For the Love of Math
On a recent return flight from a math conference with nearly 7,000 attendees, I overheard the usual polite chatter amongst strangers forced into close proximity for a cross-country trek. As this conference had only ended late the day before, there was a larger than usual number of mathematically inclined individuals around, and I heard the familiar questions posed once someone encounters a mathematician: “What can you do in math?” and “Don’t we already know everything in math?” One can easily imagine that there is more to discover in biology, chemistry, medicine and the like as results of studies from those fields regularly make their way into the popular media, but many have a difficult time believing the same is true in math. In fact, the online database of reviews of mathematical books and peer-refereed articles, MathSciNet, catalogs more than 100,000 additions per year.
Researchers in some disciplines can draw on familiar or widespread terminology, history or science to illuminate some basic ideas of research, giving us a taste of their work, but leaving us knowing that there is still great depth and detail beyond our reach. In mathematics, the language and material introduced in upper-division undergraduate math courses only hint at the basic building blocks of research, complicating a mathematician’s ability to convey his or her work. Nonetheless, there are similarities across disciplines as the nature of mathematics research also requires invested amounts of large blocks of time to investigate and pose questions, to sift through and learn other’s work, and to discuss ideas with fellow researchers.
The advancement of mathematical knowledge involves such things as extending or generalizing previously known theories, making previously unknown connections between known results, constructing or characterizing objects, and answering open questions. These are accomplished by producing new mathematics through reason, logic, proof and rigor. Furthermore, a necessary component for a math research publication is that this rigorous argument and development must be deemed significant enough by peer referees to advance the field.
While there are many exciting areas of mathematics research that are more immediately applicable to medicine, industry and government, there are just as many amazing and valuable avenues of pure mathematics research that advance our knowledge. Moreover, there is a great deal of give and take between pure and applied mathematics, making each useful to the other. My research as a pure mathematician is in the field of complex analysis with a specialization in geometric function theory. My motivation to understand and discover more mathematics is grounded in the search for knowledge itself rather than being driven by an application, a perplexing thought for my parents wondering what I was doing all those years in graduate school and how I could make a living from it!
Since I study the geometric properties of mappings, I have the advantage of being able to use mathematical software and graphics programs to explore conjectures and pose new questions, a luxury not available to many other mathematicians. However, the time comes to turn off the computer and pick up a pencil to pursue a rigorous argument. This process often requires creativity and can sometimes be unsuccessful no matter how many examples and graphs seem to indicate a conjecture’s truthfulness. For example, in 1958, at the conclusion of a groundbreaking paper in my research niche, the two authors posed an open question, now known as the Pólya-Schoenberg Conjecture. Though many outstanding mathematicians worked on this conjecture, making progress on special cases, it was not until 2003, through an imaginative and unexpected approach using a differential equation, that this conjecture was proved. In 2007, I published a paper on an extension of these results. One of the hypotheses of my main theorem involves a geometric condition that I believe will be true in a more general setting. While I have yet to produce an example that fails to support my belief, the general result remains elusive and is an ongoing work in progress. In 2008-2009, I mentored an honors student who investigated specific changes to the differential equation and the geometry of the resulting solution graphs. The work this student conducted in her honors thesis just to understand the problem goes well beyond what our typical undergraduate majors learn, and she produced an additional example related to this research problem in support of my more general hypothesis.
Surfaces from Parking Ramps to Chips
Recently, my work has led me into interesting connections between two areas that might seem disparate: minimal surfaces and planar harmonic mappings. Surfaces are two-dimensional objects in a three-dimensional space. For example, the boundaries of a ball, an empty paper towel roll, or a donut are examples of the surfaces of a sphere, a cylinder and a torus, respectively. Just as a line segment gives the minimum length amongst all curves connecting two points in a plane, loosely, a minimal surface is a surface with smallest area amongst all surfaces bounded by a given frame. In fact, dipping a wire frame into a soap solution naturally forms a soap film that is a minimal surface. There are also examples of minimal surfaces in everyday life. A spiral parking ramp is an example of a helicoid, and a Pringle™ potato crisp looks like a portion of Enneper’s surface.
Minimal surfaces are investigated from many different approaches in varying areas of mathematics. One way is through planar harmonic mappings, which I also study. Loosely speaking, planar harmonic mappings are formulas with certain mathematical conditions that transform or assign a region in a plane onto another region in a plane. Through a process called “lifting,” minimal surfaces are generated from some planar harmonic mappings where the geometry of the harmonic mapping and the surface are related. While this lifting process has been known since the 1860s, it does not always lead to a clear-cut or usable formula for the surface; thus understanding the harmonic mappings from which the surfaces are constructed has been important to the development of minimal surface results. Indeed, it is not always clear whether the surface produced is a portion or version of a known surface or a new one. Recently, a collaborator and I have provided additional identifications between the mappings and surfaces. Therefore, this work also may possibly allow us to characterize previously unknown surfaces in this manner.
Overall, mathematics research is a dynamic and creative journey often with great moments of frustration and reward. There is great beauty and breadth in mathematics