Abelian Group. Cohomology. Definition[edit] In algebraic topology, the cohomology groups for spaces can be defined as follows (see Hatcher).
Given a topological space X, consider the chain complex as in the definition of singular homology (or simplicial homology). Here, the Cn are the free abelian groups generated by formal linear combinations of the singular n-simplices in X and ∂n is the nth boundary operator. Now replace each Cn by its dual space C*n−1 = Hom(Cn, G), and ∂n by its transpose. Algebraic topology. Game theory. Game theory is the study of strategic decision making.
Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Algebraic geometry. Algebraic geometry is a branch of mathematics, classically studying zeros of polynomial equations.
Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.