# Professor Oscar Fernandez Named a Guggenheim Fellow in Mathematics

Oscar Fernandez, Class of 1966 Associate Professor of Mathematics and faculty director of the Pforzheimer Learning and Teaching Center at Wellesley, has been awarded a 2021 Guggenheim Fellowship for applied mathematics. Fernandez is one of 184 recipients selected from an applicant pool of almost 3,000 scholars from the United States and Canada. He described his selection as an “out-of-body experience.” “Now it’s not just my mom telling me I’m doing a good job,” he said. “It’s an organization empowering me to do some good. And that is what I hope to achieve through the project I proposed.”

Here, Fernandez discusses his project, his teaching, and what it means to be a Guggenheim Fellow.

**How did you react when you learned you’d been named a Guggenheim Fellow? **

It was like having an out-of-body experience. It seemed, and still seems, too high an honor to have been bestowed on me. No one in my family went to college or was awarded any prestigious fellowships; my mother was a waitress and my father a taxicab driver. So, my first reaction to receiving a Guggenheim was that same imposter syndrome that so many students experience daily. (Yes, even Guggenheim Fellows feel that.) But—like we counsel our students struggling with imposter syndrome—surely the selection committee did not pull names out of a hat. I was selected. To those of us with a lived experience of being “less than,” such an honor, such a validation and recognition of one’s work and effort, is transformational.

**Can you give a brief summary of your proposed project?**

The central aim of my fellowship project is to build mathematical bridges between disciplines that study inequality. Take, for example, the Gini Index G and the “life table entropy” H. Both G and H are measures of inequality, but G is well known to economists and not so much to demographers, while H is well known to demographers and not so much to economists. The two camps, working in their respective silos, have accumulated vast literatures describing and relating G and H to other phenomena in economics and demography, respectively. Yet lurking behind the mathematics of all that research are sure to be previously unseen connections (we know of a handful already).

Research is about discovery. I view teaching in exactly the same way—I am out to discover that special mix of support and encouragement that can help each student thrive.

Oscar Fernandez, Associate Professor of Mathematics

For example: Can we find natural habitats in which life span inequality follows the same pattern as income inequality does in some regions of the world? And what could we learn from that? Could biodemography help us come to an organic understanding of how to reduce inequality? How does our choice of an inequality measure bias the outcome—the level of inequality measured—in favor of (or against) some population? Is it possible to remove all such biases? If not, what does that say about how we ought to measure and communicate inequality?

These and other questions are too broad to be studied from one disciplinary perspective alone. Fortunately, because mathematics is fundamentally about relationships, I have hope—and some preliminary results to demonstrate—that viewing inequality through its many lenses can spawn a web of interconnections capable of revealing new perspectives and accelerating the pace of discovery for those of us working to understand inequality, its causes, and its dynamics.

**Your research focuses on geometric mechanics and the related field of nonholonomic mechanics, and you are currently researching quantum nonholonomic mechanics and the applications of the calculus of variations to problems in mathematical demography. Can you describe what those fields seek to investigate and your goals for your research? **

Geometric mechanics is the fusion of advanced mathematics with the subfield of physics known as mechanics, and nonholonomic mechanics adds constraints into that mix. Mechanics studies motion and tries to understand why objects move as they do. Why do apples not fall along a swirly curve? Why does a ball thrown in the air follow a parabolic arc? What explains these and other recurring and specific patterns in nature? Perhaps the best illustration of what transformational insights geometric mechanics is capable of lies hidden in your bedroom. Place a bowling ball at the center of your mattress, and the mattress curves down (there’s the geometry). Now place a ping-pong ball close enough to the “start” of the curved region. When you let it go, the ping-pong ball will fall towards the bowling ball (there’s the mechanics). Matter curving space to generate motion. This was one of Einstein’s fundamental insights—that gravity is the curvature of “spacetime” by matter—and it serves as a perfect example of the powerful synergy of geometry and mechanics. It’s deep insights like these that keep me captivated with mathematics and physics.

That Principle of Least Action, described by mathematician Pierre Louis Maupertuis as “Nature is thrifty in all its actions,” carries over into the world of the tiny—the quantum world. But other aspects of quantum mechanics are entirely new and not well understood. For example, how can we make sense of the quantum mechanics of constrained systems? In our macroscopic world, parallel parking is difficult because we’re not strong enough to violate the “no motion perpendicular to the wheels’ rolling direction” constraint of a car’s motion. Instead, we need to maneuver along a complicated, curvy path to fit our car into its parking spot. But does the same thing have to be true about a “quantum car”? Do such “nanocars” always have to respect similar constraints as they move, quantum mechanically? In a research article I published with Mala Radhakrishnan, associate professor of chemistry at Wellesley, we showed that they do not. Echoing other quantum weirdness, we showed that the motion of nanocars can violate the constraints one would naturally impose on the system and still satisfy all the rules of quantum mechanics.

Inequality is often hard to observe, because it requires noticing what is not there rather than passively accepting what is.

Oscar Fernandez, Associate Professor of Mathematics

My work in mathematical demography, believe it or not, stems from similar thinking. Rather than thinking about the curve a car moves along while we parallel park it, we can similarly think about the curve that describes the probability of staying alive as one ages. (Both are trajectories, just with vastly different meanings.) We each have such a curve associated with us. And it’s almost certain that no two curves are exactly the same. There is, in other words, some inequality between individuals’ survival curves. How can we measure that inequality? What are its origins? What are its dynamics? The calculus of variations and the other tools and techniques I’ve studied in mathematical physics can help study such questions and once again reveal fascinating insights and connections.

In earlier work with a colleague from UCLA—Hiram Beltrán-Sánchez—we showed, for example, how changes in one popular demographic measure of life span inequality—the “life table entropy”—can be decomposed into changes between another life span inequality measure—the “life span disparity”—and the mean life expectancy at birth. A foundational goal of the project I proposed as part of my Guggenheim application is to push out the boundary of these connections and insights into other disciplines.

**How has your teaching influenced your approach to your own research? **

Research is about discovery. I view teaching in exactly the same way—I am out to discover that special mix of support and encouragement that can help each student thrive. I firmly believe that such a mix exists for each student, and that it’s my goal to find those seeds of success and nurture them in every student I teach. These “special sauces” differ from one student to the next, often with great variation in form and substance. We return thus once more to the theme of inequality, but this time in a generative sense. Learning what works best in the classroom for different students makes me a better educator, which ultimately helps all the other students I teach succeed. Research and teaching then become a dynamic and synergistic dance that offers the promise of discovering common principles underlying those special mixes of support and encouragement I believe exist for each student. Teaching, then, is as much about research for me as it is about education.

**You co-founded the Wellesley Emerging Scholars Initiative (WESI) with the goal of increasing the number of students from underrepresented groups majoring in math. What inspired you to start the program?**

Inequality is often hard to observe, because it requires noticing what is not there rather than passively accepting what is. Who’s not seated at the table? Who’s not in your classroom? Who has not spoken yet? When I came to Wellesley in 2011, I noticed immediately that there were few students of color majoring (or minoring) in math. Yet there were such students enrolled in our calculus courses. Somewhere along the way, then, whole populations of students nearly disappeared from our mathematics curriculum—and from our department’s hallways. I was not the only one who noticed this; Stanley Chang—our current department chair—noticed it too, among others. He and I got to talking about what could be done and what could have positive results for little to no investment of funds.

Those conversations eventually led us to create the Wellesley Emerging Scholars Initiative (WESI), a weekly hour-long learning community centered around excellence in mathematics, modeled after Uri Treisman’s (University of Texas, Austin) very successful Emerging Scholars Program. We asked students to form small groups and work on challenging mathematics problems, looking first to the group for support if challenges arose, and then to us. The core point of WESI was to remove all the labels and assumptions often made of students—and certain groups of students in particular—and to treat them as we should all treat each other: as human beings capable of wondrous things if properly supported and encouraged. Indeed, a 2016 analysis of WESI by Akila Weerapana, associate professor of economics at Wellesley, showed that compared to non-WESI underrepresented students, WESI students were less likely to receive low final grades and more likely to enroll in a future math course. Today I’m very happy to be continuing to work with my colleagues—and with students—toward the much larger ideal of inclusive excellence for all Wellesley students.