# Tropical Mathematics

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. 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. 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.

**Key Researchers**

Mark is a Professor and the Head of Pure Mathematics at the University of Manchester. The main focus of his research is the theory of semigroups and automata, however he spends much of my time working on connections with adjacent fields and application areas, such as combinatorial and geometric group theory, computational complexity, quantum computation, formal language theory, and most recently tropical algebra and geometry. Some of the major topics on which he has recently worked include Geometry of Finitely Generated Semigroups, Tropical Matrix Semigroups and Small Overlap Monoids.

Marianne is a Lecturer in the School of Mathematics, her research interests cover a broad range of topics in algebra and combinatorics including free Lie algebras, representation theory, semigroup theory, and tropical algebra and geometry. Some of Marianne's previous work includes the CICADA project with Professor Dave Broomhead and Professor Steve Furber and the EPSRC funded research project 'Multiplicative Structure of Tropical Matrix Algebra' with Professor Mark Kambites.