''Introduction to Tropical Geometry'' is a book on
tropical geometry, by
Diane Maclagan and
Bernd Sturmfels. It was published by the
American Mathematical Society in 2015 as volume 161 of
Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats.
List of books
*1 ''The General To ...
.
Topics
The
tropical semiring is an algebraic structure on the
real numbers in which addition takes the usual place of multiplication, and minimization takes the usual place of addition. This combination of the two operations of addition and minimization comes up naturally, for instance, in the
shortest path problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The problem of finding the shortest path between tw ...
, where concatenating paths causes their distances to be added and where the shortest of two parallel paths is the one with minimum length, and where some shortest path algorithms can be interpreted as tropical
matrix multiplication. Tropical geometry applies the machinery of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
to this system by defining
polynomials using addition and minimization in place of multiplication and addition (yielding
piecewise linear functions), and studying the "roots" of these polynomials, the breakpoints where they fail to be linear. The field is named after the Brazilian adopted home of one of its pioneering researchers,
Imre Simon. Although past work in the area has studied it through methods of
enumerative combinatorics, this book instead is centered around explicit calculations related to the tropicalization of classical varieties. Although it is much more comprehensive than the two previous introductory books in this area by Itenberg et al.,
some topics in tropical geometry are (deliberately) omitted, including
enumerative geometry and
mirror symmetry.
The book has six chapters. Its first introduces the subject and gives an overview of some important result, after which the second chapter provides background material on
non-Archimedean ordered field,
algebraic varieties,
convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
s, and
Gröbner bases. Chapter three concerns tropical varieties, defined in several different ways, correspondences between classical varieties and their tropicalizations, the "Fundamental Theorem of Tropical Geometry" proving that these definitions are equivalent, and tropical
intersection theory. Chapter four studies tropical connections to the
Grassmannian,
neighbor joining in the space of
metric trees, and
matroids. chapter five considers tropical analogues of some of the important concepts in
linear algebra, and chapter six connects tropical varieties to
toric varieties and polyhedral geometry.
Audience and reception
This book is written as a textbook, with problems testing readers' understanding of the material. Reviewer Patrick Popescu-Pampu claims that even though it is a graduate-level book series, undergraduates with a sufficient background in algebraic geometry should be able to access it. Reviewer Felipe Zaldivar writes that it "makes the subject accessible and enjoyable" and makes "a beautiful addition" to its book series. Reviewer Michael Joswig concludes that ''Introduction to Tropical Geometry'' "will become a standard reference in the field for years to come".
References
{{reflist, refs=
[{{cite journal, title=Review of ''Introduction to Tropical Geometry'', first=Felipe, last=Zaldivar, date=August 2015, journal=MAA Reviews, url=https://www.maa.org/press/maa-reviews/introduction-to-tropical-geometry]
[{{cite journal, title=Review of ''Introduction to Tropical Geometry'', first=Patrick, last=Popescu-Pampu, journal=Mathematical Reviews, mr=3287221]
[{{cite journal, last=Joswig, first=Michael, date=February 2016, doi=10.1365/s13291-016-0133-6, issue=3, journal=Jahresbericht der Deutschen Mathematiker-Vereinigung, pages=233–237, title=Review of ''Introduction to Tropical Geometry'', url=https://page.math.tu-berlin.de/~joswig/publications/Joswig_Maclagan-Sturmfels.pdf, volume=118]
[{{cite journal, last=Draisma, first=Jan, title=Review of ''Introduction to Tropical Geometry'', year=2017, journal=Nieuw Archief voor Wiskunde, series=5th ser., volume=18, issue=2, pages=145–146, url=http://www.nieuwarchief.nl/serie5/pdf/naw5-2017-18-2-144.pdf, language=Dutch]
External links
* Elizabeth Kelley, 2020
The Fundamental Theorem of Tropical Geometry Tutorial introduction to the statement of the theorem following Maclagan and Sturmfels' treatment, with a focus on
commutative algebra.
Tropical geometry
Mathematics books
2015 non-fiction books
Publications of the American Mathematical Society