Areas of mathematics

This article supplements the main article Mathematics and the Mathematics Portal page. It serves to give an overview of the branches of mathematics.

Characteristic of mathematics is the close relationship between its sub- areas, which in many, often surprising, cross connections shows and are set by the limits of each system.

Libraries and magazines use different classifications of mathematical topics; the most common is the Mathematics Subject Classification.

• 2.1 Algebraic Geometry
• 2.2 Algebraic topology and differential topology
• 2.3 Representation Theory
• 2.4 Differential Geometry
• 2.5 Discrete Mathematics
• 2.6 Experimental Mathematics
• 2.7 Functional Analysis
• 2.8 Geomathematics
• 2.9 geometry
• 2:10 Group Theory
• 2.11 Commutative Algebra
• 2:12 Complex Analysis
• 2:13 Lie groups
• 2:14 Numerical Mathematics
• 2:15 Philosophy of Mathematics
• 2:16 probability theory
• 2:17 Number Theory

The core areas of mathematics at a glance

The following is based roughly on Bourbaki Éléments de Mathématique.

Logic and set theory

Mathematics has always required the logic, but it took a long time until they dealt with their own foundations.

It was the set theory that this changed. This had been out of employment developed with the topology, more precisely with the " paradoxes of the infinite " ( Bernard Bolzano), how she experienced in dealing with the real numbers. When they had mastered the set theory, the infinite sets, this was also the birth of a new mathematics that had emancipated themselves from the domination of numbers and geometrical figures. From the " paradise of set theory " ( David Hilbert ) wanted to no longer permitted to sell.

As the " naive " set theory proved to be untenable, suddenly won the field of mathematical logic that interest which had been denied him between Leibniz and Frege, and flourished rapidly on. The formalization of logic serves the aim of isolating the individual proof steps and evidence to be able to fully represent as sequences of elementary operations to investigate this then with mathematical (eg arithmetic ) means ( Gödel ). In the investigation of axiomatic theories one is interested in each other for their non-contradictory structure and their relationship.

Meanwhile, a variety of sub-areas and applications have emerged in and outside mathematics, among other things, include this in the computer science and proof systems.

Set theory is today supplement as lingua franca of mathematics in the category theory, which developed in the forties of the 20th century from algebraic topology.

Algebra

In modern algebra, as it is taught since the 1920s, successively developed starting from a set with only one "internal operation " (Magma ) embodies the algebraic basic structures of monoids, groups, rings and fields, which are ubiquitous, including because the different sets of numbers exhibit such structures. Closely related to this are polynomials and moduli / ideals.

The Linear Algebra has modules as object. In the simplest case, this vector space, i.e., moduli of about bodies or generally. These are the spaces of classical geometry and analysis. But there is also much more complicated situations. The multilinear algebra extends the investigation to the tensor product and related phenomena. Are closely related to ring theory and homo logical algebra; a classic issue is the invariant theory.

The Galois theory is one of the highlights of mathematics in the 19th century and the beginning of the field theory. Starting from the question of the solvability of algebraic equations, they examined the body extensions (and discovered this group theory ).

Topology

The topology is a large and fundamental area with many applications. Impetus came from the analysis ( real numbers ), the early algebraic topology and the theory of functions ( Riemann surfaces ).

First, the category of topological spaces and procedures are introduced to their construction. The closely related basic concepts are " related ", " continuity " and " limit ". Other important issues are "separation properties " and " compactness ". Uniform rooms have a topology ( metric generalization rooms) is defined by a kind of distance. Here you can define Cauchy filter and thus the notion of completeness and the method of completion of a topological space.

Topological groups, rings and fields are the corresponding algebraic objects (see above), which are additionally provided with a topology in respect of which the shortcuts are continuous (ie, rings and fields of addition and multiplication ). A historically and practically important example are the real numbers: they are constructed by the completion of the rational numbers Q with respect to the topology that comes from the standard amount. However, one can introduce the so-called p- adic amount also is a fixed prime number p, is then obtained as the completion of the body of the p- adic numbers. For this example, interested in number theory.

Metric spaces are uniform spaces whose topology is derived from a metric, and therefore particularly clear and also vivid. In addition, we know many other classes of spaces.

For applications in Analysis and Functional Analysis topological vector spaces are fundamental. Of particular interest are locally convex spaces (and their dual spaces ) for which there is a nice theory with important results.

Analysis

The Analysis examines differentiable maps between topological spaces, of the number fields R and C to manifolds and Hilbert spaces ( and beyond). She was the mathematics of natural sciences of the 17th and 18th centuries and it is still.

The focus of the analysis is the Calculus: Differential calculus describes using the derivative of a function " in the small "; Integral calculus and the theory of differential equations enable it inversely, from the derivation to deduce the function.

The algebraically defined rational functions are supplemented by the exponential function and its relatives, and many others, represented by differential equations and power series, special functions.

If we consider functions that map the complex number field in itself, so the demand for complex differentiability arises as to the far -reaching consequences. Such functions are always analytically, that is, in the small representable by power series. Their study is called function theory, it is one of the great achievements of the 19th century.

How in the small " can be represented by planar maps the Earth's surface piecemeal, or as they say, " local " or " we define manifolds as Hausdorff spaces together with an atlas of compatible cards that a neighborhood of any point in a certain model space map. With some additional assumptions about the cards can be " run analysis on manifolds ". Today, the Cartan differential form calculus of transmission analytical terms is based on manifolds; here it comes down to the new concepts of " intrinsic ", ie independently to define on which specific card you used for their realization. For a large part of the terms you can do that, although it is not always easy and leads to a series of new conceptions. As an example of Stokes' theorem may be mentioned that generalizes the Fundamental Theorem of Calculus. Plays an important role this theory in other guise, as a vector analysis and Ricci calculus in physics. Differentiable manifolds are also the subject of topology (see de Rham cohomology and differential topology ); with additional structures include Riemannian manifolds subject of differential geometry.

From the age old question on weight and until the beginning of the 20th century grew by absorbing topological terms the measure theory that underlies the current, very powerful concept integral and its applications, but also the theory of probability.

At about the same time developed from the study of integral and differential equations, functional analysis as the study of function spaces and their mappings ( operators). The first examples of such spaces were the Hilbert and Banach spaces. They proved to be accessible to investigation with algebraic topological as instruments, and an extensive theory took its origin here.

Other areas in alphabetical overview

Algebraic Geometry

A arisen from the study of conic sections and still very active area with very close relations with commutative algebra and number theory is the algebraic geometry. Covered by the earlier theory until about 1950 algebraic varieties, ie algebraic solution sets of equations in affine or projective (complex) space, now a strong generalization of the approach and methods took place.

Algebraic topology and differential topology

The algebraic topology arose from the problem of classification of topological spaces. The underlying issues were frequently quite concretely: leisure ( " Königsberg bridge problem," Leonhard Euler ), electrical networks, the behavior of analytic functions and differential equations in the Great ( Riemann, Poincaré ). Importantly, the proposal Emmy Noether, instead of numerical invariants ( dimension, Betti numbers ) was to study the underlying algebraic objects. The now very extensive area can be sharpened as describe the investigation of functors of topological in algebraic categories.

The differential topology is the topology of the ( differentiable ) manifolds. Now a manifold looks locally everywhere like the model space; so that they can investigate at all, one introduces additional structures, but have only instrumental interest.

Representation theory

The representation theory examines algebraic objects such as groups, algebras and Lie algebras, by representing their elements as linear transformations on vector spaces. One has for an object a sufficient number of such images, it can be described completely by this. Furthermore, the structure of the set of representations of properties reflects the objects themselves.

Differential Geometry

The differential geometry studied geometric objects like curves or surfaces using the methods of differential calculus. The fundamental work go back to Carl Friedrich Gauss. The branch of Riemannian geometry is required for the formulation of general relativity.

Discrete Mathematics

In discrete mathematics finite or countably infinite structures are investigated. This touches on many mathematical areas, including combinatorics, number theory, coding theory, set theory, statistics, graph theory, game theory, cryptography.

Experimental Mathematics

The Experimental Mathematics is a discipline between classical mathematics and numerical mathematics.

Functional Analysis

The functional analysis is concerned with the study of topological vector spaces, for example, Banach and Hilbert spaces, as well as properties of functionals and operators on these vector spaces. The functional analysis has provided, including with the operators an important contribution to the mathematical formulation of quantum mechanics.

Geomathematics

The term Geomathematics we group together today those mathematical methods that are used in the determination of geophysical or geotechnical sizes. Since mostly measured by satellite data are evaluated, must be particularly methods are developed which are suitable for the solution of inverse problems here.

Geometry

Historically, the Euclidean geometry was the first example of an axiomatic theory, if only Hilbert around the turn of the century to the 20th century was able to complete this axiomatization. After Descartes had set up the program to algebraisieren geometric problems, they found a new interest and developed for algebraic geometry. In the 19th century non-Euclidean geometry and the differential geometry were developed. A large proportion of classical geometry is now studied in algebra or topology. The synthetic geometry continues to investigate the classic geometric axioms with modern methods.

Group Theory

The group theory emerged as a mathematical discipline in the 19th century, is a pioneer of modern mathematics because they (for example, the real numbers ) of the internal structure represents a decoupling of representation ( laws for groups).

Commutative Algebra

Commutative algebra is the algebra of commutative rings and modules over them. It is the local counterpart of algebraic geometry, similar to the relationship between analysis and differential geometry.

Complex Analysis

During the investigation of real-valued functions of several variables is not a big problem, it is in the complex case quite different. Accordingly, the theory of functions of several variables or complex analysis very slowly developed, as they say today. Only since the 1940s, the area has unfolded, particularly through contributions to the schools by Henri Cartan and Heinrich Behnke in Paris and Münster.

Lie groups

Lie groups describe the typical symmetries in geometry and physics. In contrast to the "naked" groups they carry a topological structure (more precisely, they are manifolds ) and allow to describe continuous transformations, for example, form the rotations or translations of such a group.

Numerical Mathematics

The numerical mathematics constructed and analyzed algorithms for the solution of continuous problems of mathematics. Were the algorithms originally intended for invoice by hand, so nowadays the computer is used. Important tools are approximation theory, linear algebra and functional analysis. It mainly play questions the efficiency and accuracy of a role, furthermore, the errors occurring in the invoice must be considered.

Philosophy of Mathematics

The philosophy of mathematics, in turn, questions the methods of mathematics.

Probability Theory

In beginnings in antiquity exist, this area has fed first and for a long time from the actuarial, v. a also the special case of the theory of gambling. We distinguish:

• Probability theory, i.e., p ( Stochastic ) as a theory of stochastic experiments. The aim is to determine at any given experiment, the distribution of the random variable.
• Building on mathematical statistics, which will in imperfect knowledge of the experiment, from certain results ( a sample ) to the underlying distribution close. Two questions are central: Determination of parameters ( Estimation Theory )
• Classification of cases ( Decision Theory )

The modern theory is an important application of the measure theory since the work Andrei Kolmogorov.

Number Theory

An old, thriving even in ancient trade, whose starting point is the surprising properties of the natural numbers (also called arithmetic). Asked looks first for divisibility and primality. Many math games belong here. Many sets of number theory are easy to formulate but difficult to prove.

In modern times, the theory of numbers takes first at Fermat again and at the same time pioneering interest. Gauss ' Disquisitiones Arithmeticae form 1801 to a peak and stimulate intensive research. Today, the analytical, algebraic, geometric and algorithmic number theory have, according to the means used, the elementary joins. Long number theory was absolutely useless as a ( virtually ) until it suddenly advanced with the development of asymmetric cryptography in the center of interest.