Birkhoff polytope

Birkhoff polytope. The Birkhoff polytope is a geometric object named after the American Mathematical Society president George David Birkhoff, who made significant contributions to dynamical systems, ergodic theory, and lattice theory. It is closely related to the work of Emmy Noether on abstract algebra and Hermann Minkowski on convex geometry. The study of the Birkhoff polytope has connections to the Institute for Advanced Study, where Albert Einstein and John von Neumann worked on mathematical physics and computer science.

Introduction

The Birkhoff polytope is a polytope that arises in the context of linear programming and combinatorial optimization, fields that have been influenced by the work of George Dantzig and Leonid Kantorovich. It is also related to the theory of matrices, which was developed by Arthur Cayley and James Joseph Sylvester. The Birkhoff polytope has been studied by mathematicians such as Claus Schnorr and Alexander Schrijver, who have worked on number theory and graph theory. The University of Cambridge and Massachusetts Institute of Technology have been centers of research on the Birkhoff polytope, with contributions from Andrew Wiles and Daniel Kleitman.

Definition and Properties

The Birkhoff polytope is defined as the convex hull of the set of doubly stochastic matrices, which are square matrices with non-negative entries and row sums and column sums equal to 1. This definition is related to the work of Andrey Markov on Markov chains and stochastic processes. The Birkhoff polytope has been studied using techniques from convex analysis, which was developed by Tibor Radó and Laurent Schwartz. The Birkhoff-von Neumann theorem states that the Birkhoff polytope is the convex hull of the set of permutation matrices, which are square matrices with exactly one 1 in each row and column. This theorem is related to the work of Richard Brauer and Helmut Wielandt on representation theory.

Geometric Structure

The geometric structure of the Birkhoff polytope is closely related to the geometry of polytopes, which has been studied by mathematicians such as H.S.M. Coxeter and Branko Grünbaum. The Birkhoff polytope is a convex polytope, which means that it is the convex hull of a finite set of points in Euclidean space. The facets of the Birkhoff polytope are related to the inequalities that define the polytope, which have been studied by mathematicians such as Victor Klee and Peter McMullen. The University of California, Berkeley and University of Oxford have been centers of research on the geometric structure of the Birkhoff polytope, with contributions from Grigori Perelman and Timothy Gowers.

Combinatorial Aspects

The combinatorial aspects of the Birkhoff polytope are closely related to the theory of combinatorial designs, which has been developed by mathematicians such as Ronald Fisher and Raj Chandra Bose. The Birkhoff polytope is related to the combinatorics of permutations, which has been studied by mathematicians such as William Rowan Hamilton and Camille Jordan. The symmetry group of the Birkhoff polytope is related to the symmetric group, which has been studied by mathematicians such as Évariste Galois and David Hilbert. The Institute of Mathematical Statistics and London School of Economics have been centers of research on the combinatorial aspects of the Birkhoff polytope, with contributions from Anders Hald and Frank Yates.

Applications and Related Concepts

The Birkhoff polytope has applications in operations research, computer science, and statistics, fields that have been influenced by the work of George Box and Norman Draper. It is related to the traveling salesman problem, which has been studied by mathematicians such as Karl Menger and Merrill Flood. The Birkhoff polytope is also related to the assignment problem, which has been studied by mathematicians such as Dénes Kőnig and Jenő Egerváry. The National Academy of Sciences and Royal Society have recognized the importance of the Birkhoff polytope, with awards such as the Fields Medal and Wolf Prize being given to mathematicians who have worked on related topics, such as Andrew Wiles and Grigori Perelman. The University of Chicago and California Institute of Technology have been centers of research on the applications and related concepts of the Birkhoff polytope, with contributions from Paul Erdős and Terence Tao. Category:Polytopes