As far as Pamela E Harris knew when she was growing up, there were no Latina mathematicians. Through almost 20 years of schooling she had never met one. Then, one year shy of earning her PhD, she did. It meant a lot to know that she wouldn’t be the only one. The lack of role models who shared a similar heritage and background made Harris’ experience one of isolation. She says she feared that she wouldn’t be able to succeed as a mathematician. However, in 2012 when Harris attended a meeting of the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science (Sacnas), it changed her life. She is now part of a large, supportive community that uplifts and helps each other become leaders in their respective fields.
The making of a mathematician
Partitioning an integer involves dividing it up into smaller parts. Image: Flickr user MTSOfan, CC BY-NC-SA 2.0
Harris spent her childhood in Mexico and emigrated to California with her family when she was eight. Things were rough financially and, after a short return to Mexico, her family emigrated to Wisconsin. There, she attended Marquette University, and it’s here that she began to think seriously about becoming a mathematician. She says, “During my fourth year as an undergraduate student, my real analysis professor said, ‘When you go to graduate school…’. With this comment alone, she changed the course of my life. Her comment started me on the path to graduate school but, more importantly, her belief in my ability to succeed motivated me for years past the start of my graduate programme.”
Harris attended the University of Wisconsin-Milwaukee where she earned a masters, then a PhD in mathematics. Her research interests became algebra and combinatorics. She explains her work in this way: “Consider the following combinatorial problem: In how many ways can the positive integer n be written as a sum of positive integers (ignoring the order)?” For example, the number 3 can be written in the following three ways: 3, 2 + 1, 1 + 1 + 1. “Although this process is simple, determining a formula for the partition function, which counts the number of integer partitions of n, eluded generations of mathematicians and was only recently solved by Ken Ono, Jan Bruinier, Amanda Folsom, and Zach Kent in 2011. Their formula relied on the new and surprising discovery that partitions are fractal in nature.”
Finding formulae
Now an assistant professor in the department of mathematics and statistics at Williams College in Massachusetts, Harris researches vector partition functions and graph theory: work that has been supported through awards from the National Science Foundation and the Center for Undergraduate Research in Mathematics. A vector partition function computes the number of ways that one can write a vector, say v, by summing given vectors {a1, a2, …} in such a way that the coefficient of each ai is a non-negative integer. For example, one could ask, how many ways are there to make £5 from standard British coins? In this case the (one-dimensional) vector v is 5 and the set of given vectors ai is just the set of coin denominations: {2, 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01}.
Harris says, “Vector partition functions have many interesting properties, but finding formulae for vector partition functions is also very difficult.” In particular, Harris has worked on a vector partition function known as Kostant’s partition function which is important for representation theory. Representation theory is a branch of mathematics that tries to solve problems about abstract algebraic objects by representing their elements as matrices, which are easier to work with. In the case where the abstract object is a Lie algebra (pronounced ‘Lee’) understanding the representation turns out to involve combinatorics and Kostant’s partition function.
Inspiring the next generation
Fields medallist Artur Avila is one of the best-known Latin American mathematicians.
Harris also enjoys working with undergraduates on mathematical research. “I find that many undergraduate students do not know what mathematical research is about, or how one does research. Working to help them understand how as a mathematician we can take a problem and generalise it further to find new results, is one of the most rewarding aspects of my job.”
“Mathematics has taught me to be patient, to work hard and to be resilient. I know most times I will fail to answer the questions I pose, but I do know that along the way I will grow and develop new insights.”
Being Latina, an immigrant and the first in her family to graduate from university, Harris is firmly and actively dedicated to improving diversity and retention rates among women and minorities in science, and in mathematics in particular. She travels widely – her favourite perk of being a mathematician – to share research findings and to co-organise research symposia and professional development sessions for the national conference of Sacnas. She was a Project NExT (New Experiences in Teaching) fellow from 2012 to 2013, and is an editor of the e-mentoring blog of the American Mathematical Society. Her work has created new research opportunities for underrepresented students that support and reinforce their identity as scientists. In 2016, she helped develop and create the website lathisms.org, an online platform that features the extent of the research, teaching and mentoring contributions of Latinxs and Hispanics in the mathematical sciences.
Impact beyond mathematics
Harris (back centre) with her students. Image: Lisa Jacobs.
Harris is grateful for the support of her community and her mentors, including that first analysis professor who gave her her early self-belief. “I have been very lucky to be surrounded by peers and mentors. They often remind me that as a Latina mathematician, my work has an impact outside of the walls of my institution and that I can make a difference in the mathematical community. Their support has been invaluable throughout my career, and I am grateful to have them in my corner. I certainly wouldn’t be where I am today without them.”
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