Geometry (3)

The field of algebraic geometry is closely related to the Cartesian geometry of coordinates, and has seen periodic periods of growth and development over the years. This development has included the creation and study of projective geometry, birational geometry, algebraic varieties, commutative algebra, and other topics. In the latter half of the 20th century, the work of Jean-Pierre Serre and Alexander Grothendieck played a major role in the foundational development of the field, leading to the introduction of schemes and a greater emphasis on topological methods and cohomology theories.

Algebraic geometry is a branch of mathematics that uses concepts from commutative algebra, like multivariate polynomials, to study geometry. It has a wide range of applications, including in fields like cryptography and string theory. In fact, one of the seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry, and Andrew Wiles' proof of Fermat's Last Theorem relied on advanced methods of algebraic geometry to solve a long-standing problem in number theory.

Complex geometry is a branch of mathematics that deals with the geometric structures that can be modeled on or arise from the complex plane. It combines elements of differential geometry, algebraic geometry, and the analysis of several complex variables, and has applications in areas like string theory and mirror symmetry.

Complex geometry was first studied in depth by Bernhard Riemann, who focused on the study of Riemann surfaces. The Italian school of algebraic geometry also contributed to the development of the field in the early 1900s. More recently, Jean-Pierre Serre introduced the concept of sheaves to complex geometry and helped to clarify the connections between complex geometry and algebraic geometry.

The primary objects of study in complex geometry include complex manifolds, complex algebraic varieties, complex analytic varieties, holomorphic vector bundles, and coherent sheaves. Special examples of spaces studied in this field include Riemann surfaces and Calabi-Yau manifolds, both of which have important roles in string theory. For example, the worldsheets of strings are modeled by Riemann surfaces, and superstring theory predicts that the extra six dimensions of ten-dimensional spacetime may be modeled by Calabi-Yau manifolds.

Discrete geometry is a branch of mathematics that deals with the relative positions of simple geometric objects like points, lines, and circles. It has close ties to convex geometry and shares many methods and principles with combinatorics. Some examples of the kinds of problems studied in discrete geometry include sphere packings, triangulations, and the Kneser-Poulsen conjecture. Overall, this subject is concerned with understanding the structural and positional relationships between discrete geometric objects.

Computational geometry is a branch of mathematics that focuses on developing algorithms and their implementations for manipulating geometrical objects. It is a relatively new area of geometry, but has already found numerous applications in fields like computer vision, image processing, computer-aided design, and medical imaging.

Some of the important problems that have been studied in computational geometry include the traveling salesman problem, minimum spanning trees, hidden-line removal, and linear programming. These problems and the algorithms developed to solve them have had a major impact on the way we use computers to analyze and understand geometric data.

Geometric group theory is a branch of mathematics that uses large-scale geometric techniques to study finitely generated groups. It is closely related to low-dimensional topology and has played a significant role in important developments like Grigori Perelman's proof of the Geometrization conjecture, which included a solution to the Poincaré conjecture - a problem that was one of the Millennium Prize Problems.

In geometric group theory, the Cayley graph is a frequently studied object. This is a geometric representation of a group that allows for the study of its structure and properties. Other important topics in this field include quasi-isometries, Gromov-hyperbolic groups, and right angled Artin groups. Overall, geometric group theory provides a powerful set of tools for understanding the behavior of groups from a geometric perspective.

Convex geometry is a branch of mathematics that deals with convex shapes in Euclidean space and its abstract counterparts. It uses techniques from real analysis and discrete mathematics to study these shapes and has connections to fields like convex analysis, optimization, and functional analysis. Convex geometry has a long history, dating back to ancient times when Archimedes gave the first known precise definition of convexity. Many famous mathematicians, including Archimedes, Plato, Euclid, Kepler, and Coxeter, have studied convex polytopes and their properties.

In the 19th century, the study of convex geometry expanded to include other areas like higher-dimensional polytopes, the volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings, and lattices. These developments have had important applications in number theory and other fields. Overall, convex geometry provides a rich set of tools for understanding the properties of convex shapes and their role in mathematics and other fields.

Mathematics and art have a long history of interaction and influence on each other. One example of this is the theory of perspective, which showed that there is more to geometry than just the metric properties of figures. This theory was the foundation for the development of projective geometry.

Artists have also used concepts of proportion in their work, such as the Vitruvian ideal proportions for the human figure. The golden ratio, a particular proportion that is said to be aesthetically pleasing, has been claimed to be used in many famous works of art, though the most reliable examples were made deliberately by artists who were aware of this legend.

Tilings and tessellations, the arrangement of shapes to form a repeating pattern, have been used in art throughout history. Islamic art and the work of M. C. Escher are both known for their use of tessellations. Escher's work also made use of hyperbolic geometry.

Cézanne's theory that all images can be constructed from the sphere, cone, and cylinder is still used in art theory today, although the exact list of shapes may vary from one author to another. Overall, the relationship between mathematics and art has led to the development of new ideas and techniques in both fields.

Geometry plays a central role in architecture, with numerous applications in the field. One such application is the use of projective geometry to create forced perspective, where the use of optical illusions can make buildings appear larger or smaller than they really are. Conic sections, such as circles and ellipses, are often used in the construction of domes and other rounded structures. Tessellations, or repeating patterns of shapes, are also commonly used in architecture, as are symmetrical designs. Overall, geometry is an essential tool in the field of architecture, helping to create visually striking and structurally sound buildings.

Geometry has played a significant role in the field of astronomy, particularly in mapping the positions of celestial bodies and understanding their movements. Riemannian and pseudo-Riemannian geometry are used in the study of general relativity, while string theory and quantum information theory utilize various forms of geometry. Overall, geometry has provided a powerful set of tools for understanding the geometric structure of the universe and how celestial bodies move within it.

Calculus and geometry have a strong relationship, with calculus drawing heavily on the use of coordinates and functions to analyze geometric figures. Analytic geometry is a fundamental part of the study of calculus and is used in pre-calculus and calculus courses. Geometry has also been applied to the study of number theory, including through the use of the geometry of numbers and scheme theory. In particular, scheme theory has been used to solve problems in number theory, such as in the proof of Fermat's Last Theorem. Overall, the intersection of calculus and geometry has led to many important developments and applications in mathematics.