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Department of Mathematics

# Tropical mathematics

Tropical mathematics is an emerging field of algebra and geometry with applications in many areas

## Heilbronn Fellows

Tropical mathematics is the study of the tropical semifield, which is the algebraic structure formed by the real numbers under the operations of addition and maximum. (There are many subtle variations, such as including plus and/or minus infinity, using the natural numbers instead of the reals, or using minimum instead of maximum.)

The tropical semifield shares many of the properties of a field, with addition and maximum playing the roles of field multiplication and field addition respectively. It differs in that the maximum (field addition) operation lacks inverses; instead, the addition is idempotent (since max(a,a) = a) which leads to a radically different, but still rich, algebra and geometry.

Pure tropical mathematics divides broadly into two areas, although with substantial overlap. Tropical algebra (also known as max algebra or max-plus algebra) is the tropical analogue of linear algebra, and is largely concerned with matrices over the tropical semiring. Tropical geometry studies the (algebraic and/or convex) geometry of spaces over the tropical semiring, and what they tell us about classical algebraic geometry and related areas.

Tropical methods also have applications in a wide range of areas, including combinatorial optimisation and scheduling, microprocessor design, biochemistry, and statistics to name but a few. Many applications stem from a simple observation: if two activities must be performed consecutively then the time required to complete both is the sum of the individual times, but if they may be performed concurrently then the time required is the maximum of the individual times. Many real-world situations (e.g. railway timetables or microprocessors) involve large numbers of activities with systems of dependencies between them. Mathematically, these lead to complex systems of equations involving sums and maximums. These systems are highly non-linear, and hence difficult to analyse using classical mathematics, but working over the tropical semiring they become linear, and hence amenable to representation and analysis using matrix techniques.

Research in Manchester spans many aspects of the subject. Much of our pure research centres on the theory of modules and polytopes over the tropical semifield, and the abstract algebraic (semigroup-theoretic and group-theoretic) structure of tropical matrices under multiplication. Our applied research has so far encompassed applications in biochemistry and microprocessor design, the development of methods for understanding stochastic max-plus systems, and applications of tropical methods in numerical linear algebra.

For further information on current projects please see the websites of individual staff listed on this page.