Complex analysis proved to be a convenient language in Quantum Field Theory, Superstring Theory, and in other fields of modern mathematical physics. Research problems in contemporary study of thermodynamic and stochastic processes, and problems of bioinformatics lead to the development of methods of a new – tropical – arithmetic. In this arithmetic the sum of two numbers is the maximum of them (just as an ocean with a drop added will still be an ocean), and the product of two numbers equals their usual sum (in the same way as we multiply numbers in logarithmic scale). The latest studies demonstrate that the theory of algebraic sets in tropical setting is closely related to projections of complex algebraic sets to modular and angular components of their complex coordinates. The images of such projections are called the amoeba and coamoeba of a complex algebraic set.
We develop a new approach in the theory of distribution of complex analytic sets using the language of amoebas. In this language, we can naturally distinguish between two extreme classes of algebraic sets, the Harnack and discriminantal. Our recent result shows that cuspidal strata of a classical discriminant are birationally equivalent to certain A-discriminants. Amoebas of such strata are minimal surfaces, and the maximum likelihood degree of each strata is equal to one. This implies that they can be effective models of algebraic statistics in bioinformatics.