one of the biggest culture shocks between undergrad and PhD was about math.
during undergrad, all my friends loved discrete math – combinatorics, algebra, algorithms, number theory – and that was all we wanted to learn. we did enroll for statistics and probability as token representatives of “cool and real” math, but we deemed the majority of continuous math as “for the mechanical engineers.” we used to dub various analysis subjects by the prefix unsuitable for polite society. functional analysis was “fun,” complex analysis was “complex,” et cetera.
even the physicists – the token physics majors in my life sloughed through the differential equations just to savor Lie algebras.
after deciding to focus on machine learning, however, I realized that a lot of the relevant math is, well, continuous.
in particular, the majority of our group meetings the past year have focused on flavors of stochastic calculus. I like the first word. I don’t like the second. it evokes long equations followed by equal signs and more equations, the type of math you would LaTeX in an align environment, rather than English. it’s not that calculus (and analysis) lack elegance; it’s just that, copying long equations bored me, so I never took advanced classes in those areas, and now I’m bad at that sort of math because I don’t remember how to integrate. I was better at solving differential equations in high school, and to my detriment, despite the practical utility of this math, I don’t particularly want to remember… still, being bad at anything is a self-fulfilling prophecy, so I suppose bitter medicine is good for you?
earlier this year, I met up with a friend who ended up picking algebra and number theory for his PhD. I remarked that as undergrads, we never would have imagined that stochastic calculus might be useful. his blank, amused expression felt a little validating.