Degenerating Geometry to Combinatorics : Research Proposal for

A central appeal of algebraic geometry is that its functions are polynomials which can be written down as finite expressions. This makes the theory naturally combinatorial, in that polynomials are made up of a finite number of terms which interact through discrete rules. In practice, however, the operations performed on these polynomials are usually too complex for combinatorial methods to be of use. There are a number of strategies in modern algebraic geometry which can be seen as trying to simplify the theory enough to make combinatorial methods useful. In this paper, I describe three projects I am working on and will continue pursuing that bring combinatorics into algebraic geometry, usually via degeneration. Some technical details have been omitted; please see the cited sources. In tropical geometry, the operations of addition and multiplication are degenerated to the simpler ones of addition and minimum. This transforms the polynomial problems of algebraic geometry into piecewise linear problems of polyhedral geometry. It also has proved to illuminate hidden structures in many classical fields, such as dynamical systems [4], valuation theory [2] and compactification of algebraic varieties. My thesis is on tropical geometry in general and as applied to several major fields of algebraic geometry. the study of linear spaces, grassmannians, curves and real algebraic geometry. In my thesis, I introduce the notion of tropical linear spaces, combinatorial analogues of classical linear spaces. There are still several remaining problems about tropical linear spaces and about the tropical grassmannian which parameterizes them, I describe the most interesting of these problems in the first section. Another approach to bringing combinatorics into algebraic geometry is to study spaces that themselves have an underlying combinatorial structure. One of the most recent and exciting approaches of this kind is the discovery of cluster algebras. This subject can be viewed as the study of varieties which are covered with tori glued in a combinatorial manner. In the second section, I describe some progress I have made in understanding cluster algebras and how I hope to proceed in the future. A third approach to introducing combinatorics to algebraic geometry is to degenerate the underlying ring to a more combinatorial, often in some sense “monomial”, ring. This is the approach of the theories of Gröbner and SAGBI bases. In the third section, I describe ways I have applied and