Roughly speaking, tropical mathematics is the study of the tropical semiring, which is the algebraic structure formed by the real numbers under the operations of addition and maximum. (There are many subtle variations, such as augmenting with plus and/or minus infinity, using the natural numbers, integers or rationals instead of the reals, or using minimum instead of maximum. Sometimes the term "tropical" is used even more generally, to cover a broad class of structures which share features with the tropical semiring as defined here.)
The tropical semiring shares many of the properties of a field, with addition and maximum playing the roles of field multiplication and field addition respectively. It differs from a field because the maximum (field addition) operation lacks inverses. Instead, one gains idempotency of addition (since max(a,a) = a), which leads to a radically different, but still rich, algebraic and geometric structure.
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, being chiefly 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 and metric geometry.
Tropical methods have applications in a wide range of areas, including combinatorial optimisation and scheduling, microprocessor design, biochemistry, and statistics to name but a few. Many (although not all) of the applications stem from a simple observation: if two activities must be performedconsecutively 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 complex systems of dependencies among them. Mathematically, these lead to complex systems of equations involving sums and maximums. From a classical point of view these systems are highly non-linear, and hence difficult to analyse, but working over the tropical semiring they become linear, and hence amenable to representation and analysis using matrix techniques.