In an earlier post with the same title (and without the subtitle) I introduced some thoughts that were triggered by this simple problem:
for all , that is continuous and differentiable, and that .
Prove that everywhere.
In that post (which you can find here Fun with simple analysis problems I ), I started by presenting three solutions and then generalized and explored further.
What I did not reveal in that post was that writing it gave me an idea for a more advanced problem that could be perhaps pushed a ways. Not too long afterwards, Laramie Paxton joined my group and I gave him this problem to work on for his dissertation. We collaborated in solving the problem, since that is how I mentor all my students — their dissertations are collaborations with me — and this resulted in a paper we wrote together: A Singular Integral Measure for and Boundaries that can be found here . The paper explores that idea that was suggested by thinking about the original post I wrote in response to that simple analysis problem that started everything.
Laramie Paxton arrived at WSU quite naive with respect to analysis, having completed an online masters in mathematics that did not give him a good foundation in analysis. But he very quickly he adopted habits that led to rapid progress. He started by studying intensely the summer before arriving and passing the qualifying exam on his first try. Then he took my admittedly challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables) and courses in advanced analysis, geometric measure theory, applications in image analysis (papers he led) and his dissertation, all in the space of two years. After a year of postdoc, he landed the job he is about to start, at Marian University in Wisconsin. I believe that both the University and Laramie are lucky to have each other.
In general, I believe that small universities are good places to be nowadays, but from everything I hear, this place is better than good — it is perfect for Laramie’s talents and skills. (In addition to his impressively growing mathematical skills, he was already phenomenally skilled in logistics and organization which can be seen in his highly effective help in making the events listed here, from April 2017 to July 2018, a reality.)
While I am sure that there are other undiscovered aspects of the problem that launched these two posts and Laramie’s dissertation problem, I believe that what has been explored illustrates why it makes sense to treat simple problems as invitations to playful exploration and creativity.