Carl W. Lee
Department of Mathematics
University of Kentucky
Lexington, KY 40506-0027 USA
phone +1 (859) 257-1405
FAX +1 (859) 257-4078
My research tends to center around convex polyhedra. These are
sets that can be described as intersections of finite numbers of closed
halfspaces in Euclidean space. A bounded polyhedron is a convex
polytope; equivalently, a convex polytope is the smallest convex set
containing a given finite set of points.
My interest in polyhedra is motivated by:
- Their intrinsically interesting properties, such as
symmetry (e.g., the Platonic and Archimedean solids, and regular
polytopes in higher dimensions), combinatorial properties (e.g.,
Euler's relation V-E+F=2 for
three-dimensional polytopes and extensions and characterizations of
face numbers in higher dimensions, paths in the graph determined by a
polytope's edges), and metrical properties (e.g., formulas for volume,
numbers of interior lattice points, rigidity, triangulations).
- The tools from other areas of mathematics that can be brought to
bear on problems on polyhedra, such as commutative algebra, algebraic
geometry, enumerative combinatorics, and linear algebra. Some very
striking combinatorial results, for example, so far admit no purely
combinatorial proof, but require an algebraic perspective.
- The applications of polyhedra to other areas of mathematics, such
as linear programming, combinatorial optimization, computational
geometry, algebraic geometry, and convexity. For example, in
combinatorial optimization one often wishes to optimize a linear
functional over a finite (often large) set of points. Frequently one can
approach such problems by taking the convex hull of these points,
considering an alternate representation of this polytope as an intersection of
halfspaces, and exploiting its combinatorial/geometric structure to
develop efficient algorithms.
My graduate students are presently working on projects involving
stress and toric h-vectors of convex polytopes; triangulations of convex
polytopes; and three-dimensional aperiodic tilings.
Some recent publications:
- C.W. Lee, P.L.-spheres, convex polytopes, and stress,
Discrete and Computational Geometry 15 (1996) 389-421.
- C.W. Lee, Subdivisions and triangulations of polytopes, Chapter 14
in Handbook of Discrete and Computational Geometry, J.E. Goodman and J.
O'Rourke, eds., CRC Press LLC, Boca Raton, 1997, pp. 271-290.