"On Life and Math", Chapter 2: A Little Brown Pond
When it came time to apply to college, I got the same question over and over: “It’s great that you’re so good at math, but what do you want to do with it?” Growing up and getting a real adult job means abandoning the pursuit of math for its own sake.
Pivot towards what, then? Through high school, I’d done well enough in my science classes (all Advanced Placement, of course): easily the best in physics, excellent but not standout in biology, and struggling but nevertheless at the top in chemistry.1 But none of them really moved me the way math did.
At a loss, I eventually started telling people that I wanted to major in engineering. That sounded mathy enough, and it was sufficiently broad that it didn’t require me to really contemplate what I was signing up for; I figured I’d figure out the specifics later. But most of all, it gave me the feeling of being settled on a path towards a real adult job.2
I ended up at Brown. Like all incoming engineering majors, I began with Intro to Engineering. I quickly developed tremendous disdain for it: I recall feeling as though all we ever did was solve the same basic linear algebra problem over and over,3 having been fed thusly contrived oversimplifications (i.e. down to linear algebra) of real-world problems.4
Meanwhile, in math-land, I was flourishing. I was taking a course on differential geometry: the study of curves and surfaces in 3-dimensional space. As engineering class sputtered along, my differential geometry professor brought us from zero to the perfectly miraculous Gauss—Bonnet theorem.56
At some point during that first semester, I suddenly realized: Wait a sec, math professors are people too! And this is their job!! And if this can be their job, then it can be my job too!!! And so, becoming a math professor became my biggest goal: a goal that would last for the next 17 years of my life.
And what a noble goal it was: the creation and dissemination of pure beauty and truth. Or even better, of pure beauty derived from pure truth. I had found my life’s purpose.7
As the semesters rolled on, I became ever more deeply engrossed in mathematics. I quickly finished the bachelor’s degree coursework, and took enough grad-level classes to earn a concurrent master’s degree as well. I joined the Math Department Undergraduate Group, and soon became its president. We hosted weekly math seminars with samosas, and an annual day-long conference uniting undergraduate math-lovers across the Northeast. I spent all my summers doing math research, too.
And I was still the best. More or less. I placed at or near the top of all my classes,8 and found ways to convince myself that I outshone each of the three other math majors in my year that I viewed as competitors.
But then there was Dave. Dave was a year younger than me, and off in his own league. He came into college knowing an outrageous amount number theory. I think I considered him a different species. And I ended up entirely avoiding number theory throughout undergrad, I think as a defense against having to come to terms with how much he knew: with how incredibly good he was.
Dave would keep an eye on the number theory seminars taking place over at Harvard and MIT, and often took the train out to Boston to catch the ones he found particularly intriguing. On some level, I think it didn’t even compute for me that there was more math happening elsewhere: at Brown, I was a big fish in a small pond, and I loved it that way.
Actually, chemistry was the hardest class for me in all of high school. I’d regularly stay up until 12 or 1am to finish my homework, despite needing to be back at school by 7:30am for orchestra rehearsal (an optional “0 period” elective). It was then that I began running in order to blow off steam — midnight sprints up a tall steep hill.
I’m reminded of a friend who told everyone throughout college that she was planning to go to law school afterwards, only to realize upon graduating that she actually hated the idea. In both of our cases, claiming a societally acceptable career path prevented us from contemplating its on-the-ground realities, since few people pressed us for more details.
Namely: given that Ax = b, solve for x.
Most real-world phenomena are nonlinear, and so trying to model them using only linear algebra generally entails dramatic oversimplification. It makes sense that basic linear algebra was the only math the instructors felt they could use, though, because multivariable calculus and linear algebra were merely concurrent course requirements.
Well, zero beyond the prerequisites of multivariable calculus and linear algebra.
The Gauss—Bonnet theorem states that for any surface, a particular geometric quantity miraculously only depends on the overall shape of the surface.
Precisely: given a compact Riemannian surface M with Gaussian curvature K (a smooth real-valued function on M determined by its Riemannian metric), it states that
In particular, the total Gaussian curvature (the surface integral on the left) only depends on the topology of M — specifically on its Euler characteristic χ(M).
As a first example, all spheres have total Gaussian curvature 4π (since they have Euler characteristic 2): a tiny sphere has little area but high curvature, whereas a larger sphere has more area but lower curvature — and these differences exactly cancel out in the integral.
But this also applies not just to round spheres! Even a sphere with all kinds of geometric irregularities — for instance the surface of the earth (with its innumerable mountains, valleys, and saddle points) — is still guaranteed to have total curvature 4π.
And again, the Gauss—Bonnet theorem applies not just to spheres, but to all compact Riemannian surfaces! What an absolute delight.
I’ve come to associate that feeling with a particular instant I experienced at a monastery in upstate New York on a crisp autumn morning: the sun shining brilliantly across the valley; the leaves radiating a thousand different colors, from bright yellow to deep red; a bell ringing cleanly through the air, inviting us all into the transcendent fullness of sheer presence. What a gift to be alive, suspended in this utterly perfect moment.
One of my professors didn’t like giving out perfect scores as a matter of principle, and so when I aced his midterm he jokingly took off 3 points for spelling my name correctly.