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David Futer, who joined the Mathematics Department in 2008, studies knot theory and other areas of geometric topology. Mathematical knots differ from many physical knots in that they are closed loops; they cannot be unraveled without breaking their continuity. Knotted loops of this type can be found in DNA molecules and the magnetic fields of stars. Futer's research develops methods for characterizing knots without actually going through the process of attempting to unravel them.
Knot theory is one aspect of geometric topology, the study of manifolds. Local areas on manifolds follow the rules of traditional Euclidean geometry but cannot be described by Euclidean geometry on larger scales. For instance, the surface of the earth is a curved manifold. For small regions of the earth, we can draw accurate maps on a flat piece of paper. But on a larger scale, an airplane traveling directly between three different distant cities will trace a triangle whose interior angles add up to more than 180 degrees. Euclidean geometry cannot account for triangles of this sort.
In Euclidean geometry, the parallel postulate states that for any line continuing indefinitely in a two-dimensional plane, there is only one line that can be drawn through a point not on the line that will not intersect it — the line exactly parallel to the original line. Non- Euclidean geometries, in contrast, are self-consistent geometric systems based on different versions of the parallel postulate. In hyperbolic geometry, for instance, there are infinitely many lines that can be drawn through a point not on the line that will not intersect the given line.
Using non-Euclidean geometry, Futer is developing methods that ignore incidental twists and crossings in certain classes of immensely complicated knots, allowing him to reduce them to their essential knotted nature with a single mathematical operation. His work is uncovering new relationships between the volume of the space surrounding knots and their other invariants. He has recently published articles in the Journal of Differential Geometry, Mathematical Research Letters and the Proceedings of the London Mathematical Society.
Along with Professor Igor Rivin, an internationally-known topologist whose work Futer has used in his own research in the past, Futer forms the core of the Mathematics Department's growing Geometry Research Group.
Futer earned his BA in Mathematics and Philosophy and his MA in Mathematics from the University of Pennsylvania in 1999 and his PhD in Mathematics at Stanford University in 2005. He was a NSF Research Training Group Postdoctoral Instructor at Michigan State University from 2005 to 2008.
(posted July 2009)