Kristine Jones, PhD – Senior Data and Applied Scientist, Microsoft
Opinions expressed are my own and not necessarily those of my employer or any institution that I may be affiliated with.
When I was first approached about writing this post, I was asked to try to convey what I know now that I wish I had known as I was applying for jobs out of grad school. My response to questions such as this is always that there is no one course I wish I had taken, no one skill I wish I had acquired, no one opportunity that would have pushed my early career down a dramatically different path. This is not to say that I haven’t reaped the benefits of broad exposure to numerous skills commonly bullet-pointed on tech industry data scientist job descriptions. I would not be doing my due diligence if I pretended otherwise. That being said, hiring decisions for mathematicians based exclusively, or even primarily, on those bullet points are poorly considered. (Learn some coding, stats, and optimization methods, though … it can only help you).
Ultimately, the most valuable skill that a mathematician brings to any team is her ability to abstract the core technical problems the team is facing and provide the basis for solving these problems across all of the scenarios in which they assert themselves. The crux on which this skill rests is an education that teaches students to think deeply and critically about mathematics, both independent of and relative to the breadth of contexts in which it appears. What follows is a personal history of how I arrived at this view.
From my vantage point as an undergraduate, the University of Chicago embodied this notion of linking independently considered critical thought with the scope of its applications.
The opposite criticism is often levied against the University of Chicago: that it is overly devoted to abstraction in place of the concrete. Perhaps the most telling supporting evidence of this criticism is the slogan “That’s all well and good in practice, but how does it work in theory?”, text often displayed along with the University’s logo on t-shirts sold by student organizations. Other examples abound, even (perhaps especially) from within the mathematics curriculum. As a senior studying abroad at the University of Chicago Paris Center, I took a course in representation theory. On the first day, the instructor, trying to assess the background of his students, asked the class who could define a vector space. Several hands shot up, mine among them, all with the same answer ready – a vector space is a module over a field. The basis for the criticism is not unfounded.
But, while the University does certainly value abstraction, it does not do so without purpose. Even their motto declares this intention: Crescat scientia; vita excolatur. Let knowledge grow from more to more; and so be human life enriched.
The goal of all the theory, of all the abstraction, of all the critical thought, is to enrich human life. To make meaningful changes in the way scenarios play out and issues are addressed. To enhance the ways people think and people act.
This idealistic motto filtered down to the mathematics undergraduate curriculum in two ways that were evident to me.
First, mathematical abstraction was not left to the math majors. Proofs and notions of generalizability were at the core of even the most introductory of mathematics coursework, both calculus and non-calculus based. To these students especially, the value to be found in the mathematics coursework was not in a particular formula or tool that they may or may not ever need, nor in seeing jargon from their chosen field strewn out across rote problems. Instead, they left their math classes with the ability to reason deeply and strongly about quantitative questions they would encounter in the future.
Second, if you wanted a B.S. in mathematics instead of a B.A., you had to take three non-introductory courses in a related minor field. Physics, chemistry, and computer science were common choices, in addition to a more structured option in economics, which was my selection. This was not for an applied math degree, where the minor field requirement was even larger, but for a degree in pure mathematics. Amidst the modules over the fields, you had to understand what was real about the math you were studying.
I left the University of Chicago not only with lofty examples of abstract concepts and their footprints across mathematical theory, but also with an ability to “suss” out broad connections to this theory across all manner of problems I encountered. I headed off to graduate school to pursue a passion for what I thought was mathematics, with no particular career path in mind. I saw, and still see, a graduate education as an end in itself.
Let’s be honest here, though. Grad school is not for everyone. Grad school is hard.
That’s ok, it’s supposed to be. At its very best, graduate school asks that the student be lost in a sea of seemingly unconnected examples, looking for problems and finding solutions, ultimately weaving everything together into a single theoretical fabric. It is from this process that graduate education derives its worth. Completing this work is an incredible demand on any person. I was lucky to have a truly amazing graduate advisor. He excelled at guiding his students from examples to theories and back again, all the while allowing them to find their own way.
Which is not to say I was always successful in seeing that path. I spent one summer doing hundreds of matrix computations only to reach the conclusion that I would need to do thousands more to see if there was any pattern that might indicate the presence of an underlying theorem. My advisor’s comment upon hearing this was that I needed to come up with an “exit strategy for the project,” without saying much in the way of how I might do that. That was the “finding my own way” part.
Moving forward out of that moment was one of the hardest things I have ever done. It was only when that work was nearly completed that I saw the value of my education was in the struggle to see connections between specificity and generality, regardless of any career decision I would make. More than that, I realized that my passion for math was really a passion for executing this skill. This insight came at a great moment because I was about to graduate, and seemingly out of nowhere, previously unconsidered options abounded.
I sent out resumes and went on interviews. Microsoft stuck. Since I’ve been here, I’ve designed large-scale machine learning systems, implemented component-scale algorithms, coordinated projects across research and engineering teams, and performed executive-facing analyses in high-value areas. Very little of this has much relation to the content my thesis (although I did run a lecture series on the theoretical underpinnings of homomorphic encryption that one time), but I doubt I would have been able to accomplish much if I hadn’t gone through the process of writing it. All of my solutions for Microsoft draw their impact from seeing connections between many smaller problems, with many commonalities, and solving them all at once – quite similar to academic mathematics in form, if not in function.