# Tropical arithmetic and Graph Theory

## Class

Article

## Department

Mathematics and Statistics

## Faculty Mentor

Dave Brown

## Presentation Type

Oral Presentation

## Abstract

Recall your linear algebra class and the "dot-product". The dot-product is the basis of matrix multiplication and a mechanism by which angles between vectors are analyzed. A graph (a vertex-edge graph, not a Cartesian- y = f(x)-graph) is used to analyze the nature of the dot-product in a nonstandard, but highly applicable setting, and, conversely, vectors are used to construct graphs (again the vertex-edge graphs). This unusual setting is made more interesting by changing the way addition and multiplication -- the two main operations of any useful number system -- are performed. Namely, we look at graphs whose vertices are vectors and whose edges are determined by the result of the dot-product of the vectors in what is called the "tropical semiring". In the tropical semiring (whose name is a nod to of one of the original developers of the concept who is from Brazil) addition is replaced with minimum, and multiplication is replaced with addition. For example 3 "+" 4 = 3, and 3 "x" 4 = 7. Perhaps this seems strange, but this number system has many uses in computer science and modeling discrete and continuous systems. Interestingly, the way classic things like linear algebra and algebraic geometry are done in this system has yet to be determined. Studying the tropical semiring is one of the frontiers of Mathematics. Attending this presentation will equip you with the ability to tell someone, anyone, about one of the frontiers of Mathematics and help dispel the myth that "Math is all done".

## Start Date

4-9-2015 11:00 AM

Tropical arithmetic and Graph Theory

Recall your linear algebra class and the "dot-product". The dot-product is the basis of matrix multiplication and a mechanism by which angles between vectors are analyzed. A graph (a vertex-edge graph, not a Cartesian- y = f(x)-graph) is used to analyze the nature of the dot-product in a nonstandard, but highly applicable setting, and, conversely, vectors are used to construct graphs (again the vertex-edge graphs). This unusual setting is made more interesting by changing the way addition and multiplication -- the two main operations of any useful number system -- are performed. Namely, we look at graphs whose vertices are vectors and whose edges are determined by the result of the dot-product of the vectors in what is called the "tropical semiring". In the tropical semiring (whose name is a nod to of one of the original developers of the concept who is from Brazil) addition is replaced with minimum, and multiplication is replaced with addition. For example 3 "+" 4 = 3, and 3 "x" 4 = 7. Perhaps this seems strange, but this number system has many uses in computer science and modeling discrete and continuous systems. Interestingly, the way classic things like linear algebra and algebraic geometry are done in this system has yet to be determined. Studying the tropical semiring is one of the frontiers of Mathematics. Attending this presentation will equip you with the ability to tell someone, anyone, about one of the frontiers of Mathematics and help dispel the myth that "Math is all done".