Details about Timothy D. Comar's Current Interests
Mathematical Biology
I am interested in the applications of mathematics to biological problems. Mathematics, computation, and quantitative analysis have become increasingly important in many areas of biological research including disease modeling, epidemilogy, drug design, ecology, molecular and cellular biology, and physiology. This is a great area for student research projects! The new interdisciplinary area of bioinformatics involves biology, mathematics, computer science, and statistics. From an educational viewpoint, I am committed to introducing and training undergraduate students in the biological sciences with appropriate and relevant mathematical and computuational skills so that they can succeed in future endeavors in modern, quantitative biological research. I am currently heading a team of mathematicians and biologists at Benedictine University and the College of DuPage that is writing a new biocalculus (calculus for the life sciences) textbook and lab manual, teaching introductory biomathematics courses (including biocalculus), and organizing activities for students to become aware of the intersection between mathematics and biology and learn about interdisciplinary research opportunities.
The biocalculus project is now funded by NSF CCLI Grant #0633232, "Biocalculus: Text Development, Dialog, and Assessment." The project summary for the grant proposal is located here.
I am co-editing a proposed volume in the MAA Note Series entitled "Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions" along with Glenn Ledder and Jenna Carpenter. This solication for the annoucement can be found here and the detailed instructions for submission can be found here.
Knot Theory
A knot is a tangled up circle in space that does not intersect itself. Topologists are interested in determining properties of knots that do not change as the knot is moved about in space. Knot theory is a part of low dimensional topology. (Topologists are concerned with properties of an object that do not change when the object is deformed without tearing.) Knot theory and the closely linked area of topological graph theory have many applications in biology, chemistry, and physics. I am most interested in questions that can be approached combinatorially using "hands-on" techniques, particularly questions about using geometrical and topological techniques knot theory and topological graph theory to model molecules. I am currently working on "stick number" problems. That is, I am trying to determine the minimal number of sticks required to build a knot in space under a list of geometric restrictions.This is a great area for students research projects! Stop by my office to discuss or pick up a list of references.
Hyperbolic Geometry and Kleinian Groups
Hyperbolic Geometry is a non-Euclidean geometry. That is, it is very much like the Euclidean geometry studied in high school, but there is one very important catch. Euclidean geometry postulates that there is exactly one line M that can be drawn through a point P that does not lie on a given line L such that the line M does not intersect L. This concept is known as the Parallel Postulate. In hyperbolic geometry, this postulate is modified so that there are more than one such non-intersecting lines. In fact, there are infinitely many such non-intersecting lines! (Analogously on a sphere, there are no such non-intersecting lines.) This minor change gives rise to a most beautiful form of geometry. It turns out that most two-dimensional surfaces naturally have hyperbolic geometry living on them. (The sphere has spherical geometry, and the torus (surface of a doughnut) has Euclidean geometry.)
I am most interested in studying three-dimensional hyperbolic manifolds (three dimensional objects with hyperbolic geometry) and groups of hyperbolic rigid motions (isometries) that give rise these manifolds, which are called Kleinian groups. My research is concerned with how these manifolds can be deformed. On a much more basic level, I am interested in making this material more accessible to undergraduate students by developoing activities to use basic analytic geometry and computer algebra systems to illustrate the behavior of these rigid motions also known as Mobius transformations.
Educational issues
I am very interested in developing excellent future mathematics teachers. I currently teach the mathemematics for elementary teachers course and am always looking for ways to make the mathematics content for future teachers enjoyable and applicable to their future careers.
I am currently developing a biocalculus textbook and lab manual as described above.
I am also interested in developing new activities, projects, and approaches for all of the classes I teach. I am also interested in developing new avenues for undergraduate mathematics courses or projects particularly in the general areas of geometry and topology. If you have interest in these areas, please let me know.