Wednesday, April 11, 2012

Matt Mizuhara ’12 writes on nearly-inner automorphisms and the mysteries of mathematics

Matt Mizuhara ‘12
Mathematics Major
Hometown:  Allentown, PA

What writing project(s) are you working on right now?

I have been spending the past academic year completing an honors thesis in the Mathematics Department.  I spent parts of the summer and the majority of the fall semester conducting research on a problem in an area called group theory.  A group is a fundamental object in mathematics that captures the essential properties of symmetry.  Groups arise naturally in the physical sciences: chemists study the symmetries of crystal structures and physicists verify conservation laws by observing symmetries of space-time.  Groups themselves contain symmetry in the form of special functions called automorphisms.  Many mathematicians have conducted research in order to understand the structure of these automorphisms in order to more fully understand the underlying groups themselves.  The project on which I have been focusing deals with a special class of automorphisms, called nearly-inner automorphisms, in order to understand how common an occurrence they are.

After discovering some interesting results, my thesis adviser and I have been writing a formal report to the Honors Council so as to share our work with the greater mathematical community.  This report is theoretically self-contained, requiring minimal formal mathematical knowledge, however quickly builds the requisite tools to state and prove our main result.

What do you love about it?

I love the feeling of conducting independent research, which, very fortunately, leads to new results.  It's an incredibly unique and rewarding sense of accomplishment.  This has been the most demanding and frustrating venture I have attempted in academia; however, the end result is something of which I am very proud.

What about it (if anything) is driving you nuts?

Accepted language and syntax is very distinct in mathematics, and is something I am still developing.  I owe a lot to my adviser, Professor Pete Brooksbank, for his patience and support. As the thesis reports were due April 1st, we had several busy weeks of intensive writing and editing.  My inability to succinctly and accurately describe certain aspects of our mathematics was quite frustrating.  However, Professor Brooksbank's advice and mentorship guided our project from imprecise writing to a properly formatted report.

How would you describe your writing process?

For me, it is most important to understand the organization of the mathematics.  Writing general ideas and broad schemes is most important so that I introduce all relevant information in the most succinct and natural manner possible.  Once a first draft is written, my edits serve several purposes.  I aim to streamline the exposition as well as rewrite the mathematics in the most "balanced" way possible.  It is important to find a balance between writing too many and too few details.  In the former case, the reader can be bogged down by trivialities which can cause him/her to lose track of the overall ideas.  In the latter case, the reader could be left confused and uninspired.

What kind of feedback on your writing do you find most helpful?

Generally, it is easy for me to lose track of those aspects that need explanation.  After a year of study, many concepts become natural to a researcher while they remain esoteric to even their most qualified peers. As such, it is often requisite to develop mathematical intuition and ideas slowly in one's writing.  The most useful feedback to me is when people simply explain at which points of a paper they become less comfortable with the information or they feel I have explained a concept too quickly (or perhaps too slowly).  In that sense, the most useful feedback is more global, rather than local.  The overarching flow and rhythm of my writing is more difficult to judge and write, while the specific details are much easier to modify independently.

What would you like your peers to know about you as a writer?

Mathematics is hardly as exotic and inaccessible as popular culture or
perceptions imply.  The same thought process which allows a computer scientist to write a program, a philosopher to conjecture a new idea, or a Spanish major to learn a new language is precisely that which allows mathematicians to discover new and exciting results.  Math research is as alive and varied as any other science, and the results are, in some instances, more groundbreaking and beautiful than any physical experiment or art form.  Problems can go unsolved for hundreds of years, only to be cracked by an enlightened new approach by a fresh mind, and once a result is proved, it will remain so forever.  As Hardy said, mathematical results are, in some sense, the closest form of immortality that we mortals can attain. Further, Times magazine once wrote of a mathematical novel that "to read it is to realize that there is a world of beauty and intellectual challenge that is denied to 99.9 percent of us who are not high-level mathematicians." In my mind, it is unfair that not everyone can share this same passion and understanding of mathematics. It takes some time to understand the language of mathematics, but I only ask that everyone at least devote some patience to unravel its mysteries and to give it a try before turning up their noses at a rewarding and beautiful art.

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