Tropical linear algebra and its applications
Alexander Guterman

This is a joint work with Marianne Akian and Stephane Gaubert.
Tropical algebra (sometimes called  max algebra) is a set  of real numbers
equipped with the maximum operation instead of usual addition and addition
instead of usual multiplication. Under these operations this is an
algebraic structure called a semiring. Such structures naturally appear in
modern scheduling theory, control theory, optimization, dynamical
systems and networks. Tropical arithmetic allows to reduce difficult
non-linear problems to the linear problems but over tropical semiring.
Therefore, to investigate these problems it is necessary to develop linear
algebra in the tropical case. This subject is very actual nowdays. We plan
to introduce and investigate tropical linear algebra and to connect it with
several applications, in particular, to the game theory.