Theories Taking Shape

COLLEGE OF NATURAL SCIENCE & MATHEMATICS

An early-career grant for a theoretical mathematician brings real-world advantages to students.

Assistant Professor Lihan Wang received a Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS-MPS) grant from the National Science Foundation (NSF) in August 2023. This prestigious grant is awarded to assistant professors who have never served as principal investigator on an NSF grant. Dr. Wang, of California State University, Long Beach’s College of Natural Sciences and Mathematics, was awarded $250,000 for two years to study the spectral geometry of Steklov eigenvalues.

Purely theoretical mathematics is the foundation for many technologies and experiences that most people take for granted. The Global Positioning System (GPS), for example, is only possible because of mathematical concepts, such as manifolds and geodesics, that have been developed by mathematicians for hundreds of years. But, says Dr. Wang, “Most people don’t need to know the math. They just use it.”

Dr. Wang’s research grant allows her to spend time on her own areas of inquiry and mentor more students, who could be more likely to pursue careers in advanced mathematics as a result.

Dr. Wang was not satisfied to merely use applications of theoretical mathematics, however. As an undergraduate studying earth science, she was taught what equipment could tell researchers about the composition of the earth, but she couldn’t understand the theory behind that equipment until she began to study mathematics. “I felt I should learn math first, especially since it has a lot of applications,” she says. Now, she can’t say what’s inside the earth. Instead, she develops the mathematical analysis that enables other scientists to see what would otherwise remain unknown. These applications range from electrical impedance tomography, which displays the shapes of things inside the earth, to space telescopes that reveal the shape of objects billions of miles away.

The focus of Dr. Wang’s current research and her grant is the spectral geometry of Steklov eigenvalues. Steklov eigenvalues are complicated and abstract, but in general, eigenvalues can be interpreted as tones of sounds, and Steklov eigenvalues are a special type of eigenvalue. Dr. Wang’s research is so theoretical that real-world examples are the best way to explain it, and she offers the example of how a coffee cup and a glass of wine differ. “When you walk with a coffee mug, the coffee spills or sloshes out, but when you walk with a wine glass the wine does not,” she says. This is because, she says, “the shape of the mug or the glass really matters, and the math behind that relates to Steklov eigenvalue problems.”

With the funding from the grant, Dr. Wang seeks to increase the participation of underrepresented groups in the field of mathematics and better prepare CSULB students for a career in math or science.  The grant allows Dr. Wang to focus on mentoring undergraduate and graduate students more than she otherwise could, and she mentored four students during the fall 2023 semester. Dr. Wang assigns her mentees papers and other materials to read and discuss that are related to her research area or that explain new applications as a way to motivate them to consider fulfilling careers in mathematics. She also selects assignments that they can work on to introduce them to more advanced theoretical mathematics than they access in their regular courses.

Dr. Wang is thrilled to receive the LEAPS-MPS grant.

Mentees meet regularly with Dr. Wang to discuss their progress and ask questions. The grant also helps Dr. Wang bring visiting speakers and workshops to campus. CSULB has a high percentage of Latino students and first-generation students, and introducing these students to excellent mathematicians with backgrounds similar to theirs is critical to increasing their confidence and inspiring them to seek advanced mathematics degrees. This is how Dr. Wang’s LEAPS-MPS grant is launching not just her academic career, but the potential careers of many more future researchers.