[triangulations book] [AGTO book]
In joint work with Joerg Rambau and Francisco Santos, I wrote a book on Triangulations, Triangulations: Structures for Algorithms and Applications which is volume 25 of the series Algorithms and Computation in Mathematics of Springer Verlag. It appeared in 2010 (539p. 496 illus., 281 illus. in color!). We think it contains most of the key results related to the theory of secondary polytopes, regular triangulations, flips, and related subjects. Triangulations of polyhedra appear in combinatorics, commutative algebra, optimization, and other subjects.

Jointly with Raymond Hemmecke and Matthias Koeppe, I wrote a book on the emerging theory of algebraic techniques in optimization. The title is Algebraic and Geometric Ideas in the Theory of Discrete Optimization and was published as volume 14 of the SIAM and Mathematical Optimization Society joint Series on Optimization. We hope to promote these algebraic ideas to more optimization workers. The book shows several applications of algebra and geometry to problems with non-linear constraints and hopefully demonstrates the need to use algebraic ideas in the transition from linear discrete optimization to nonlinear discrete optimization.

Both books are available as E-books, and often are freely available like that to all your students if your library subscribes to Springer and SIAM journals.



  • Here is a link to our popular software LattE for counting integer lattice points in convex polytopes and computing Ehrhart polynomials. In its latest version, you can also compute integrals of polynomials over a rational polytope (exact), in particular it can do volumes.

  • The program universalbuilder, written together with Samuel Peterson, is a C++ program that generates CPLEX readable integer programs to compute minimal and maximal triangulations of convex polytopes of arbitrary dimension. It can also compute chirotopes of point configurations.

  • The program PUNTOS allowed you to compute regular triangulations of point sets. It also has useful subroutines for checking regularity and computing the GKZ coordinates of the secondary polytope. BUT you should REALLY use TOPCOM instead! This is here now for pure historic reasons