Age: 17Irvine, CA
Project Title: The Rational Cherednik Algebra of Type A_1 with Divided Powers
My project studied a class of mathematical objects called rational Cherednik algebras, which appear in the intersection of algebraic geometry, representation theory, and mathematical physics. These objects were extensively studied in zero characteristic (essentially one of two perspectives in algebra), but in recent years, a theory of Cherednik algebras started to develop in positive characteristic. In this latter case, a big part of these algebras disappears, which causes them to lose a lot of their interesting structure. In my research, I worked on a way to recover this lost structure using divided powers and investigated what the new recovered object looks like. I proved a number of useful structural results about these extensions, and I constructed a generalization which unifies the zero characteristic perspective with the divided power version. Overall, the theory of Cherednik algebras in positive characteristic is still in its early stages, and since Cherednik algebras are so ubiquitous, my work could lead to many unforeseen results.
My name is Lev Kruglyak, and I love to ask questions. From my earliest years in an orthodox Jewish elementary school, I learned about my ancestors — generations of scholars and Rabbis who dedicated their lives to asking and answering their own theological questions in a never ending cycle of intellectual pursuit. This philosophy of persistent curiosity has lived on in me, and it remains my most defining characteristic. The best example of this is my passion for theoretical mathematics. To me, doing mathematics is analogous to exploring a complex landscape woven out of ideas. Precisely formulating and answering the right questions allows us to soar high above these landscapes, and to see elegant structures emerging from the chaos below. Last year, I was thrilled to be accepted into the MIT PRIMES USA program, which gave me an opportunity to take part in true mathematical research. Now, over a year later, I feel immensely grateful to be named a Davidson Fellow and humbled to be included in a community of such brilliant peers.
A question I often get asked when talking about my research is: Why? What has humanity gained with this incremental discovery about some obscure structures in theoretical mathematics? A simple argument one could make is that even the most abstract mathematical concepts will eventually find applications in the real world. However, from my point of view, this perspective completely misses what mathematics is truly about. I don’t love math because of its real world applications, I love math because it is the purest form of creative expression, an art form removed from the physical limitations of the real world. Theoretical math is as useful as literature or music — it represents humanity’s desire to take the chaotic world and transform it into something of beauty. For thousands of years, humans have posed questions of imaginary objects, weaving a complex web of mathematical truths. Gradually, with every new proof, a node or connection is added to this web until a critical mass is achieved and a powerful result emerges. My research is the perfect example of this gradual process.
I began my work in an active area of math research: the study of Cherednik algebras in positive characteristic. This area looks to be incredibly promising, with numerous applications to important areas of mathematics, such as representation theory and mathematical physics. My advisor suggested to take a novel perspective on this topic — rather than directly study Cherednik algebras in positive characteristic, we introduced “divided powers” to enhance the structure of the algebra. I was one of the first researchers to investigate Cherednik algebras from this perspective, so I had very little prior research to base my results on, and I had to develop most of the techniques from scratch. After a year, my efforts were rewarded with concise, elegant theorems which revealed key structural information about Cherednik algebras with divided powers. The novel approaches I developed serve as a foundation for researchers to build upon in the future.
It’s very rare in mathematics to discover the correct proof of a difficult problem shortly after encountering it. In the first months of my research, I stumbled a lot. The ideas of representation theory were still very new to me, and the original research question was quite vague. My mentor and I were the first researchers to study Cherednik algebras with divided powers, so we had little to no definitions, theorems, and even conjectures to start from. My initial explorations yielded some results, but they were convoluted and not very interesting. In fact, practically all of the results I proved between February and June weren’t relevant enough to make it into the final paper. Oftentimes when reading a finished paper, or an elegant proof, it’s easy to forget that this polished surface is the end result of many hours of effort, filled with roadblocks and dead ends. I learned this valuable lesson first-hand while working on my project.
Although it might not seem like it, I haven’t always been interested in math. During my ninth grade year, I was an avid coder, caterpillar farmer, or chemist, depending on the year. Even when math joined my expanding roster of interests, I still found myself exploring many non-mathematical things. For instance, when I learned about Newton’s law of gravitational attraction in school, I spent the rest of class wondering if such a simple rule really could lead to the rich emergent behavior of galaxies and stars, forming and colliding into each other in a perfect balance. A few weeks of coding later I was able to simulate the slender arms of a spiral galaxy emerging naturally from millions of random, chaotic particles. This same passion drove me to explore fractal graphics, neural networks, electrical engineering, and dozens more unrelated yet endlessly fascinating topics. Aside from these theoretical explorations, I enjoyed wrestling as part of my high school varsity team, tutoring math at the Russian School of Mathematics, and hiking and biking the various mountain trails in Orange County. During the summers, I attended the Ross Mathematics Program at the Ohio State University where I made amazing friends and learned some beautiful mathematics.
In the Fall of 2021, I will be attending Harvard University where I will be concentrating in pure math, with maybe a secondary in computer science. Due to the unusual circumstances surrounding the upcoming school year, I decided to take a gap year. During this time I plan to intern as a software engineer, travel, and work on my latest passion project: building a 1970s era computer entirely out of several thousand discrete transistors.
Where do you see yourself in 10 years?
Working on a PhD in theoretical math, after that I’ll be figuring out what to do with a PhD in theoretical math.
If you could have dinner with the five most interesting people in the world, living or dead, who would they be?
Alexander Grothendieck, Evariste Galois, Ada Lovelace, Isaac Newton, Paul Erdos
In the News
IRVINE TEEN AWARDED $25,000 SCHOLARSHIP FOR UNMATCHED ACHIEVEMENT IN MATHEMATICS RESEARCH
Lev Kruglyak to be Named a 2020 Davidson Fellow Scholarship Winner
Irvine, Calif. – The Davidson Fellows Scholarship Program has announced the 2020 scholarship winners. Among the honorees is 17-year-old Lev Kruglyak of Irvine. Kruglyak won a $25,000 scholarship for his project, The Rational Cherednik Algebra of Type A_1 with Divided Powers. He is one of only 20 students across the country to be recognized as a scholarship winner.
“I feel immensely grateful to be named a Davidson Fellow and am humbled to be included in a community of such brilliant peers,” said Kruglyak.
For his project, Kruglyak studied a class of mathematical objects called rational Cherednik algebras, which appear in the intersection of algebraic geometry, representation theory, and mathematical physics. The novel approaches developed by Kruglyak will serve as a foundation for researchers to build upon in future research.
“I love math because it is the purest form of creative expression, an art form removed from the physical limitations of the real world,” said Kruglyak. “Theoretical math is as useful as literature or music — it represents humanity’s desire to take the chaotic world and transform it into something of beauty.”
Kruglyak will be attending Harvard University in the fall of 2021 to study pure math, following a gap-year internship in software engineering.
Click here to download the full press release
Click the image to download hi-res photos:
The following disclosure is provided pursuant to Nevada Revised Statutes (NRS) 598.1305:The Davidson Institute for Talent Development is a Nevada non-profit corporation which is recognized by the Internal Revenue Service as a 501(c)3 tax-exempt private operating foundation. We are dedicated to supporting the intellectual and social development of profoundly gifted students age 18 and under through a variety of programs. Contributions are tax deductible.
Profoundly gifted students are those who score in the 99.9th percentile on IQ and achievement tests. Read more about this population in this article.