Getting to Know You - with Michael Zhao
Michael Zhao, currently a sophomore in mathematics, was encouraged to go into mathematics by his parents. His mother had left China to pursue graduate studies in physics at Utah State University, and his dad had received a PhD in materials science from Jilin University. His primary interest was in history until he took a number theory class in eighth grade, which piqued his interest in mathematics.
When he came to the U, he was insistent upon learning number theory. His first year, he did a reading course with Dr. Savin on algebraic number theory, a branch of number theory that studies algebraic structures like groups, rings and fields and their relation to algebraic integers, which are roots of monic polynomials (polynomials with leading coefficient equal to one) with integer coefficients.
In general, algebraic number theory focuses on using algebraic techniques to study number theory. One relatively well-known class of problems in number theory are Diophantine equations. These are equations with usually many more unknowns than equations, where one seeks integer-valued solutions for those unknowns. One question, for instance, that he has thought about is whether there are two Pythagorean triples with the same product, i.e. numbers a, b, c, d, e, f with a2 + b2 = c2, d2 + e2 = f2, and abc = def.
He says that he has really enjoyed the reading course, but also hopes to explore other areas of mathematics such as analytic number theory. Analytic number theory seeks to use analytic methods to study the integers. One well-known result, for instance, called the prime number theorem, states that for large enough N, the number of primes less than or equal to N is roughly N/log N. In fact, this was the area of mathematics he was intending to learn more of when he first came to the U.
Currently, he is taking the math department’s intro to research class, where they are studying the electric and elastic breakdown criteria of composite materials. It is often only possible to measure, say, the electric field from the boundary, and the question is, given measurements of electric field and current at the boundary, is it possible to determine if the internal fields are high enough to create a hazard (i.e. cause the material to breakdown)? Next semester, he says, he will likely be doing an REU related to random graph generation.
He says that mathematics, contrary to what many people think, is essential to modern conveniences. It provides a clear and succinct way to express how the world works; without that, it would be difficult to imagine that we’d have a theory of relativity, which is used in GPS calculations, or quantum mechanics, which one needs to build a transistor, hence a personal computer. Even pure mathematics often drives discoveries in applied mathematics, but barring practical applications, one should not need a good reason to study pure mathematics, just as one does not try to justify hiking a difficult mountain in terms of practical applications!