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.:: Ph.D::.PureMathematicsIntroductionPure mathematics is the study of mathematical concepts independent of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around (the year) 1900. It is worthy to know that almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. .:: MS.c.::.Pure Mathematics/ branches: Analysis, Algebra, Mathematical Logic, Geometry-TopologyIntroductionPure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around (the year) 1900. It is worthy to know that almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. Algebra: Algebra is the branch of mathematics that studies algebraic systems and the manipulation of equations within those systems. It is a generalization of arithmetic that includes variables besides regular numbers and algebraic operations other than the standard arithmetic operations like addition and multiplications. Algebra is relevant to many branches of mathematics, like Geometry, Topology, Analysis and also other fields of inquiry like Logic and the empirical sciences. Mathematical Logic: Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory )also known as compatibility theory). Research in mathematical logic commonly addresses mathematics. Properties of formal systems of logic such as their expressive or deductive power. It can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Mathematical logic like other parts of modern mathematics mainly has begun since 19th century and it has been evolving and making progress since then. Analysis: Analysis is the branch of math dealing with continuous functions, limits, and related theories such as differentiation, integration, measure, infinite sequences, series and analytic functions. Analysis can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). The modern foundations of mathematical analysis were established in the 17th century Europe and it has been evolving till now and is currently a main branch of mathematics. Geometry-Topology Geometry is a branch of math. Concerned with properties of spaces such as the distance, shape, size and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. It has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Modern geometry has evolved mainly since 19th century and it is evolving continuously. Topology In mathematics, Topology is concerned with properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. The ideas underlying topology go back to the 17th century and the motivating insight behind it, is that some geometric problems depend not on the exact shape of the objects involved, but rather on how they are put together. Topology as a well-defined math. discipline originates in the early part of the 20th century. Now it is a well-established branch of mathematics with many connections and applications to/in other parts of mathematics. |
