Mathematics Professor to Join Four-Month Program at the Mathematical Sciences Research Institute

“Number theory is amazing. The problems are stated simply but solving them is challenging.”

Min Ru, professor of mathematics at the University of Houston, will share his expertise in Diophantine geometry, a branch of number theory, through a four-month research professorship at the Mathematical Sciences Research Institute in Berkeley, California.

According to the institute, professorships are reserved for distinguished researchers who can make key contributions to their programs. His assignment-in-residence begins January 17, 2023, and ends May 26, 2023.

Diophantine geometry is one of three programs the institute will organize next year. The aim of the program is to bring together experts and young researchers to gain knowledge from one another, begin and continue collaborations, and to further advance the field by solving mathematical problems.

## Origins of Diophantine Geometry

Ru, a faculty member at UH’s College of Natural Sciences and Mathematics, researches number theory, which mathematicians have wrestled with for more than a millennia. Number theory is the study of the properties of integers, or whole numbers, rational numbers and other number structures that share similar features.

Diophantine geometry evolved from mathematician Diophantus of Alexandria who was born in the third century. He authored Arithmetica and introduced algebraic concepts through it. The 1621 edition of Arithmetica gained fame after Pierre de Fermat, a French mathematician, wrote a theorem, known as Fermat’s last theorem, in the margins of his copy.

Somewhat comically, Fermat never proved his theorem, because he wrote that the margin of his Arithmetica copy was too small to fit his proof, and he later died.

The problem of finding a proof for the theorem went unsolved for centuries, until mathematician Andrew Wiles provided a proof of it in a 1995 paper published in the journal Annals of Mathematics.

These proofs influenced Diophantine geometry, which involves the use of algebraic geometry as a tool to solve Diophantine equations.

## Applications on Cryptography and Coding

Ru studies these intriguing topics because it is interesting and challenging for him.

“People think mathematics is useless,” Ru said. And while there are few direct applications of the work, “for these algebraic questions, they can have implications on coding or cryptography. I’m teaching a course on number theory’s application on cryptography.” Cryptography protects information through codes.

Number theory was the subject of Ru’s Ph.D. thesis, and he has pursued the study since then.

“Solving the problems isn’t always a straight line.”

- Rebeca Trejo, College of Natural Sciences and Mathematics