In an earlier post with the same title (and without the subtitle) I introduced some thoughts that were triggered by this simple problem:

Suppose that

                         (1)

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. 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. This resulted in a paper we wrote together: A Singular Integral Measure for and Boundaries that can be found here.

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 challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables), pushed through courses in advanced analysis, and geometric measure theory, and worked on applications in image analysis (generating papers he actually led) and finished 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.)

A major point of both the original post on the problem and this present post, is that the paper with Laramie, as well as the results in the first post, flowed from taking time to think about a simple analysis problem that would usually be viewed as a not-too-hard exercise, not worthy of more thought than it takes to find one solution.

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.