Mathematica Eterna
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Short Communication on Geometric Topology

Johnson H^{* *}Correspondence: Johnson H, Associate Professor, Faculty of Science and Technology, University of Patras, Greece, Email:

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Pure mathematics can tell you ways long and what direction a path between 2 points are, Topology can tell you whether or not or not there's a path between 2 points.

Visibly in native to world theorems in geometry, and results similar to the Gauss–Bonnet theorem and Chern–Weil theory. Sharp distinctions between pure mathematics and topology is drawn, however, as mentioned below. it is also the title of a journal pure mathematics & Topology that covers these topics.

The terms are not used fully consistently: symplectic manifolds area unit a boundary case, and coarse pure mathematics is world, not local.

Geometry has perpetually been tied closely to mathematical physics via the speculation of differential equations.

It uses curvature to differentiate straight lines from circles, and measures symmetries of areas in terms of Lie teams, named when the far-famed Norwegian man of science Sophus Lie.

Topology, in distinction, is that the study of qualitative properties of areas that area unit preserved below continuous deformations.

The areas in question could also be tame kind of a sleek manifold, or wild and laborious as rock.

Topological ideas arise in sensible issues, and analysis in topology still finds new applications, specifically to mathematical issues that do not appear to be directly phrased in terms of

(i) algebraical pure mathematics – may be a branch of pure mathematics finding out zeros of the variable polynomial.

It includes linear and polynomial algebraical equation used for resolution the sets of zeros.

the applying of this kind includes Cryptography, string theory, etc.

(ii) distinct pure mathematics – thinks about with the relative position of easy geometric object, like points, lines, triangles, circles etc.

(iii) Differential pure mathematics – Uses techniques of pure mathematics and calculus for problem-solving. The varied issues embody general theory of relativity in physics etc.

(iv) parabolic geometry – The study of plane and solid figures supported axioms and theorems as well as points, lines, planes, angles, congruence, similarity, solid figures.

it's a good vary of applications in engineering science, fashionable arithmetic downside resolution, physical science etc.

(v) convex pure mathematics – Includes convex shapes in Euclidean space victimisation techniques of real analysis.

it's application in improvement and useful analysis in range theory.

(vi) Topology – thinks about with properties of area below continuous mapping.

Its application includes thought of compactness, completeness, continuity, filters, perform areas, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric areas, nets, proximal continuity, proximity areas, separation

In geometric analysis: heat kernel analysis on Lie teams and mathematician manifolds, metric diophantine properties of the geodesic flow on a hyperbolic Bernhard Riemann surface, mathematical general theory of relativity

Geometry has evolved considerably throughout the years, there area unit some general ideas that area unit elementary to pure mathematics.

These embody the ideas of purpose, line, plane, distance, angle, surface, and curve, similarly because the a lot of advanced notions of topology and manifold.

Euclidean areas, and, a lot of typically, metric areas area unit samples of a set, as any distance or metric defines a topology.

The deformations that area unit thought of in topology area unit homeomorphisms and homotopies.

Basic samples of topological properties are: the dimension, that permits identifying between a line and a surface; compactness, that permits identifying between a line and a circle; connectedness, that.