Mathematical Induction (The Good Kind)

When I arrived at UCSB as a transfer student, my concerns mostly converged on one question: will I be able to do anything in my field ever? I psyched myself out well before arriving by looking ahead to the material I would inevitably learn and the work that professional mathematicians do. In short, I found myself daunted by the complexity that I faced. Our undergraduate years are fraught with episodes of imposter syndrome, even when it has been shown repeatedly that we arrive at such a stage in our lives on merit alone, and I am no exception to feeling this way.

Upon entering the INSET-V/STEEM program, I was suddenly on the frontline of math. There was no longer any question as to whether I could do anything with my skill set in my field; it was now required of me. So what exactly was required of me? I’ll boil it down to three categories.

First, doing research on these topics requires complete mathematical fluency. You can’t shy away from analysis arguments and bury your head in the sand when scary proofs come along. I have had to prepare myself by being comfortable with deltas and epsilons as well as being able to extract every wonderful detail from research papers and advanced textbooks. But that’s expected.

Second, studying dynamical systems requires an analytic eye for detail. Any mistake in my Matlab code has disastrous consequences for the validity of the subsequent mathematical work. Having to depend on semi-empirical evidence like the graphical output of a program seems like I could manufacture whatever results I want, but the end result comes from thousands of calculations performed in just five seconds. Needless to say, thank goodness for computers. And so, it’s necessary to pay attention to any errors in the output that arises from, say, a missing comma or coefficient in the code.

Lastly, being a computational mathematician requires a vast supply of coffee. A good strong home-brewed cup is best, but an iced latte in the afternoon serves as a crucial pick-me-up after scouring through the results of dozens of nearly identical programs. Should any land on earth prove to have an endless supply of coffee beans, I may have to move there permanently.


The ancient Romans worshiped the goddess of coffee, Java. (I am no authority on mythology)

Most importantly, though, I don’t need reassurance as an undergraduate student that my math tool set will someday be put to use, where “someday” approaches infinity. What I have done in the past weeks is the creation and betterment of mathematics, in theory and in practice. Real-world applications in this field exist in general, but my specific work might not be of any direct benefit for many years or it might directly apply right now, relevant for some climatologist in Norway or economist in New York. The STEEM branch program has been my mathematical induction.

When Life Imitates Research

The laboratory of a mathematician is an office, a mind, pen and paper. The components are so easy to carry with you, whether it’s while you’re eating dinner or riding your bike home. Some may think that it’s a sign of inherent weakness: how could anyone possibly work on their research project while pedaling away or eating a sandwich? I contend that it’s a sign that I picked the right field to get into. The essential truth value of my research isn’t confirmed in a physical place but rather in the rigor of my proofs and how clear I communicate these results mathematically. At least, that’s the idea.

Much like a software engineer, as I should know because my father was guilty of this, I find myself taking my work home with me. At any hour of the day or night, a realization washes over me and I take out a scrap of paper and write it down lest I forget it in the morning. I find that receipts are valuable in this respect, and food wrappers are not. It’s exceptionally fitting considering what my project actually concerns: the unexpected and the unpredictable. There is no way I could predict when the stubborn knot of knowledge gives way; I can only hope to work diligently at the task and be well prepared for enlightening moments.

Though all-consuming is somewhat of a harsh term, it’s a vaguely appropriate description all the same. Studying how systems and models decay into disarray and chaos is a mathematical challenge that demands conscious and subconscious attention . So, while I am usually found in a quiet space working at discovering more and more details during regular business hours, the work carries on well past that.

The work itself is fascinating: given a map that is iterated by composing it with itself endless times, it is my task to find the exact values of the parameter that make the map tick. That is, finding numbers for which the map behaves chaotically. Not much can be done once the dynamical system enters a region of chaos. A trajectory can get lost in an infinite complicated cycle around a sink or oscillate uncontrollably in an interval of values. The benefit of investigating this map lies in determining what values to avoid – hopefully ensuring that the system behaves somewhat predictably for a certain interval of time.

When I approached Dr. Birnir, my faculty advisor, I hardly gave this topic a thought and proposed working on something like modeling a quantum state or an arrangement of magnetic dipoles and a metal pendulum (that particular example is wild – I played with a simulator for a solid hour). However, he suggested studying chaos in a more controlled environment: a one-dimensional discrete dynamical system. Even this is enough work to make my calculator refuse to calculate polynomial zeros. Seriously, try plugging in a one-hundred-term polynomial equation in your calculator. That’s one hundred terms and about eighty terms too much for early 21st century technology to compute in a few minutes.

Now well into the analysis of the system, ideas on how to isolate the problematic values for the system come in periodic bursts. These values are the parameters that in real life models can cause a stock market simulator to sell all your healthy stock options or deliver a lethal electrical pulse to your heart. Chaos in the real world is far from friendly. Even on paper, I contend that chaos is still not a mathematician’s best friend.

Meanwhile, I think I’ll avoid oscillating uncontrollably in an interval in the office by getting some more coffee.