Friday, June 01, 2007

Existence, Uniqueness, and Continuous Dependence on the Data

I was inspired to contribute to the Carnival of the Postdocs by Sciencewoman's theme. The theme is uniqueness, and the word immediately triggered what has become an automatic reflex for me: reciting the definition of a well-posed problem. A well-posed problem has a solution that
  1. Exists,
  2. Is unique, and
  3. Depends continuously on the data (meaning that a small perturbation in the input results in a small perturbation in the output).
A problem is ill-posed if its solution fails to meet one or more of those criteria. A problem with a non-existent solution is boring, so it's usually problems that don't meet the second and/or third criterion that people study.

One of the major themes of my dissertation was ill-posed problems. But it could just as easily be a theme of life, not just for me, but for everybody. Do our lives depend continuously on the data, or do small perturbations result in big changes?

They say that the beating of a butterfly's wing can lead to a hurricane, and likewise, the smallest things in our lives can make a big difference. I often think of one very pivotal moment in my life, a moment that at the time seemed so mundane, a moment that wasn't really even my life's moment, but that drastically changed the trajectory of my professional life.

It was a weekday afternoon, the same as any other. Jeff was at work, taking calls about the cable modem service, and solving people's problems over the phone. He helped a man who worked at the supercomputing center on campus, and the man thanked him profusely. They got to chatting a little bit, and the customer asked Jeff about himself, and Jeff told him that his wife was a grad student in computer science at the university. The man told Jeff about an opening for a Research Assistant position at the center, and told him to encourage his wife to apply.

I applied for that job, and I got it. I didn't realize it at the time, but it was the best job I've ever had. It opened so many doors for me! The experience I acquired by doing that job made it possible for me to lose my fear of parallel computing, enabling me to apply for and use supercomputing resources for my dissertation. Ultimately, it led me to the place I am right now. If I'd never heard of that job, I wouldn't be nearly as employable as I am today. If it weren't for that job, I wouldn't have had four interviews and two job offers straight out of grad school. If it weren't for that job, I might feel less optimistic about my future at this lab, where the focus is on leadership computing. So I'm really glad that my husband just happened to be working that afternoon and took that call. Who knows where I'd be, otherwise?

I tend to think that I would have just gone in a different direction, and things would have worked out well anyhow. But this little perturbation really influenced the outcome. Life is an ill-posed problem.

What pivotal event(s) changed your life?


Tony said...

Ah, the strength of weak connections.

Jane said...

Great story! It's funny how many pivotal events are a result of random conversations between people. I started on my current research project as a result of a random conversation with a colleague, in which the colleague realized that he knew someone who was interested in the same types of problems that I was interested in (and in fact was working on the same problem from a totally different angle), and introduced the two of us....and the rest is history!

Laura said...

"Life is an ill-posed problem."

Lovely. You could make an inspirational poster out of that saying. (Though only computer science people would get it...)

rachel said...

Seeing an ad in Student Life (the Wash U paper) that said "Your comic strip here!"

elswhere said...

Staying at a friend's place in Seattle on my way up to Alaska to can salmon for the summer in 1989. I thought Alaska would be the big adventure, and it was, but it was the stop on the way that changed the trajectory of my life.

Flicka Mawa said...

I like the conclusion sentence: Life is an ill-posed problem.