Mathematics professor will explore the Einstein equations of general relativity, among other topics
Justin Corvino, assistant professor of mathematics, was awarded an $114,570 National Science Foundation grant to help fund his current research program focusing on “Problems in Geometric Analysis and General Relativity.”
“The problems in my proposal involve the application of differential geometry and partial differential equations, or geometric analysis, to mathematical problems often motivated by the theory of general relativity,” explains Corvino. “In particular, much of the proposal involves problems about the space of initial data for the Einstein equations of general relativity, a theory which models gravitation and forms the foundation for much of cosmological modeling. It turns out there is a rich interplay between geometric structures and models of physical quantities in this theory. Several of the problems I discuss are related to the asymptotic structure of isolated gravitational systems, stemming from my earlier work, which led to a resolution of a question of Roger Penrose that was open for many years.”
A key element to receiving the grant was showing how students would be involved with the research through a Research at Undergraduate Institutions (RUI) Impact statement. The RUI Impact statement is a chance to describe how the grant might impact the undergraduates in a way that might not be part of the project at other institutions.
“One tricky aspect of an RUI proposal is often how to include problems that undergraduates can handle, but are also interesting and relevant to the larger questions in the field and tied to the rest of my research program – much of which is too advanced for most undergraduates,” says Corvino. “I didn’t want to propose ‘toy’ problems for the students, but meaningful ones that are interesting to me. I then had to give evidence that we can have success with such problems, the bulk of which was provided by my past work on EXCEL projects, honors thesis and Research Experience for Undergraduates (REU) projects.”
The project includes several problems that could involve students, such as problems related to the isoperimetric problem (enclosing an amount of volume with a surface of smallest area), as well as problems related to the famous Hawking-Penrose black hole singularity theorems. All of these problems stem from Corvino’s prior work with students. Corvino hopes to have two students each summer, one funded by the grant, and another through the EXCEL program.
“These students would work with me during the summer, and be part of the summer research atmosphere in the mathematics department, working alongside other EXCEL scholars, as well as the REU students,” says Corvino. “I hope that the project can attract students to the field, and lead to honors theses, independent study projects, and/or special topics or seminar courses.”
“In general, the hope is that the students will be motivated to learn advanced material, often at the graduate level, all the while drawing on all the course work they will have had at Lafayette, and applying and extending it. Also, students can get a glimpse of what it’s like to do research when you try to answer a problem whose solution is not known,” says Corvino. “It should help form a solid foundation for graduate work, and also to help students decide if they want to pursue research.”
Corvino looks forward to starting research on this three-year grant.
“I am completely thrilled to receive the grant,” says Corvino. “In the past five years, only five grants were awarded in Geometric Analysis to RUI proposals from faculty at institutions similar to Lafayette. When I looked at the entire list of those who also received awards this year, I was humbled to see so many big names in the field and to have received an award alongside such esteemed colleagues is truly an honor.”
Corvino joined the faculty in fall 2004. He received an M.S. and Ph.D. in mathematics from Stanford in 1996 and 2000, respectively, and a B.S. in mathematics with a minor in physics from MIT in 1994.