The research by David Cruz-Uribe, professor of mathematics, can be difficult to understand if you don’t have a Ph.D. in mathematics. Recently, though, Cruz-Uribe was inspired by a colleague to collaborate with his undergraduate students, and he has been awarded a three-year grant of $105,772 by the National Science Foundation (NSF) for his work.
Cruz-Uribe’s research is a continuation of the approach taken by 19th-century French mathematician Joseph Fourier, who used simple functions with smooth graphs to approximate the graphs of complicated functions with sharp corners while studying the flow of heat in solid objects. Exploring this approach, mathematicians have developed powerful tools and discovered solutions to differential equations that arise in various branches of science. With the support of the NSF, Cruz-Uribe hopes to continue that trend.
“My work is devoted to developing new tools within the field of harmonic analysis, and then applying them to the study of more abstract differential equations,” he says. “This kind of mathematical research deepens our understanding of a wide range of mathematical ideas, and these, in turn, can have surprising and unforeseen applications in many different areas.”
The grant from the NSF, in addition to supporting the continuation of Cruz-Uribe’s research into weighted norm inequalities and partial differential equations, will support him in engaging advanced undergraduate students to join in his work.
He was inspired by Scott Rodney, an associate professor of mathematics at Cape Breton University in Canada, who has successfully involved undergraduates in his research, something that can be a challenge for researchers in pure mathematics.
“When students used to ask me what I did, my standard answer was to tell them to go to graduate schools, and when they had passed their qualifying exams to come back and ask me again!” he says. “However, this year my senior thesis student, [Mathematics Scholar] Philip Cho ’15, is writing his thesis on Sturm-Liouville theory, an area I want to learn more about. And I am currently working on a project with [George A. Kellner ’64 Presidential Scholar] Greg Convertito ’16 that is related to my research.”