Igor Tolkov

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I've been interested in mathematics since a preschool logic puzzles book I read for fun. While a middle and high school student I took advanced classes and excelled at local and regional math competitions. In summer of 2006 and 2007 I attended a 6-week residency camp for high school students interested in mathematics where we were exposed to college-level math.

In the fall of 2007 I started college. At the University of Washington (UW) there are two years of accelerated calculus sequence. The first covers calculus, differential equations, and linear algebra; the second covers topology, multivariate calculus, and complex analysis. At the end of the second year everyone wrote a paper on a topic of their choice. By this time I've come to be fascinated with the Sato-Tate conjecture, so I wrote about it in my term paper.

Honors Advanced Calculus Term Paper | |

Mentor | Dr. Jim Morrow |

Period | 11 May - 3 June, 2009 |

Abstract | We introduce the the elliptic curve and the problem of counting the number of points on the curve when it is reduced modulo a prime. For any such curve that is nonsingular, Hasse's theorem provides a bound for the number of points. We prove Hasse's theorem and discuss more recent developments, namely, the exact formula for the number of points on a special class of curves (the ones with complex multiplication), as well as the Sato-Tate conjecture for the distribution of Hasse error terms on curves for which no exact formula exists. |

Full text | Counting points on elliptic curves: Hasse's theorem and recent developments (PDF, 9.3 MB) |

In my third year I studied abroad in Budapest. It was a great experience to do nothing academic other than math. (The only exception was a Hungarian language class.) There was a little lounge in the college building where our classes were held, and often we occupied the lounge and discussed mathematics. One of the senior students lectured us in category theory and abstract algebra. We also went on a lot of weekend trips.

Quickly the four-month adventure was over, and I was back at UW, where I took number theory in Winter and a graduate course in the Spring. Budapest was my bridge to graduate math. After I came back, I took mostly graduate math courses.

In my fourth year I took graduate algebra. We covered modules over PIDs, modules in exact sequences (injective, projective, etc.), some cohomology, some representation theory, Galois theory, and ended with Noetherian rings and local rings. In my fifth year I took graduate real analysis, where we covered Lebesgue measure theory, both concrete and in terms of abstract measures, topology, Hilbert and Banach Spaces, dual spaces, and Fourier analysis.

My genuine interest, though, was in preparing for research as quickly as possible. I sat in or took relevant graduate courses in algebraic number theory. In my sixth year at UW I was finally ready for more advanced math. I took algebraic number theory in the fall and elliptic curves in winter and spring. For my fall quarter class I had to write another course paper. Long before I started exploring a problem related to elliptic curves, so I wrote about it.

Algebraic Number Theory Term Paper | |

Mentor | Dr. William Stein |

Period | November - December 2012 |

Abstract | In this paper we consider the question of whether an elliptic curve over Q(√5) can have infinitely many points with primes in the denominators, that is, whether for infinitely many P=(x/z², y/z³) with z relatively prime to x and y, z is prime. We confirm that this question is related to the rank of E, and that the number of primes on rank-1 curves admits a uniform bound, whereas the number of primes on higher-rank curves is unbounded. We also describe an algorithm for computing the rank of an elliptic curve after a list of generators is given by 2-descent. |

Full text | Primes on Elliptic Curves over Q(√5) (PDF, 385 KB) |

However, I felt like this project was too computational, and I always wanted to do something more theoretical. I am currently exploring other topics, also related to the Sato-Tate conjecture.

I enjoy competing, so I took advantage of the undergraduate student math competitions. I took the Putnam Competition the maximum number of four times, once in Budapest. This is a 6-hour competition with 12 problems. The annual median score ranges in 0-2 out of 120 maximum points.

Putnam Competition | |

2008 | 11/120. The UW team with Nate Bottman, and William Johnson, and me placed 15th, and we were featured in the UW math department news. |

2009 | 21/120. |

2011 | 17/120, which placed me in the top 500. |

2012 | 20/120. |

The Mathematical Contest in Modeling (MCM) is another competition. As part of it, students spend 72 hours building a model and writing up a solution to a real-life problem. I participated twice.

Mathematical Contest in Modeling | |

2009 | The team with Chris Raastad, Ian Zemke, and me placed meritorious.Our paper: Traffic Circles (PDF, 361 KB) |

2010 | The team with Alex Leone, Chris Raastad, and me placed successful.Our paper: Yet Another Approach to Geographic Profiling (PDF, 1.0 MB) |

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