Before you do research

Not many students really think about researching as an undergraduate because graduate school is something of a mysterious land that they have yet to even imagine exploring. However, it’s something I have considered pursuing since matriculating into UCSB but the thought itself is very scary. That is why undergraduate research is a step in the right direction because it allows for you to experience a small portion of what could be if you do decide to head toward this direction. But before you start research, I want to talk about a misconception I had previously about research.

Not all research experiences are the same. I came in thinking that I would immediately be in the lab designing and testing circuits and working side by side with my mentor; instead, I had papers upon papers to read. The big mistake I made there was thinking that research was going to be easy. The material that doctoral students had to understand and digest cannot be understood in a few weeks by undergrads. The reality of the situation was that I needed a strong physics base knowledge in electromagnetic fields to start comprehending the design parameters of my project. But like I said before, not all research experiences are the same. There are research projects where the majority of the job is spent testing samples because the project is nearing completion or the bulk of the work lies in repetitive tasks. These usually require less technical knowledge but more lab skill. For example, a friend of mine was looking at plants under a microscope to sort them for a professor. The research value gained from that was mainly the networking and exposure to the lab atmosphere. Each experience is valuable in its own respect and you have to make the best of it.

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.

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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.

The M in STEM

I’m Robin Allison. I’m doing research this summer in the STEEM program here. My research is in a very interesting branch of math called Differential Topology. This subject roughly deals with studying an abstract notion of space where it is possible to do calculus. Most days I spend around campus doing the research and it has been an immensely rewarding experience. Moreover the program provides several career development workshops such as a writing course which has been great.

One of the things that struck me the most is the resemblance math takes with the empirical sciences. To many this may seem obvious (does it?) but being entrenched in math for the past two years made this easy to forget. The STEEM summer research this summer is being run alongside three other programs (EUREKA, INSTET-V, and MARC) which shares only one other math major. This means that most of my encounters with other undergraduate researches are in biology, chemistry, engineering, or interdisciplinary to name a few.

In science one roughly forms hypothesis, forms experiments and runs tests, and analyzes data and draws conclusions. Math is quite similar. In fact if one adopts the perspective of math  as being in its own plane of existence (or however the philosophers put it) then mathematicians are in fact doing the same things as scientists. The world of math just happens to be so simple that you are able to prove certain phenomena.

Math even has its analogue of machines, big powerful theorems that allow one to compute what many would have little idea to even define. For instance integral calculus allows one to compute volumes of shapes that the greatest minds couldn’t even imagine for centuries. Of course the list goes on and on and Differential Topology is full of such machines. My research might be described as making one of these machines better. Then others can use it to prove still deeper results.

One can also draw the parallel between gathering data in science and math. In math data, data is likely examples which show the trend of a subject. The mathematician Paul Halmos once said, “A good stack of examples, as large as possible is indispensable for a thorough understanding of any concept.” This rings true on every level. Examples are the matter of the mathematical realm. In math we get to cheat a little bit because our theories are retrofitted to describe a particular phenomena.

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.