Integrals of Polynomials over Convex Polytopes: Combinatorics and Algorithms

The volumes and integrals of over polyhedra are perhaps the most fundamental and useful basic concepts in the history of mathematics. Integrals over polyhedral regions find applications from algebraic geometry to computer graphics, from combinatorics to probability and statistics.

But, how does one go about actually computing an integral over a convex polytope if one cares to compute the number exactly? In this talk we survey why exact integral computation is relevant, why calculus techniques fail miserably for the goal of computation, and end with the latest results about efficient computation of integrals of polynomials over convex polytopes. If time allows I will demonstrate our new Software LattE Integrale.

New results are joint work with: V. Baldoni, N. Berline, B. Dutra, M. Koeppe, M. Vergne.

Speaker Bio: Jesus De Loera received his B.S. degree in Mathematics from the National University of Mexico in 1989, a M.A. in Mathematics from Western Michigan in 1990, and his Ph.D in Applied Mathematics from Cornell University in 1995. An expert in the field of Discrete Mathematics, his work approaches difficult computational problems in Applied Combinatorics and Optimization using tools from Algebra and Convex Geometry. He has held visiting positions at the University of Minnesota, the Swiss Federal Technology Institute (ETH Zurich), the Mathematical Science Institute at Berkeley (MSRI), Universitat Magdeburg (Germany), and the Institute for Pure and Applied Mathematics at UCLA (IPAM). He arrived at UC Davis in 1999, where he is now a professor of Mathematics as well as a member of the Graduate groups in Computer Science and Applied Mathematics. His research has been recognized by an Alexander von Humboldt Fellowship, the 2010 INFORMS computer society prize, and a John von Neumann professorship at the Technical University of Munich. He has received over three million dollars in national and international grants. He is associate editor of the journals SIAM Journal of Discrete Mathematics and Discrete Optimization. For his dedication to outstanding mentoring and teaching he received the 2003 UC Davis Chancellor's fellow award, the 2006 UC Davis award for diversity, and the 2007 Award for excellence in Service to Graduate students by the UC Davis graduate student association. He has supervised seven Ph.D students, five postdocs, and over 20 undergraduate theses.

Presentation Slides: PDF

Note: Visitors from outside NIST must contact Cathy Graham; (301) 975-3668; at least 24 hours in advance.