myself in the pursuit of my dream to do math that was born from a random math lecture in Krasnoyarsk Polytechnic University, and which I still pursue. 

I got several offers from the PhD math programs. I chose Stanford. 

After two impossible years of constant pressure, I moved to California. The PhD program was to be five years long. I started to have a consistent salary. And I worked so hard for the last two years in Moscow. I could finally relax, right?

Wrong. 

At Stanford, math PhD students have to pass the qualifying exams on Algebra and Real Analysis. Many of my classmates passed the exams at the beginning, before the start of the academic year. Since my education had many gaps, it wasn’t as easy for me. 

And it was hard to concentrate on the exams. For the first time in my life, I was living abroad. I started to have spare money. I felt that I finally deserved rest after everything I’d done in Moscow. That I could relax. Needless to say, I didn’t do a good job preparing for the quals. 

I failed them. And failed miserably. I had one more attempt to retake the quals, and if I failed the second time, I would be expelled from Stanford. Of course, everyone assured me that it never happens—that everyone passes the second time. 

But for me, it was an existential threat to my journey. It meant that I couldn't relax. I had a summer to prepare for the retake. And that’s exactly what I did. It was a hard time full of guilt. Every day I didn’t spend on the preparation felt like self-betrayal. It was a hard time for me. 

I passed the retake. Barely. My scores were just above the cut-off line. I remembered the algebra exam with my 6/10. The dean of graduate studies asked me to meet with them privately to discuss my poor performance on the quals. I was told that my low score can affect how my potential advisors could see me as their student. I was reminded that I have a lot more to do. That I still have many gaps in my education. That I shouldn’t forget about them. And I mustn’t relax. The quals were over, but the feeling of despair remained. 

When I called my mother and told her that I failed the first quals, she said that maybe it was right — that maybe it was too hard for me, that it was my mistake to think that I could go to a good American college. Maybe I should go back to Russia. 

I still can’t forgive her. 


In the next year, I started to work with Daniel Bump, an excellent and supportive advisor. We began to work on the topics of solvable lattice models and their connections with the representation theory of p-adic groups. 

After almost two years of studying and work, I published my paper on Combinatorics of Iwahori Whittaker Functions. Now I run the Solvable Lattice Models Seminar and continue working on the representations of p-adic groups and exactly solvable lattice models. 

It took me another two years before I started to feel comfortable with my math background. To stop feeling like I am that guy who transferred to the math department from the polytechnic university. That guy somehow got to Moscow without knowing basic math. That guy who had a talk at that conference without a good reason. That guy who accidentally got accepted to Stanford. That guy who failed the quals and should have returned to Russia. 

Conclusion.

It was one teacher. Just one teacher at a Polytechnic Institute in Siberian Federal University in Krasnoyarsk teaching engineers the “History of algebra and geometry”. One initiative and ambitious teacher who decided to bring life to the class and teach it passionately. One teacher was enough to ignite genuine curiosity and desire to understand things better. 

In personal conversations with Mikhail, I learned about math, about my academic career, about how to enter this world. I learned that the mathematical world was not out of reach. In fact, it sat right at the tip of my pen. It was accessible and, in a way, real. And it was waiting for me. 

Thank you, Mikhail! 

After I started studying mathematics, I fell in love not only with math but with understanding. It became interesting to me to understand and be aware of different things in mathematics and beyond. Alas, I became interested in all school subjects many years after graduation. And now I regret that my parents did not transfer me to a good school where I could learn physics, chemistry, biology, literature, history, and everything, everything, everything else that interests me now but I have so little time for.

Please, if you teach (anything at all!), bring your soul and passion to your class. Your example, your attitude, and your love can touch one of your students, and you can transform someone’s life. Even a course with the idiotic name History of Algebra and Geometry taught for engineers in a random university in Siberia can radically change someone’s life. My example shows that it’s possible. 

Thank you!

The mistakes of teachers are not so noticeable, but ultimately they are no less expensive [than doctors']. - The Irony of Fate, or Enjoy Your Bath!

My math journey.

July 2014: I graduated high school thinking that I would never do math in my life

September 2014: I fell in love with math during a math lecture

January 2015: I transferred to the math department at Siberian Federal University

February 2016: I published my paper on entire functions

August 2016: I transferred to the math department at HSE in Moscow

December 2017: I applied to schools for a Ph.D.

January 2018: I published my paper on random matrices

January 2018: I got an offer from Stanford University

July 2018: I graduated from HSE with honors

September 2018: I moved to California to study at Stanford University

September 2019: I passed qualifying exams on the second attempt

November 2021: I published my paper on representations of p-adic groups