Associate Professor of Mathematics Stephan Ramon Garcia recently received a $199,017 National Science Foundation (NSF) grant for his project Operators on Hilbert Space. This NSF grant, which is Garcia's third while at Pomona, will support multiple undergraduate researchers over the next three years.
Garcia's research is focused on complex symmetric operators and truncated Toeplitz operators, aiming to clarify the intriguing relationship between these two interrelated classes of Hilbert space operators. Garcia's prior NSF grants were for "Complex Symmetric Operators - Theory and Applications" ($164,890, 2010-14) and "Complex symmetric operators and function theory" ($90,274, 2006-10).
Operator theory can be described as linear algebra in infinite dimensional spaces. The origins of this subject lay in Joseph Fourier's work on heat conduction and the Hilbert-Schmidt theory of integral operators in the 19th century, and in John von Neumann's mathematization of quantum mechanics in the early 20th century.
Since these foundations, explains Garcia, the study of operators on Hilbert space has proven invaluable in electrical engineering and quantum physics, in addition to being foundational to many other branches of mathematics. As such, Professor Garcia's research is wide-ranging and often takes him and his students into related fields such as matrix analysis and representation theory. Garcia says he always writes his grant proposals so that they include funding for undergraduate research that might not directly relate to the main topic.
"One can't always assume that the best projects for students will be directly related to one's main research," says Garcia, who usually has three to four senior thesis students each year as well as a few younger research students. "I find that when working with students, one needs to be on the lookout for new opportunities for undergraduates. Sometimes my regular research simply requires too many prerequisites for a student to get up to speed, so if there are novel topics which are within reach (although not directly related to grant research), I lead my students down that path."
One recent example of the branching nature of this research was the senior thesis of Bob Lutz '13, which presented long-studied exponential sums in a new visual form, discovering patterns no one had ever seen before. The work led to a co-authored paper with Garcia and Bill Duke (UCLA) in the Proceedings of the American Mathematical Society -- which was accepted for publication the morning after it was submitted. The paper was the second for the pair of researchers. A third forthcoming paper, coauthored with three other students (Bryan Brown '15, Michael Dairyko '13, and Michael Someck '15), is in press at the American Mathematical Monthly. Lutz is now attending graduate school for mathematics at the University of Michigan, Ann Arbor.
Several other articles that are attached to this particular grant, many with student co-authors, have already been submitted. All four of his current senior thesis students have been co-authors on papers, including Amy Shoemaker '14, who started Pomona thinking she'd be an English major and is now planning to be a math teacher for a few years, possibly get a masters in education, and then work on a Ph.D. in mathematics.
"Doing research with Professor Garcia and taking his Real Analysis class were probably the two main factors that helped me not to do math but to think math," says Shoemaker, who began doing research with him during her freshman year after taking his Linear Algebra class.
"He really helped me see the beauty and elegance of true math. Professor Garcia has been monumental in challenging and encouraging me and in finding ways for me to immerse myself in the field."
Her thesis topic is another example of Garcia offering a breadth of research opportunities for students. "[After] a summer program I did last summer, I became really interested in a subset of math that was not in his main research field," recalls Shoemaker. "And Professor Garcia, with his knowledge so expansive it doesn't seem human, managed to find a research topic that weaves together the two disparate fields of Logic/set theory (the stuff I was interested in) and complex analysis (the stuff he's interested in)."
Garcia has been at Pomona College since 2006, teaching classes such as Linear Algebra, Advanced Linear Algebra, Calculus II, Functions of a Complex Variable, and Principles of Real Analysis I and II. In 2009, he won the College's highest honor for faculty, a Wig Distinguished Professor award for excellence in teaching. He earned his BA and Ph.D. in mathematics from UC Berkeley, and taught at UC Berkeley and UC Santa Barbara before arriving at Pomona.