*September 24, 2024*

# Robin Neumayer Awarded NSF Career Award

Carnegie Mellon University mathematician Robin Neumayer has received a Faculty Early Career Award (CAREER) from the National Science Foundation. One of the most prestigious awards for young faculty, CAREER awards recognize and support those who exemplify the role of teacher-scholars through their outstanding research and teaching.

Neumayer, an assistant professor of mathematical sciences, received the five-year grant to work on geometric aspects of isoperimetric and Sobolev-type inequalities.

“Isoperimetric inequalities and Sobolev inequalities are ubiquitous in the mathematical fields of analysis and geometry, both as a topic of study and as tools to understand other mathematical problems,” Neumayer said. “Some of the most exciting problems are ones where a result about these inequalities has connections or applications to other parts of mathematics.”

The classical isoperimetric inequality, whose study dates back to antiquity, asserts that among all surfaces that enclose a given volume, the sphere has the least surface area. This mathematical fact reflects the round shape of soap bubbles! More generally, isoperimetric inequalities, along with closely related Sobolev inequalities, relate a measurement of the “energy” (like surface area) of a shape or function to a measurement of its “size” (like volume).

Often these inequalities provide a mathematical framework to describe optimal configurations for various engineering problems and physical systems. They are also central to the mathematical fields of analysis and geometry.

Neumayer’s NSF project investigates several geometric questions related to isoperimetric and Sobolev-type inequalities, including the following: if one only has measurements of a given rod's resistance to twisting forces, how much geometric information can be recovered about the shape of the rod's cross-sections?

Questions of this type have powerful and sometimes unexpected applications in other branches of mathematics.

Recently, two collaborators and Neumayer proved a theorem about an isoperimetric-type inequality called the Faber-Krahn inequality. Following this, another group of mathematicians applied their theorem to prove a conjecture in the field of geometric measure theory.

“We had suspected that such an application was possible, but we lacked the harmonic analysis expertise to carry it out. This outcome was an exciting instance of tools and ideas being exchanged across different mathematical subfields to make progress that otherwise would not have been possible,” she said.

The award also supports teaching and outreach efforts by Neumayer. As part of her activities, she will organize a workshop for women in analysis at Carnegie Mellon, and integrate research and education through minicourses, research talks and other programs.

In addition, she plans to initiate a joint Directed Reading Program between Carnegie Mellon and the University of Pittsburgh.

“Sharing ideas through seminars, discussions, and research collaboration is central to our job as mathematicians,” she said. “We are lucky to have Pitt Math as our neighbors, and I hope the joint Directed Reading Program will be a step toward building more collaborative interaction between the CMU and Pitt math departments. The joint nature of the Directed Reading Program also will broaden mathematical opportunities for students, since there are research areas represented at Pitt but not CMU and vice versa.”

Neumayer joined Carnegie Mellon in 2021 and was awarded the 2024 Association for Women in Mathematics Sadosky Research Prize at the Joint Mathematics Meetings in San Francisco in January for outstanding contributions to calculus of variations, partial differential equations (PDEs) and geometric analysis.