I primarily study the kinematics of robots and mechanisms. This field is concerned with collections of rigid bodies with geometric constraints between them. Examples of such constraints are rotational hinges, prismatic (linearly sliding) joints, and spherical (ball-and-socket) joints. Mathematical models based on ideal joints and rigid bodies closely approximate the motion of many practical devices, ranging from steering and suspension systems on vehicles to multi-limbed robots.
The most common geometric constraints involve entities that are algebraic (points, lines, planes, cylinders, and spheres), and since squared distances are algebraic also, the mathematical models are algebraic. Thus, the questions to be answered fall within the domain of algebraic geometry. In the late 19th century and early 20th century, these questions were actively pursued in mathematical circles, and such well-knowns as Cayley, Chebychev, Kempe, Schönflies, Study, and Sylvester made significant contributions. Subsequently, the main thread of algebraic geometry moved to a higher level of abstraction and the study of kinematics became mainly the province of engineers. Cross-fertilization between the fields resumed in the late 20th century as fast computers and the bloom of robotics inspired engineers to ask new and difficult questions that have once again drawn the attention of applied mathematicians.
To answer questions from kinematics, as well as algebraic questions from other disciplines such as chemistry and computer graphics, one needs to describe and manipulate the solution sets of systems of polynomial equations. One of several computational techniques for addressing these systems is polynomial continuation. In 1995, Andrew Sommese (Notre Dame) and I coined the term numerical algebraic geometry to describe a new class of algorithms to deal with positive-dimensional solution sets, built on top of existing techniques of polynomial continuation for finding isolated solutions. Past work with Andrew and Jan Verschelde (UIC) includes algorithms to compute irreducible decompositions, membership tests, and the intersection of algebraic varieties. Since about 2003, I have been working with Andrew Sommese, Daniel Bates, and Jon Hauenstein on extensions to these methods and on the software package, Bertini. This includes regeneration methods to solve systems equation by equation, which is particularly advantageous for solving large, sparse systems. Bertini is now open source, and a book on its use will appear in Fall 2013.Charles Wampler homepage.