Chern classes are a part of algebraic topology, as well as other math groups, and are characteristic classes related to complex vector bundles. Let \(X\) be a topological space of closure-finite weak CW complex and let \(V\) be a line bundle. The first chern class is the only nontrivial Chern class and is an element of the second cohomology group of \(X\).... read more

Abstract Algebra or modern algebra can be defined as the theory of algebraic structures. For the most part, abstract algebra deals with four algebraic structures: groups, rings, fields, and vector spaces. We will look at and examine these four algebraic strucutres in this page. The three most commonly studied algebraic... read more

Let us first start with a formal definition of a 2 vector convex combination. Then we will break down the definition into parts and analyze the definition. Then we will formally define and analyze a convex combination with a finite number of vectors in the same manner. A subset \(S \subseteq \mathbb{R}^{n}\)... read more

An \(n\)-dimensional manifold is a topological space where each point has a neighborhood that is homeomorphic to an open subset on \(n\)-dimensional Euclidean space. Let \(M\) be a topological space. A chart in \(M\) consists of an open subset \(U \subset M\) and a homomorphism \(h\) of \(U\) onto an open subset of \(R^{m}\). A... read more

\(G = (V, E)\) \(V\) is a set of vertices \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\) A simple undirected graph \(G\) is an ordered pair or tuple \((V, E)\) where \(V\) and \(E\) are finite sets. \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\)... read more