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 to obtain the cochain complex Then the nth cohomology group with coefficients in G is defined to be Ker(δn+1)/Im(δn) and denoted by Hn(C; G). Note that the above definition can be adapted for general chain complexes, and not just the complexes used in singular homology. Given an element φ of C*n-1, it follows from the properties of the transpose that as elements of C*n. It can be shown that this map h is surjective, and that we have a short split exact sequence History[edit] There were various precursors to cohomology. . .
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."[1] An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.[2] Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant or participants. Today, however, game theory applies to a wide range of behavioral relations, and has developed into an umbrella term for the logical side of decision science, including both humans and non-humans (e.g. computers, animals). Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Representation of games[edit] Extensive form[edit] [edit]
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. In the 20th century, algebraic geometry has split into several subareas. Much of the development of the main stream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. Basic notions[edit] Zeros of simultaneous polynomials[edit] Sphere and slanted circle