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.