Research Advisor: Anthony Várilly-Alvarado
To Jack Petok, numbers like π and i are pieces of a mathematical puzzle waiting to be put together. Jack deals with many different number systems, equations, and potential solutions. The intersection of these has been allowing Jack to work on some problems related to the centuries-old study of polynomial equations.
“I think about how to find rational number solutions to polynomial equations in many variables. I am particularly interested in how geometry affects the behavior of these solutions if they exist, or explains the lack of any solutions. A good part of my research right now relates to how the geometry of polynomial equations over the complex numbers informs the geometry over the rational numbers”
To relate complex numbers, rational solutions, and polynomial equation, Jack uses “the modern tools of number theory and algebraic geometry. Powerfully, the theory makes sense not just over the real numbers, but over the complex numbers and rational numbers as well.”
It is Jack's love of math that has led him to delve deeper and deeper into research. For his love of math, Jack credits the great teachers he has had who got him interested in solving difficult problems for their own sake. “The classes I enjoyed most in my undergraduate years were those with significant geometric and algebraic components, so once I got to grad school I gravitated towards research areas involving those subjects.”
Although Jack enjoys solving difficult math problems, he admits that it can be “very easy to get stuck for months working on a difficult problem by yourself.” To get unstuck, you get help from others. “I love the collaboration-friendly math environment here. Our department has a culture of helping each other out, mathematically and otherwise.”
In his free time, Jack enjoys choral music, cooking, and has recently picked up baking bread. After graduation, Jack hopes to get a postdoc position where he can continue his research in algebraic and arithmetic geometry.
If you run into Jack, feel free to talk to him about math and ask him for a recommendation on mathematics books!