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Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts.

Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While...

This volume consists of surveys on topics to which Bernd Sturmfels has contributed over his mathematical career: invariant theory, Grobner bases, toric ideals and varieties, algebraic methods in discrete and convex optimization, hypergeometric systems, algebraic statistics, likelihood geometry, tropical geometry, chemical reaction networks, numerical methods in algebraic geometry, sums of squares, tropical geometry, tensors, and algebraic vision. Each article gives a gentle introduction to the topic. Many contributions summarize the state of the art in each subject. The volume is perfect for...

This volume contains the proceedings of the conference on tropical geometry and integrable systems, held July 3-8, 2011, at the University of Glasgow, United Kingdom. One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition,...