# Sub-areas of mathematics

A characteristic of mathematics is the close connection between its sub-areas, which shows itself in many, often surprising, cross-connections and by which each systematics are limited.

Libraries and journals use different classifications of mathematical subjects; the most widespread is the Mathematics Subject Classification .

## An overview of the core areas of mathematics

The following is roughly based on Bourbaki's Éléments de Mathématique .

### Logic and set theory

Mathematics always needed logic , but it took a long time to get to grips with its basics.

It was set theory that changed this. This had developed from the preoccupation with the topology , more precisely with the paradoxes of infinity ( Bernard Bolzano ), as one experienced them in dealing with real numbers . When the infinite sets had been mastered with set theory, this was the hour of birth of a new mathematics that had emancipated itself from the rule of numbers and geometrical structures. People no longer wanted to be driven out of the “paradise of set theory” ( David Hilbert ).

When the so-called naive set theory proved untenable, the field of mathematical logic suddenly gained the interest that had remained with it between Leibniz and Frege , and it flourished quickly. The formalization of the logic serves the goal of isolating the individual proof steps and being able to present proofs completely as sequences of elementary operations in order to then examine them with mathematical (for example arithmetic ) means ( Gödel ). When examining axiomatic theories, one is interested in their consistent structure and their relationship to one another.

Meanwhile, a variety of sub-areas and applications have emerged in and outside of mathematics, including including in the computer science and proof systems .

Set theory is now supplemented as the lingua franca of mathematics in category theory , which developed from algebraic topology in the 1940s .

### algebra

In modern algebra, as it has been taught since the 1920s, the algebraic basic structures of monoids , groups , rings and solids , which are ubiquitous, are developed one after the other , starting from a set with only one internal operation ( called magma ) , because the different sets of numbers have such structures. Polynomials and modules / ideals are closely related to this .

The linear algebra has modules as object. In the simplest case these are vector spaces , i. H. Modules over bodies, mostly or . These are the spaces of classical geometry and analysis . But there are also much more complicated situations. The multilinear algebra extends the investigation to the tensor product and related phenomena. There is a close connection to ring theory and homological algebra ; a classic question is the invariant theory . ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$

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 , she investigates body extensions (and in doing so invents group theory ).

Further areas: representation theory , group theory , commutative algebra , lattice theory , universal algebra

### Analysis

Analysis examines differentiable mappings between topological spaces, from number fields and up to manifolds and Hilbert spaces (and beyond). It was and still is the mathematics of the natural sciences of the 17th and 18th centuries. ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$

The focus of the analysis is the infinitesimal calculus : the differential calculus describes a function in the small with the help of the derivative ; Conversely, integral calculus and the theory of differential equations make it possible to infer the function from the derivative.

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

If one looks at functions that represent the complex number field, the demand for complex differentiability arises , which has far-reaching consequences. Such functions are always analytical ; H. can be represented in small areas by power series . Their investigation is called function theory , it is one of the great achievements of the 19th century.

How the earth's surface can be represented piece by piece, or as it is said, locally by plane maps, one defines manifolds as Hausdorff spaces together with an atlas of compatible maps that depict the surroundings of each point in a certain model space. With some additional assumptions about the maps, one can do analysis on manifolds. Today Cartan's differential form calculus is the basis for the transfer of analytical concepts to manifolds; It is important to define the new terms intrinsically, i.e. independently of which specific maps are used to implement them. This can be done for the majority of the terms, although it is not always easy and leads to a number of new concepts. One example is Stokes ' theorem , which generalizes the fundamental theorem of analysis . This theory plays an important role in a different guise, as vector analysis and Ricci calculus in physics. Differentiable manifolds are also the subject of topology (cf. De Rham cohomology and differential topology ); With additional structures, Riemannian manifolds are among other things the subject of differential geometry .

From the age-old question of measure and weight, the theory of measure emerged only at the beginning of the 20th century with the inclusion of topological terms , which underlies the current, very powerful integral term and its applications, but also the calculation of probability.

At around the same time, the study of integral and differential equations developed into functional analysis as the study of function spaces and their mappings ( operators ). The first examples of such rooms were the Hilbert and Banach rooms . They turned out to be amenable to investigation by algebraic as well as topological instruments, and an extensive theory originated here.

Other areas: ordinary differential equations , partial differential equations , complex analysis , operator algebras , global analysis

### topology

Topology is a large and fundamental area with many uses. Impetus came from analysis ( real numbers ), early algebraic topology and function theory ( Riemann surfaces ).

First, the category of topological spaces and procedures for their construction are introduced. The closely related basic terms are connection , continuity and limit value . Other important issues are separation properties and compactness . Uniform spaces have a topology which (in generalization of metric spaces ) is defined by a type of distance. Here one can define Cauchy filters and thus the concept of completeness and the method of completing a topological space.

Topological groups , rings and bodies are the corresponding algebraic objects (see above), which are additionally provided with a topology with respect to which the links (i.e., addition and multiplication for rings and bodies) are continuous. A historically and practically important example are the real numbers : they are constructed by completing the rational numbers with respect to the topology that comes from the standard amount. However, one can also introduce the so-called p-adic amount for a fixed prime number p, which then results in the completion of the fields of the p-adic numbers . Number theory , for example, is interested in this . ${\ displaystyle \ mathbb {Q}}$

Metric rooms are uniform rooms, the topology of which is derived from a metric, and are therefore particularly clear and clear. Many other classes of rooms are also known.

Topological vector spaces are essential for applications in analysis and functional analysis . Locally convex spaces (and their dual spaces ), for which there is a nice theory with important results, are particularly interesting .

Further areas: Algebraic Topology

## Other areas in alphabetical order

### Algebraic Geometry

Algebraic geometry is a very active area that emerged from the study of conic sections and has the closest relationship to commutative algebra and number theory . The subject of the older theory is algebraic varieties until around 1950 , i. H. Solution sets of algebraic systems of equations in affine or projective (complex) space, in the meantime there has been a strong generalization of the questions and methods.

### Algebraic topology and differential topology

The algebraic topology arose from the problem of classifying topological spaces . The underlying questions were often very specific: leisure activities (Königsberger bridge problem , Leonhard Euler ), electrical networks, the behavior of analytical functions and differential equations on a large scale ( Riemann , Poincaré ). Emmy Noether's suggestion became important to study the underlying algebraic objects instead of numerical invariants ( dimension , Betti numbers ). The now very extensive area can be described as the investigation of functors from topological to algebraic categories .

The differential topology is the topology of the (differentiable) manifolds . Now a manifold looks locally everywhere like the model space ; in order to be able to investigate them at all, additional structures are introduced, but these are only of instrumental interest.

### Representation theory

The representation theory examined algebraic objects such as groups , algebras or Lie algebras , by representing the elements as linear images on vector spaces. If you have a sufficient number of such representations for an object, it can be completely described by them. Furthermore, the structure of the set of representations reflects properties of the objects themselves.

### Differential geometry

The differential geometry studied geometric objects such as curves or surfaces using the methods of differential calculus . The fundamental work goes back to Carl Friedrich Gauß . The sub-area of Riemannian geometry is required for the formulation of the general theory of relativity .

### Discrete Math

In discrete mathematics , finite or countably infinite structures are examined. This affects many areas of mathematics, including combinatorics , number theory , coding theory , set theory , statistics , graph theory , game theory , and cryptography .

### Experimental math

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 , such as Banach and Hilbert spaces , as well as properties of functional and operators on these vector spaces . Functional analysis has made an important contribution to the mathematical formulation of quantum mechanics with the operators .

### Geomathematics

The term geomathematics today covers those mathematical methods that are used to determine geophysical or geotechnical quantities. Since data measured by satellites are mostly evaluated, methods have to be developed that are suitable for solving inverse problems.

### geometry

Historically, Euclidean geometry was the first example of an axiomatic theory, even if Hilbert was not able to complete this axiomatization until the turn of the 20th century. After Descartes set up the program to algebraize geometric problems , they found new interest and developed into algebraic geometry. In the 19th century, non-Euclidean geometries and differential geometry were developed. Much of classical geometry is now explored in algebra or topology. The synthetic geometry continues to investigate the classical axioms of geometry with modern methods.

### Group theory

The group theory , originated as a mathematical discipline in the 19th century, is a pioneer of modern mathematics, since it. Decoupling of representation (for example, the real numbers) from the internal structure is (laws for groups)

### Commutative algebra

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

### Complex analysis

While the investigation of real functions of several variables is not a big problem, it is completely different in the complex case. Accordingly, the function theory of several variables or complex analysis , as they say today, only developed very slowly. This area has only developed since the 1940s, primarily through contributions from the schools of Henri Cartan and Heinrich Behnke in Paris and Münster.

### Lie groups

Lie groups describe the typical symmetries in geometry and physics. In contrast to “naked” groups, they have a topological structure (more precisely: they are manifolds ) and make it possible to describe continuous transformations, for example the rotations or the translations form such a group.

### numerical Mathematics

The numerical analysis designed and analyzed algorithms for solving continuous problems of mathematics. While the algorithms were originally intended for manual calculations, nowadays the computer is used. Important tools are approximation theory , linear algebra and functional analysis . Above all, questions of efficiency and accuracy play a role, and errors that occur must be taken into account in the calculation.

### Philosophy of mathematics

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

### probability calculation

This area first and for a long time developed from actuarial mathematics , v. a. also fed the special case of the theory of gambling . One differentiates:

• Probability theory i. e. S. ( Stochastics ) as a theory of stochastic experiments. The aim is to determine the distribution of the random variables for a given experiment.
• Based on this, the mathematical statistics , which, with incomplete knowledge of the experiment, want to deduce the underlying distribution from certain results (a sample). The focus is on two questions:
• Determination of parameters (estimation theory)
• Classification of cases (decision theory)
These tasks are presented as optimization problems, which is characteristic of statistics.

Modern theory has been an important application of measure theory since the work of Andrei Kolmogorov .

Other areas: ergodic theory , statistical mechanics , information theory , operations research

### Number theory

An old subject that flourished in antiquity , the starting point of which is the surprising properties of natural numbers (also called arithmetic ). The first question asked is divisibility and primality . Many math games also belong here. Many theorems of number theory are easy to formulate but difficult to prove.

In modern times , number theory first found renewed and forward-looking interest in Fermat . Gauss ' Disquisitiones Arithmeticae culminated in 1801 and stimulated intensive research. Today, have according to the used resources for elementary the analytical , algebraic , geometric and algorithmic number theory joins. For a long time, number theory was considered (practically) absolutely useless until it suddenly became the focus of interest with the development of asymmetric cryptography .

## literature

• Oliver Deiser, Caroline Lasser, Elmar Vogt, Dirk Werner : 12 × 12 key concepts in mathematics. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2297-2 .