Most of the frontier disciplines of contemporary mathematics were formed in the first half of the 20th century, it mainly involves abstract representative Mathematics (including group theory, ring and algebra theory, domain theory, lattice theory, overall Li Qun theory, algebra group theory, simultaneous modulation algebra and various derivative structure theories) general topology, Point Set Topology, measurement and integral theory, and functional analysis (including linear topological space theory and operator algebra theory), composite topology and algebraic topology, overall differential geometry, multi-variant function theory, dynamic system theory, and random process theory. The new fields created in the 19th century, such as algebra number theory, algebra and geometry, column-related geometry, and local Li Qun theory, have also made major breakthroughs in the framework of structured mathematics and have become the frontier of contemporary mathematics. Some fields formed in the early 20th century, such as differential topology, large-scale analysis, k theory, and non-switching ry, can also be seen in the sprout.
In addition to the expansion and deepening of the field of pure mathematics, the appearance of applied mathematics and Computational Mathematics in the 20th century has also undergone fundamental changes.
On the one hand, the application scope of mathematics has been expanded from classical mechanics, astronomy and geology, mathematics physics, and other fields before the 20th century to almost all branches of natural science, engineering technology, social science, and humanities, in addition, it plays an increasingly important role. On the other hand, a group of new application mathematical fields have become relatively independent branches and constitute an integral part of the big mathematical science. On the one hand, they are closely related to actual problems, on the other hand, they also form an independent mathematical research direction. Among them, the most typical is the mathematical statistics formed at the end of the 19th century and the beginning of the 20th century. Together with the probability of application, they have become a subject field that is equal to the classical Applied Mathematics in nearly half a century. Another field of mathematics-composite mathematics has almost the same long history as mathematics, but it has only been mature and developed independently for more than half a century.
After the Second World War, some new fields of applied mathematics were independent, especially the branches of operational research, system Science, control theory, automation science, and information science, which are closely related to engineering technology, have also been unprecedentedly developed.
The first important event in the history of science and technology in the 20th century was the birth of electronic computers, and its impact on society as a whole was not overestimated. From computer design and manufacturing to large-scale application, there is no need for mathematics everywhere. At the same time, a new field of mathematics has been opened up. They can be divided into two parts: computer science, it guides the future development of computers. First, it points to large-scale computing in scientific computing and engineering technologies. The continuous popularization and improvement of computers also have an important impact on mathematics. It raises a series of algorithm problems for mathematicians and forms a set of effective algorithms, such as the simple method and its various improvements, the finite element method and its derivative algorithms, and analyzes the algorithms, problems such as convergence speed, error propagation, and stability form a Numerical Analysis Branch.
In recent years, computers have transitioned from numerical operations to symbolic operations, forming an important branch of computer algebra. In particular, Wu Wenjun's mechanized Mathematical Program is a major breakthrough in Machine Proof.
From the end of the 19th century to the beginning of the 20th century, mathematics, like physics, ushered in a period of intense change. On the one hand, people began to accept the Set Theory of G. Conway as the basis of unified mathematics, but soon found a paradox in it, resulting in a serious Mathematical Crisis. On the other hand, as the main method of mathematics in the future-the Internet physical and chemical method laid by Hilbert, he published the "geometric Foundation" in 1899, which gave great inspiration to mathematics in the 20th century. Under his impetus, he formed a small public physical and chemical boom. On this basis, the new fields of structured mathematics and metamathematics are formed. At the beginning of the 20th century, mathematics became more and more abstract. Abstract group theory research, French mathematician leberger's distance theory and point theory, Hilbert's integral equation theory, French mathematician frerest's abstract Space Theory, and some of the theory of physical and chemical theory of modern mathematics have emerged one after another, together with the establishment of composite topology at the end of the 19th century, it represents a dramatic change in modern mathematics centered on modern mathematics. The emergence of functional analysis greatly changes the appearance of analysis, and provides ready-made tools for quantum physics. Compared with previous mathematics, mathematics in the 20th century has the following features:
1. Mathematics is no longer just a few relatively independent parts of number theory, algebra, ry, and analysis. Rather, with the emergence of set theory, a large number of new disciplines, new branches, and new theories have emerged. For example, mathematical basics and mathematical logic (as well as the resulting model theory, progressive analysis, and proof theory) abstract representative Mathematics (including group theory, ring theory, domain theory, simultaneous adjustment representative mathematics, algebra K theory, lattice theory, and various algebra structures) general topology, algebraic topology, differential topology, topological group theory (and other topological algebra, including Li Qun), algebraic group theory, measure and integral theory, functional analysis, random process theory, and so on. Almost all mathematical departments related to mathematics and computers are the product of the 20th century. Even the classic mathematical department has completely changed its face. For example, in the 19th century, the Representative mathematics mainly studied the problem of solving the algebraic equations and the algebraic equations. in the 19th century, the galova theory, linear representative mathematics, and non-variant theory were studied, modern modern mathematics is already a branch of group theory, ring theory, domain theory, and simultaneous adjustment of mathematics, and those classic contents have already accounted for less than a few percent.
2. Mathematics is no longer just a discipline that solves special problems and seeks special algorithms as it used to do, but under the concept of structure, there is a unified object, a unified method, and an independent discipline with its own independent problems. It is not only about the number and shape, but mainly about the various structures, in particular, the algebraic structure, topological structure, ordinal structure, and various multi-structures produced by the combination of these structures bring rich and profound contents to mathematics in the 20th century. The concept of structure is further developed into the concept of category and correspondence, which plays a great role in unifying the idea of mathematics. The unification of ideas and the deepening of methods promote the solution of many classic problems.
3. The content of mathematics is becoming more and more complex and abstract. It is not only not separated from reality, but also many concepts developed from mathematics itself provide many powerful tools for physics, chemistry and biology. For example, for the theory of general relativity, the theory of Functional Analysis for quantum mechanics and quantum field theory, and even the theory of fiber bundle, differential ry and algebraic ry in recent years, the group sequence theory is like a customized tool for atomic structure, nuclear structure, and basic particle classification, which not only surprised physicists. Even the Laidong transform discovered in 1917 provides a theoretical basis for medical examination of essential X-ray analyzers for tumors 40 or 50 years later. Before and after the Second World War, the advent of electronic computers and the development of many applied mathematics have opened up a very broad prospect for the application of mathematics. In turn, practical and applied mathematics has brought forward many new concepts and problems for pure mathematics, and even promoted the solution of many classical problems. For example, use the normative field theory to promote research on Four-Dimensional topology and make significant breakthroughs.
4. With the invention of electronic computers, both pure mathematics and applied mathematics are strongly influenced by electronic computers. Numerical Analysis has formed an independent branch of mathematics, if the current mathematical calculation method cannot be used on the computer, it will greatly reduce the color. Many methods (such as simple form method, Monte Carlo method, finite element method, Kalman filter, etc) their Superiority lies in their good integration with computers. In this way, many mathematical problems can be used for computer experiments and gradually solved. In addition, many purely mathematical problems have been proved with the help of computers. The most prominent of these problems were the use of computers to prove the four-color conjecture in 1976. The proof of mechanization is expected to reduce the repetitive and tedious work of mathematicians and focus on solving more important mathematical problems.
Mathematics in the 20th century can be divided into two stages by World War II. The period from 1870 to 1940 was the beginning of modern mathematics. From computing to research structure, mathematics is unified on the basis of set theory, it marks the birth of five major disciplines, including mathematical logic, abstract substitute mathematics, measurement and integral theory, topology, and functional analysis. By 1950s, The burbaki school used the concept of mathematical structure to unify mathematics and published multiple books on mathematical principles, which became a classic of mathematics in the future. After the 1940 s, it was a prosperous period of modern mathematics, with pure mathematics taking topology as the center of rapid development. At the same time, with the emergence of computers, the application of mathematics and computational mathematics has also made unprecedented progress, it plays an increasingly important role in science and society.
Next we will discuss the progress of pure mathematics from four aspects.
I. metamathematics
In the early 20th century, the internal contradictions of the set theory began to be exposed, and the paradox that Russell discovered in 1901 shocked the field of mathematics. In order to solve this contradiction, Russell proposed the branch theory, and on this basis, he co-authored three major volumes of mathematical principles (1910-1913) with whitesea ). One way to solve the paradox is the generalization of the set theory proposed by cemetaro in 1908. His system of justice was supplemented and modified later to become a recognized basis of the set theory. At the same time, there was a heated debate on the basis of mathematics, which resulted in three schools of opposition: logical, intuition, and formalism. The formalism, represented by Hilbert, attempts to establish all mathematics on the basis of a minority of them, and then gives an absolute proof of the non-contradictions of them. This is the so-called theory of proof. In 1931, Godel proved his well-known Incompleteness Theorem, making it impossible to prove the absolute completeness of the form system that Hilbert expected, so that the mathematical logic is completely switched to a new direction.
In 1931, the Incompleteness Theorem of Godel led to the development of mathematical logic. The first is the concept of general recursive functions developed in 1930s. in 1936, Turing proposed the concept of a Turing Machine to give a specific description of the calculability. Due to the incompleteness theorem, the undefinable Problem in the form system occurs, especially the undefinable problem of group words and the negative solution of the question of Hilbert 10th. In 1938, Godel proved that the hypothesis of continuity was relatively non-contradictory. In 1960s, it discovered the relative independence of the choice principle and the hypothesis of continuity, which produced a series of mathematical consequences. Especially since 1950s, the birth of the model theory has had a great impact on Mathematics itself. Among them, the generation of non-standard analysis and the publication of topology theory. Due to incomplete systems of the set theory, we naturally consider adding some new ones. The selection principle is important and is indispensable in many proofs of algebra and analysis. However, there are also some theorems, such as the principle of a large base, which can be used to export the subsets of all real numbers that are measurable by leberger. The Research of mathematical logic has been re-emphasized by mathematicians.
Ii. Structured mathematics
In the first half of the 20th century, the branches of abstract algebra, general topology, measurement and integral theory, functional analysis, and so on were laid. In the second half of the 20th century, structural mathematics focused on algebra topology.
1. Abstract Algebra
Since the end of the 19th century, the appearance of modern mathematics has undergone a fundamental change. At this time, the structure theory and Representation Theory of abstract groups have developed. In 1910, stynitz unified the abstract processing of domain theory, and the most important development was the structural theory that evolved from the end of the 19th century combining algebra and non-combated algebra, in particular, Wade born proved the structure theorem of linear concatenation algebra in 1907. Before and after this ,. jia dang completed the structure theorem of semi-single Lie algebra in the complex number field and extended them to the real semi-single Lie algebra. At the same time, they studied their representation theory, which constitute the initial sprout of abstract algebra. However, the development of abstract algebra comes from. e. NORT's ideal theory,. e. NORT develops the general ideal theory through the public physical and chemical method, establishes the ideal structure theory of NORT ring and did golden ring, and establishes the basis of the combined algebra. A ting first popularized the structure theory of algebra to the ring, leading to the birth of the ring theory. The real-domain research by aiting et al. solves the question of Hilbert 17th and reflects the power of abstract methods. In 1930, van der Walden published his book "Modern modern mathematics", marking the birth of abstract mathematics.
2. Functional Analysis
At the same time, functional analysis was born as a discipline. In addition to functional calculus in Italy and France, there is also a study of the integral equations that Hilbert and his students studied at the beginning of the 20th century. They introduced L2 space and L2 space to prove the Risan Fisher theorem. RISS also introduced abstract linear operators and defined the operator's Norm. He extended the fully continuous concept of Hilbert on the integral equation to the Abstract Operator. In this way, the theory of Hilbert Space and Its linear operator was basically built, but it was not until 1928 that Feng nuiman made it public. The third line of functional analysis comes from the work of those people, such as banner and so on. They mainly study the Norm Space and introduce operators on it, especially promoting the work of RISS, the concept of dual space is established. The emergence of functional analysis not only promoted the spectral theory in the early 20th century, but also became a useful mathematical tool for quantum mechanics. The emergence of quantum mechanics further promotes the study of functional analysis and the generation of operator theory.
3. Finite Group Theory
The main goal of the finite group theory is to classify all finite groups. Therefore, we can take two steps to find all single groups (that is, the basic unit of all groups ), the other is to assemble these single groups into various groups.
We know a lot about a limited single group a long time ago. Apart from the cyclic group of prime numbers, galova knows the correct group. Many typical groups of matrices were known around 1900, but they did not make much progress before 1955. In 1955, Xie walai systematically created all the known single groups (except a few exceptions) using the Lie algebra method. Later, others used his method to create many new infinite single group series, these are called Li-Type Single-group. However, these groups do not fully include a limited single group. In addition to the unlimited single-group, there are 26 scattered single-group groups. As early as And, Matthieu knew five of them, and 21 were found successively in-years. By the beginning of 1980, all the 26 scattered single groups had been created. Is the classification of a single group complete? Most of the group theory experts think this is the case, but it is completely proof that it is still being published.
4. Topology
In the first 30 years of the 20th century, topology went through a long period of chaos. Many simultaneous adjustment theories emerged and some application fields were explored, it is particularly worth noting the relationship between topology and analysis. In 1925, Morse established a wide-range variational method, namely the Morse theory of the variational problem. This theory associates the critical point (singularity) index with the number of Betis. In 1931, D. Ram proved the d. Ram formula to associate the differential form with the same tone. At this time, due to the development of abstract algebra, under the influence of A. E. NORT, the concept of the same group is formed, and the geometric structure is linked with the algebraic structure. By the end of the Second World War, Alan Berger and stinrod will discuss the public physics and chemistry with the same tone, thus ending the chaos of the same tone. Later, I discovered many generalized simultaneous tuning theories (such as K theory), which provided many powerful weapons for topology and even the entire mathematics.
One of the topics in topology is the homeme theory. pongalai has already put forward the concept of the basic group. Later, chech and Hu reevich successively proposed the concept of the same group. A group includes rich information about the topology space. However, it is a group that is extremely difficult to calculate. The one-sided progress often brings great impetus to topology and even the entire mathematics. For example, Bott uses the Morse theory to obtain the periodic theorem of typical group colons, which becomes a source of k theory.
A natural object of topology is a manifold, which can be seen as a piece of Euclidean Space bonded together. If these blocks are bonded together through linear ing, they become Piecewise Linear Manifold. If these blocks are bonded together through micro- ing, a differential manifold is obtained.
The main conjecture, a well-known conjecture that has not been solved for a long time, claims that the existence of any Piecewise Linear Manifold must be essentially the only triangular Division (that is, the linear bonding method ). Obviously, the primary conjecture is true for many manifold, but there are still counterexamples. In addition, in-, it is also proved that there is no piecewise linear structure in the topological manifold.
A differential manifold is an object with a wide range of applications. In 1956, milno discovered that the 7-dimensional spherical S7 of the differential manifold had different differential structures. This was a major achievement of topology, marking the birth of differential topology. Soon afterwards, we found that some Piecewise Linear Manifold had no differential structure, and in turn, any differential manifold was essentially the only piecewise linear structure that we knew long ago.
Another important conjecture is that the single-connected, directed, and closed three-dimensional manifold must be a three-dimensional sphere. So far, this conjecture has not been confirmed and has not been denied. However, in the broad sense, the equivalent of a 5-dimension conjecture was confirmed around 1960. In 1982, the four-dimensional conjecture was also proved.
In recent years, the singularity theory, the dynamic system (ordinary differential equations) theory, and the leaf structure theory have achieved great development. Tom started from the singularity theory and developed the mutation theory, which can explain many natural and social phenomena to varying degrees.
Iii. Impact of structured mathematics on classical mathematics
The algebra and Topological methods developed in the 20th century have played a significant role in promoting ancient disciplines. Structural mathematics plays a decisive role in transformation of branches such as algebra number theory, algebra, multiple complex variable function theory, abstract harmonic analysis, and large-scale differential ry, this greatly expands their scope. As a result, many classic problems have been broken through and even completely solved.
1. the geometry of the differential Manifold
The introduction of group concepts in composite topology has become an algebraic topology. In 1940s, the same tone theory reached the public, unified the basis of the same tone theory, and opened up the development path of the same tone theory in the broad sense. At the same time, the rise of the question has enriched the contents of topology and made topology an important tool for the development of mathematics. The introduction of the concept of fiber bundle and layer plays a decisive role. After 1950s, we made an important breakthrough in the study of manifold. In 1956, we found the unequal differential structure on the sphere, proved the generalized panggalai conjecture, and solved the primary conjecture, and developed a wide range of dynamic system theories. The Study of the differential manifold promotes the development of singularity theory, solves a series of topology problems related to the differential geometry, and develops the theory of the leaf structure.
2. Classical Analysis
The development of new disciplines provides an important tool for classical analysis, including the concept of fixed point theorem and topological degree. In particular, generalized function theory has greatly promoted the development of partial differential equations. On the differential manifold, the theory of huochi is promoted by the consideration of the differential operator. This theory combines the topological nature of the manifold with the analytical nature, it is deepened together with the riman-lohe theorem into the theory of Atya-Singh. The theory of Atya-Singh is the main driving force for the introduction of pseudo-Differential Operators. Pseudo-differential operators not only contain Linear Differential Operators, furthermore, it includes the Singular Integral Operators previously studied to systematize the theory of linear partial differential equations. This theory was later extended to the theory of Fourier Integral Operators.
3. Algebra
The switching ring theory lays a solid foundation for algebra and ry. From Van der Walden, Wui, zrisky to SEL and grodongdick, not only has abstract algebra and ry been developed, but also has solved a series of classical problems, in particular, the wide, medium, and dry condition solves the Singularity Elimination problem of the algebra cluster with feature 0, and establishes the frontier discipline of arithmetic algebra and ry, which leads to a series of important conjecture solutions. In 1974, Deline successfully proved Wei Yi's conjecture, which is the greatest achievement of the theory of indefinite equations. In 1983, faltines proved model's conjecture that this is one of the central problems of the lost graph ry. The centuries-old achievements of falls in 1994 prove the ferma theorem.
4. Algebraic Number Theory
At the end of the 19th century, Hilbert had sorted out the most important achievements of algebraic number theory in his number theory report (theory of the number field) and developed the concept of domain-like, A series of domain-based conjecture is given and many special cases are proved. These results and conjecture have become a guide to the development of the number theory of the half-leaf algebra in the 20th century. Such as the promotion of the Hilbert domain, the unique category of Alibaba Bell's expansion, and the spring dream of clonek will be gradually solved by Gao Mu zhenzhi and others in 1920. By 1927, aiting proved the General inverse law, and thus completed the theory of Abel's domain theory. From the 1930s s to the 1950s s, with the help of tools such as abstract algebra and homophone algebra, domain-like theories can be expressed with beautiful algebra theories and homophone theories, making them a shining pearl of the mathematical kingdom.
The domain theory is not only in the range of the original algebra field, but also in the single-variable algebra function field in the closed algebra field. In addition, henzer discovered the number of p-adic, and also had a corresponding "local domain" for various algebra Field Numbers. Accordingly, he established the Abel Extension Theory for various local domains, that is, the local domain theory. In 1960s, local domain theory can be expressed in a concise manner using the form group tool.
Afterwards, the domain theory developed towards the non-Alibaba Cloud domain theory. Here, the self-defense form, algebraic ry, group embedding theory, and the above-mentioned simultaneous adjustment are mixed. Ranz and others developed a set of systems called the ranz philosophy, which greatly affected the development of the entire mathematics.
Iv. classical mathematics
Many classic issues have made significant progress in the 20th century. Some important projects are listed below.
1. Analytical Number Theory
(1) riman Conjecture
(2) elementary proof of Prime Number Theorem
(3) The question of hualin and the conjecture of godbach
(4) density method and Screening Method
(5) triangle and Method
2. lost image approximation and Number Theory
(1) Solve the Problem of Hilbert 7th
(2) optimum forcing of the number of algebra
(3) Gaussian conjecture on the virtual quadratic field of Class 1
(4) katalan Equation
(5) ε (3) is an irrational number.
3. Single-variant function theory
(1) nai wanglinna Theory
(2) anthropomorphic ing
(3) bibbbach's Conjecture
4. Theory of Real-Time Variable Functions
Fourier series is a problem of convergence and divergence almost everywhere, as Lu Jin guessed.
5. Differential Equations and Variational Methods
(1) Minimal Surface and Prato Problems
(2) KdV Equation
(3) Existence and Uniqueness of Solutions for Linear Partial Differential Equations
Although most of them have nothing to do with structural mathematics, the progress of some of these problems still shows the influence of structural mathematics.
Of course, classical mathematics is not limited to the above several branches, but another active branch is probability theory.
Although probability theory has existed in the 300 s, by the beginning of the 20th century, people had only a vague understanding of probability, and there was no strict foundation for probability calculation. At that time, there were only some basic concepts of Classical probability and the original forms of the big number law and central limit theorem. At the beginning of the 20th century, we strictly proved the central limit theorem. In 1909, polar came up with a powerful law of numbers and began the study of Markov chains. By 1920s, we established the necessary conditions for the establishment of the law of big numbers and the central limit theorem, which can be said to be the final completion of Classical probability theory. However, the understanding of probability is quite different in this period, and there are also different views on the mathematical basis of probability. polar consciously builds the probability theory on the basis of the measurement theory, establishes the probability method of the number set, fills the gap between the classical probability and the geometric probability, and the probability theory has a reliable mathematical foundation. In 1933, Kolmogorov put probability theory into the public, and probability theory became an independent discipline. From 1920s to 1940s, it was the heroic age of probability theory. During this period, it formed the French school represented by Levi, the former Soviet school represented by cormorgorov and sinchin, and the American school later, in this period, we studied the Limit Law of independent random variables and the promotion of the big number Law and the central limit theorem in the case of related random variables. The most important aspect is the random process. The most typical example of a random process is the Brown Movement. Based on Einstein's physical interpretation in 1905, N. inner first establishes a theoretical model of the Brown motion from the mathematical perspective. Afterwards, Levi studies the Brown motion from the perspective of Markov process and proposes a strongly Markov property that assumes that the future has nothing to do with the past. Later, we found that the transfer probability of the Markov process satisfies the differential integral equation. After the Second World War, probability theory developed important branches such as random process and random analysis, and played a major role in theory and practice. For example, in the popular universal mathematics, the random differential equation plays a decisive role.
In the 20th century, applied mathematics and computational mathematics have also achieved great development. In addition to classical Applied Mathematics, many new fields have emerged in the 20th century, especially statistical mathematics, operational research, control theory, and Computer-Related Computer Science and computational mathematics.
The Application of probability theory is mathematical statistics, which is derived from the optimization algorithm. In the early 20th century, Pearson's theory of structural correlation established the basis of biometrics. He introduced distribution, opened up the parameter test theory, and later gosett opened up the small sample test method, which is based on classical probability. In 1920s, Fisher's series of theoretical and practical activities promoted the development of mathematical statistics. His main contribution was the hypothesis test and experimental design. He also developed the analysis of variance analysis methods. His mathematics was not strict enough. Later on the basis of the theory of probability, Neman and others laid the foundation for Statistical Hypothesis testing. Experimental Design is a widely used method. It is closely related to the combination theory, especially the existence of the orthogonal Latin Square and the block design theory, which have become an independent branch of the combination theory. Another development of mathematical statistics is the statistical judgment function theory created by Wald. in the Second World War, he developed the sequential analysis method, which has great practical value.
The Application of mathematics in the 20th century not only continued to develop deeply in physical science, but also expanded to biological science, economic science, management science and other aspects. After 1940s, with the development of computer science, the application of mathematics and computational mathematics developed closely, solving a series of important problems. It applies not only basic mathematics, but also emerging abstract disciplines, such as topology, abstract algebra, and functional analysis. In turn, the application of mathematics also promotes the development of pure mathematics, and even directly promotes the solution of pure mathematical problems, such as the proof of four-dimensional pangalai conjecture caused by Yang zhenning's one Mills Theory in 1982. The most important branch of the operational science developed during the Second World War is the planning theory, especially the linear programming theory, which is closely related to the complexity of algorithms. It is also applied in many aspects. In 1948, a complete set of information theories were developed. The coding problem is closely related to the problem of algebra. Control Theory, abstract algebra, functional analysis, random process theory, and even differential and algebraic geometry are generated due to the control problems of satellites and rockets. In 1958, pontriyajin and others proposed the necessary conditions for optimal control, and formed the optimal control theory of the centralized parameter system. In 1960, Kalman and others proposed a Recursive Filtering algorithm, which is suitable for computer computing and is easy to use. The development of the game theory was earlier. In 1944, the book "game theory and economic behavior" by Feng nuoman and others was published to summarize the research on the countermeasure model of the predecessors. It defines the countermeasure with a public concept, it laid the foundation for this science. At the same time, they have also developed a direction of mathematical economics.
With the development of Mathematics Applications, the requirements for computers are getting higher and higher. A large number of numerical calculations make people pay more attention to the research of numerical calculation methods. In 1947, Feng nuoman and others published the "inverse of the values of the high-order matrix", marking the birth of the field of numerical analysis. The most common method for numerical analysis is to solve linear equations. In addition to Gaussian elimination, iterative methods are also developed, and a wide range of sparse matrices and generalized inverse concepts are studied. To understand partial differential equations, common difference methods, and the finite element method that laid the foundation of kulang et al. in 1920s, this method is the most widely used method. There are also splines, Fast Fourier transformation, and pure shape methods for linear programming and their improvements. In 1960s, with the practice of computing mathematics, the branch of computing complexity emerged, and it made a quantitative evaluation of the algorithm.
At the beginning of the 20th century, most math jobs were done in Germany and France. The French math leader was panggalai, and later. polar, leberger, Adama and others have important international influences on function theory. German mathematics is mainly based on the geding root school headed by Hilbert. The research of Hilbert has been affecting the 1930s s, and there are many small centers in Germany, active mathematical research is being conducted in areas such as Berlin and Hamburg. Before Hitler came to power, Germany had been in the world's leading position. After the First World War, A. E. NORT's school promoted the birth of abstract algebra and the algebra of topology. At the same time, in France, only. Cart studied the isolation of Li Qun and differential geometry. After the Second World War, French mathematics with the burbaki school as the core occupies a dominant position in the world, especially playing a decisive role in the development of structural mathematics.
In the 19th century, although there were some famous mathematicians in Britain, they fell behind the continental Europe. For a long time, the development of mathematics has stagnated. At the end of the 19th century, Yang and his wife went to gdinggen to introduce new mathematics to the UK. At the beginning of the 20th century, he and Li terwood began to make international contributions in terms of classical analysis. After 1930s, British mathematicians began to make outstanding contributions in topology, algebra, and abstract algebra.
After 1870s, after Italian mathematicians went to Germany and France to study, their mathematics had a huge development. From the end of the 19th century to the beginning of the 20th century, many Italian mathematicians made contributions to the first stream in terms of differential ry, algebraic ry, functional analysis, and real-variable function theory. Italian mathematics tends to decline from 1930s to 1950s. After that, it is revived and has made many contributions in various fields. The most prominent part is in analysis, especially partial differential equations.
In the 20th century, a series of national schools emerged. American mathematicians first learned from Germany and France and created some excellent mathematicians, such as N. Vina and G. D. berkhoff. After 1930s, a large number of European mathematicians moved to the United States, making the United States an important mathematical power after the Second World War. Another big mathematical country, the former Soviet Union, was headed by lujin's Moscow school. In 1920s, many people went to Germany and other places to study abroad. A group of excellent topology scholars, mathematicians and analysts appeared. The original Soviet Union's probability theory is particularly outstanding. After that, some emerging disciplines were once lagging behind and have been restored and revitalized since 1960s, forming a complete class of mathematical system. Since the First World War, Polish mathematics has formed its own school of thought. They focus on collection theory, logic, topology, functional analysis, and real analysis, and are in the leading position in these fields. However, in the Second World War, more than half of mathematicians were killed, causing a serious damage to Polish mathematics. Since the end of the 19th century, mathematics in Japan began to learn from Europe. At the beginning of the 20th century, mathematicians such as Kochi have emerged. After that, Japanese mathematics has gradually formed a complete mathematical system. After the Second World War, many world-class mathematicians emerged. The Nordic countries have produced a group of important mathematicians, but their interest is mostly focused on classical mathematics. Hungary provides the world with a large number of excellent mathematicians, such as RISS and von noiman. After the Second World War, there were some important mathematicians in Latin America, India and other countries.
Before the Second World War, exchanges between mathematicians began to become active. Apart from four years of international mathematician conferences, regional and professional conferences were also held. After the Second World War, due to the improvement of transportation tools, various conferences emerged one after another, greatly promoting the internationalization of mathematics. The mathematics publications in the 20th century also increased exponentially. The number of mathematical papers published each year increased from about 1900 in 1500 to 1980 to 40 thousand in 50 thousand, this large amount of literature makes it difficult for mathematicians to master, so some new digest magazines are gradually generated. Mathematicians are increasingly communicating over words rather than reading documents and books.
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