FAYETTEVILLE, Ark. — The movement of a robot’s arm, the pattern of spilt milk on a table and the act of parallel parking share something in common-they can all be described by mathematical equations. Mathematicians use these equations to explain why things happen and why they don’t happen-for instance, how it is possible to parallel park a car even if the car can’t be shoved sideways.

University of Arkansas mathematics professor Luca Capogna studies equations like these, focusing on geometric models in which there are "forbidden" parameters-directions in which a robot arm or a car cannot move. His research has won the attention of the National Science Foundation, which recently awarded him a Career Award for $355,000 dollars.

The equations Capogna studies interest engineers, physicists and chemists, who use them to model real-life situations. An example might be looking at how a substance spreads: It may take the shape of a sphere at first, but impurities and irregularities can cause changes in its geometry-and also change the equation that describes its movement.

The equations can be used to reflect changes in a situation and also to mirror its constraints. Scientists use computers to calculate complex equations, but computers have their own constraints when it comes to solving these equations. Capogna looks at ways to show that the solutions to certain differential equations are as "regular" as possible. For instance, the solution could be continuous, or its derivatives might be continuous. Once one knows that a certain equation yields "regular" solutions, then it becomes feasible to use computers and numerical methods to explicitly evaluate such solutions.

In addition to furthering his research, the funds from the career award will provide stipends for graduate and undergraduate research positions, allowing students to study the same problems that Capogna examines in his research and work under his supervision on original research projects.

Capogna has been working with groups of undergraduate and graduate students, having them work on research projects that require them to learn something new and tackle unsolved problems. Tackling equations together works for both groups of students, Capogna said. The undergraduate students benefit from the stimulation of working with graduate students on research projects. Graduate students have to explain complex subjects to the undergraduates and test their knowledge.

"There are many very talented undergraduate students here who could be involved in this kind of work," he said.

NSF program title: Integration of Research and Education in the Study of Analysis and Partial Differential Equations in Carnot-Caratheodory Spaces.

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How to model an equilibrium status (e.g. temperature or density of a liquid) in the “Parallel Parking Geometry”?

Let us denote by x and y the usual coordinates of a point in the plane. The “Parallel Parking Geometry” (better known as Grushin Plane) describes a metric structure in the plane, in which an hypothetical particle standing at a point (x, y) can only move away from this point along the directions (1, 0) and (0, x). This geometry provides a simple setting to describe motion under constraints. In particular, it can be used to model real-life situations as different as parallel parking (a car standing on the y . axis cannot move vertically) or the motion of a robot arm.

To give an example of how the geometry of the space is re ected in the equations which model the physical system, we consider the motion of a liquid particle or the diffusion of temperature in the plane equipped with the “Parallel Parking Geometry”. Denote by u(x, y) a function which represents the temperature (or
the density of a liquid) in its equilibrium status. A simple computation shows that such function can be obtained as a solution to the equation

Notice that on the y axis one has x = 0, so the second term in the sum drops and there is no more information about the vertical (along y) direction. This is an indication that the vertical direction is ”forbidden”.

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