Solving problems in algebraic geometry

Lawrence Ein “has been one of the foremost algebraic geometers in the U.S. and worldwide for the past 30 years.” — Photo: Roberta Dupuis-Devlin

Lawrence Ein’s work solves problems in algebraic geometry, a branch of pure mathematics that has important applications in theoretical physics.

“Algebraic geometry is the study of geometric objects defined by algebraic equations,” said Ein, LAS Distinguished Professor of mathematics, statistics and computer science. “An application of algebraic geometry is cryptography. One can use geometric object such as an elliptic curve to create secured codes.”

His research, which has been continuously funded by the National Science Foundation since 1983, has solved fundamental problems in the field of algebraic geometry.

“People have spent a lot of time — the last 200 years — understanding geometry of algebraic curves and surfaces. Only in the last 40 years or so, we are beginning to have methods and ideas to work on higher dimension geometry,” he said.

Ein’s recent work focuses on describing equations and relations among equations, called syzygies. He and his collaborators resolved the Gonality Conjecture of Green and Lazarsfeld — a 30-year-old problem about the equations of curves in space. He’s also proven Shokurov’s Ascending Chain Condition conjecture for smooth varieties,
a key question in birational geometry.

“Any one of these achievements would suffice to establish Ein as one of the foremost mathematicians of our times,” Brooke Shipley, professor and head of mathematics, statistics and computer science, wrote in Ein’s award nomination.

“He has been one of the foremost algebraic geometers in the U.S. and worldwide for the past 30 years. He has proved numerous groundbreaking results, resolved many fundamental questions that had been open for decades or even centuries, and has introduced crucial techniques that are widely used by other researchers.”

Ein was invited to speak at the International Congress of Mathematicians in 2006, one of the highest distinctions in mathematics. He is a Fellow of the American Mathematical Society and he has more than 70 journal publications.

“You get to where you are with help from a lot of people, and you’re grateful to be so fortunate,” he said.

Ein, who joined UIC in 1984, teaches mainly graduate students and currently supervises two doctoral students.

“Research allows you to have deeper understanding and appreciation of knowledge, and with this knowledge, it helps you to prepare graduate students,” he said.

Ein was happy to be recognized for his work with a Researcher of the Year award.