each cell of dimension after quotienting out the affine span of the cell. The property that V is connected in codimension one means for any two points lying on dimension d cells, there is a path connecting them that does not pass through any cells of dimension less than . Tropical curves The study of tropical curves (tropical varieties of dimension one) is particularly well developed and is strongly related to graph theory. For instance, the theory of divisors of tropical curves are related to chip-firing games on graphs associated to the tropical curves. Many classical theorems of algebraic geometry have counterparts in tropical geometry, including: Pappus's hexagon theorem. Bézout's theorem. The degree-genus formula. The Riemann–Roch theorem. The group law of the cubics. Oleg Viro used tropical curves to classify real curves of degree 7 in the plane up to isotopy. His method of patchworking gives a procedure to build a real curve of a given isotopy class from its tropical curve. Applications A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007. Yoshinori Shiozawa defined subtropical algebra as max-times or min-times semiring (instead of max-plus and min-plus). He found that Ricardian trade theory (international trade without input trade) can be interpreted as subtropical convex algebra. Tropical geometry has also been used for analyzing the complexity of feedforward neural networks with ReLU activation. Moreover, several optimization problems arising for instance in job scheduling, location analysis, transportation networks, decision making and discrete event dynamical systems can be formulated and solved in the framework of tropical geometry. A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design. The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry can show self-organized criticality. See also Tropical analysis Tropical compactification Notes References Maslov, Victor (1986). "New superposition principle for optimization problems", Séminaire sur les Équations aux Dérivées Partielles 1985/6, Centre de Mathématiques de l’École Polytechnique, Palaiseau, exposé 24. Maslov, Victor (1987). "Méthodes Opératorielles". Moscou, Mir, 707 p. (See Chapter 8, Théorie linéaire sur semi moduli, pp. 652–701). Further reading Tropical geometry and mirror symmetry External links Tropical Geometry, I