New Geometries for a New Era in Mathematics

    VIENNA, AUSTRIA, November 06, 2023 /24-7PressRelease/ -- Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.

Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined.

In 2021, a new geometry "Dynamic Geometry on Time Scales" was published by Svetlin Georgiev, professor emeritus of mathematics at Sorbonne University, Paris, France, in memory of Professor Marcel Berger.

https://www.routledge.com/Dynamic-Geometry-on-Time-Scales/Georgiev/p/ ... 1032070780

The new geometry unifies classical differential geometry and discrete differential geometry. With the help of the new geometry, a number of open problems in the field of discrete differential geometry can be solved.This year International Society for Geometry and Graphics recognized "Dynamic Geometry on Time Scales" as a new compulsory university course for bachelors. From this academic year, "Dynamic Geometry on Time Scales" is studying at the world's leading universities. Under the leadership of Svetlin Georgiev, a number of other new branches in geoemtry have been developed, such as "Multiplicative Analytic Geometry", "Multiplicative Euclidean and Non-Euclidean Geometry", "Multiplicative Differential Geometry" which find applications in physics, chemistry, biology and computer science.

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