The dimension of algebraic family V with respect to base field K can be defined as the transcendental times of rational function field of V on K. One-dimensional algebraic family is called algebraic curve and two-dimensional algebraic family is called algebraic surface.
The simplest example of an algebraic cluster is an algebraic curve on a plane. For example, the famous Fermat conjecture (also known as Fermat's Last Theorem) can be summed up as the following questions: On the plane, the equation
The curve defined by algebraic geometry (called Fermat curve) has no point whose coordinates are all non-zero rational numbers when n≥3.
On the other hand, the following homogeneous equation
Algebraic geometry defines a curve in the projective space over the complex number field. This is an elliptic curve.
The study of algebraic clusters is usually divided into local and global aspects. The local research mainly discusses the singularity in algebraic clusters and the properties of algebraic clusters around the singularity by commutative algebra.
As an example of singularity, we can check the origin (0,0) in the plane curve defined by equation xy. This is a difference.
Algebraic clusters without singularity are called nonsingular algebraic clusters. 1964, the mathematician Nakatomi proved the singularity elimination theorem when the characteristic of the base field K is 0: any algebraic cluster is an image of a nonsingular algebraic cluster under a birational mapping.
The mapping from one algebraic cluster V 1 to another algebraic cluster V2 is called bi-rational mapping, if it induces isomorphism between rational function domains. Two algebraic clusters V 1 and V2 are called doubly rational equivalence, if one of the dense open sets in V 1 is isomorphic to that of V2. This condition is equivalent to the isomorphism of rational function fields of V 1 and V2. Because of this equivalence relation, the classification of algebraic clusters can often be attributed to the classification of doubly rational equivalence classes of algebraic clusters.
At present, the focus of algebraic geometry research is the whole problem, mainly the classification of algebraic clusters and the properties of subgroups in a given algebraic cluster. Homology algebra plays a key role in this kind of research.
The classification theory in algebraic geometry is established as follows: for each related classification object (such classification object can be a certain kind of algebraic cluster, such as nonsingular projective algebraic curve, or a doubly rational equivalent class of related algebraic cluster), people can find a set of corresponding integers, which are called its numerical invariants. For example, in the case of projective algebraic clusters, the dimension of homology space in each order is numerically invariant. Then, we try to establish a natural algebraic structure on the set of all classified objects with the same numerical invariants, which is called their parameter clusters, so that when the points in the parameter clusters change in an algebraic structure, the corresponding classified objects also change in the corresponding algebraic structure. At present, only some algebraic curves and surfaces and a few special high-dimensional algebraic clusters have established a relatively complete classification theory. The most in-depth study is the classification of algebraic curves and Abelian clusters.
Closely related to the sub-cluster problem is the famous Hodge conjecture: Let X be a nonsingular projective algebraic cluster over a complex number field, and P be a positive integer with a dimension less than X ... then any (p, p)-type homology class in X has algebraic representation.
1On April 26th, 935, the famous scientist Einstein said at the memorial service for Nott: "According to the judgment of modern authoritative mathematicians, Ms. Nott is the most important and creative mathematical genius since women began to receive higher education. In the field of algebra, the most talented mathematicians have been busy for centuries. She found a set of methods, and the growth of the current generation of young mathematicians proved its great significance. In this way, pure mathematics becomes a poem of logical concepts.
Emmynother (1882-1935),1882 was born in a Jewish family in the German university town of Ireland on March 23rd, and his father was Max Nott (1844- 192655).
My brother FritzNoether (1884 ~) is also a mathematician. He first worked as a professor at Blythe Institute of Labor Studies in Germany. 1935 After being persecuted by the Nazis, he fled to the Soviet Union and became a professor at Tomsk Institute of Mathematical Mechanics in Siberia. He was quickly put into prison and never heard from again.
When Nott 12 years old, she was a middle school student at Roots Girls' High School in Ireland. She is not interested in religion, piano, dance and other courses for girls, but only interested in language learning. After graduating from high school, she successfully passed the French and English teacher qualification examination in April. 1900. She had intended to be a teacher. In the autumn of the same year, she changed her mind and decided to study mathematics at the University of Ireland, where her father taught.
But at that time, German universities did not allow women to register, so they could only be audited and pay tuition fees. Only in rare cases can they get the consent of the lecturer and take the exam to get a diploma. Luckily, Nott passed the exam on July 1903. That winter, she came to the University of G? ttingen and was directly inspired by lectures given by famous mathematicians such as Hilbert, Klein and Minkowski. 1904 german universities were reorganized to allow female students to register. In June of that year, she officially returned to Ireland to register and study. By the end of 1907, she had passed the doctoral examination. Her doctoral thesis is entitled "Trivariate Biquadratic Invariant Complete Systems" and her tutor is Gordan (1837 ~ 657).
Gordan, a colleague and best friend of Nott's father, had a great influence on Nott's early life. Nott's doctoral thesis completely inherits the characteristics of Gordan's work, and is full of Gordan's formula and symbolic calculus. Later, although Nott left Gordan's research direction, she always had deep respect for her tutor, and a portrait of Gordan hung in her study. Godan died in 19 12, and his successors were Schmidt and Fisher. Under the guidance of Fisher, Nott gradually realized the transformation from Gordan's formal concept to Hilbert's research method. In this sense, Fisher's influence on Nott's academic development may be deeper than that of Gordan.
19 15, Klein and Hilbert of the University of G? ttingen invited Nott to go to G? ttingen. They were keen on the study of relativity at that time, and Nott's strength in invariant theory was very helpful to their research. 19 16, Nott left Ireland and settled in G? ttingen. Hilbert wanted to help her get the teaching qualification of the University of G? ttingen, but at that time, a linguistics professor and a history professor in the philosophy department of the University of G? ttingen strongly opposed it because Nott was a woman. Hilbert said angrily at the school affairs meeting: "gentlemen, I don't understand why the gender of the candidate is the reason to prevent her from obtaining the qualification as a lecturer." After all, we are a university here, not a bathhouse. " Perhaps because of this sentence, his opponent was even more angry, and Nott was still not allowed to pass.
But she still gave lectures to the students on the platform in G? ttingen, but in the name of Hilbert. After the end of World War I, the German Republic was founded and the situation changed. Nott became a lecturer on 19 19. From 1922 to 1933, she obtained the position of "supernumerary associate professor", which is an unpaid title. Just because she teaches algebra, she gets a little salary from the tuition fees paid by the students. In this difficult situation, under the influence of Hilbert and Klein's theory of relativity, Nott published two important papers in 19 18. One is to change the differential invariants commonly used in Riemannian geometry and general relativity into algebraic invariants, and the other is to link conservation laws with invariants in physics, which is called "Nott Theorem".
After 1920, Nott began to establish his own "abstract algebra". Starting from the similar phenomena in different fields, she abstracted and axiomatized different objects, and then unified them to reach a general theory, which can also deal with the particularity of different fields. Nott's theory is the system theory of "ring" and "ideal" in modern mathematics, which was completed in 1926. It is generally believed that the time of abstract algebra is 1926. Since then, the research object of algebra has changed from studying the calculation and distribution of the roots of algebraic equations to studying the algebraic operation rules and various algebraic structures of numbers, characters and more general elements, thus completing the essential transformation from classical algebra to abstract algebra. Nott is one of the founders of abstract algebra. Nott's academic papers are only over 40, and her great influence on the development of abstract algebra is not entirely from her papers, but more importantly from her contacts, exchanges, cooperation and lectures with colleagues and students. Her lecture skills are not brilliant, and she is in a hurry and incoherent. But she often describes her new ideas in detail, which are full of profound philosophy and extraordinary creation. She loves her students very much and has formed a bustling "family" around her. These students are called "Nott's children". Among them, a dozen students later became famous mathematicians. 1928 At the International Congress of Mathematicians held in Bologna, Italy, Nott was invited to give a 30-minute group report. At the International Congress of Mathematicians held in Zurich on 1932, Nott gave a one-hour lecture. Her report was praised by many mathematicians and won a high international reputation. Some elderly mathematicians saw with their own eyes the problems that they could not solve with the old calculation method, while Nott solved them beautifully and simply with abstract algebra, which had to be convincing. In the same year, because of her outstanding achievements in algebra, Nott and Aiding both won the Ackerman Sebner Prize. However, just a few weeks after the press conference, bad luck came. 1933 65438+ In October, Hitler came to power and persecuted Jews crazily. On April 26 of that year, the local newspaper published a notice that six Jewish professors at G? ttingen University were ordered to leave the school, one of whom was Nott. At the moment, Nott was deprived of her low-paid position at the University of G? ttingen, and she was almost cornered. At first, she wanted to go to the former Soviet Union. Because in the winter from 1928 to 1929, she visited Moscow University, where she taught abstract algebra and directed a seminar on algebraic geometry, which had a good influence on mathematics and mathematicians in the former Soviet Union and made a friendship with Aleksandrov, a famous mathematician in the former Soviet Union. Aleksandrov immediately welcomed Nott to teach at Moscow University, but failed to do so for various reasons. Later, with the introduction and help of the famous mathematician Weil,1September 1933, Nott was able to move to the United States, teach at Brinmar Women's College in the United States, and work at Princeton Institute for Advanced Studies.
During his stay in America, Nott went to Princeton to give lectures every week. Professor Quinn, who was listening to her lecture at that time, recalled that Nott was short, slightly fat, with dark skin and short black hair with a few strands of gray silk. She wears a pair of thick short-sighted glasses and gives lectures in incoherent English. She likes walking and often goes hiking with her students. On the way, she was always so absorbed in talking about math that the students had to protect her safety, regardless of pedestrians and vehicles. In Nott's life, perhaps he has never received such respect, sympathy and friendship as he did at Brinmar College and Princeton Institute for Advanced Studies. However, she still misses her motherland and G? ttingen. 1934 In the summer, she returned to G? ttingen, and she was sincerely pleased to see that Hasse was still trying to rebuild the glorious and long-standing mathematical tradition in G? ttingen.
1935 In the spring, when Nott returned to the United States, the doctor found that she had been riddled with cancer, and the tumor had caused serious harm to her body. Only surgery can save her life. After the operation, her condition improved once, and everyone expected her to recover. Unexpectedly, I got complications from the operation.
/kloc-in April of 0/4, this great female mathematician who was unmarried for life and devoted all her energy to her beloved mathematics career died at the age of 53. On April 26th, Brinmar College held a memorial service for Nott, Einstein wrote an obituary for her, and Weil wrote a long eulogy for her, deeply remembering her life, work and personality:
She used to be an energetic model,
With her firm emotions and courage to live,
Standing firmly on our planet,
So everyone is unprepared for this.
She is at the peak of her mathematical creativity.
Her far-reaching imagination,
The skills accumulated by her long experience,
Has reached a perfect balance.
She enthusiastically began to study new problems. Now that it's all suddenly over,
Her work was suddenly interrupted.
Fell into a dark grave,
Beautiful, kind, kind,
They all went quietly;
Smart, witty, brave,
They all went quietly;
I know, but I'll never recognize it.
I will not obey.
Algebra Geometry Our memory of her scientific work and her personality will never disappear soon. She is a great mathematician, and I firmly believe that she is also one of the greatest women in history.
The origin of developing algebraic geometry naturally begins with the study of algebraic curves on the plane. For a plane curve, the first numerical invariant that people notice is its degree, that is, the degree of the equation that defines this curve. Since the first and second order curves are rational curves (i.e. curves isomorphic to straight lines in algebraic geometry sense), it is generally believed that the study of algebraic geometry began with the study of cubic or more plane curves in the first half of19th century (the algebraic clusters studied in the early days were all defined on complex fields). For example, N.H. Abel found the double periodicity of elliptic functions in the study of elliptic integrals from 1827 to 1829, thus laying a theoretical foundation for elliptic curves (both of which can be expressed as cubic curves on a plane). On the other hand, C.G.J Jacoby considered the inverse function of elliptic integral, and his work is the basis of many important concepts in algebraic geometry (such as Jacobian cluster of curves, θ function, etc.). ).
B Riemann introduced and developed algebraic function theory in 1857, which made a key breakthrough in the study of algebraic curves. Riemann defined his function on some kind of multi-layer overlapping plane of complex plane, thus introducing the concept of so-called Riemann surface. In modern languages, compact Riemannian surfaces correspond to abstract projective algebraic curves one by one. Using this concept, Riemann defines one of the most important numerical invariants of algebraic curves: genus. This is also the first absolute invariant in the history of algebraic geometry (that is, an invariant that does not depend on the embedding of algebraic clusters in space). For the first time, Riemann also considered the parameter cluster problem of the doubly rational equivalence class of all Riemannian surfaces with genus G, and found that the dimension of this parameter cluster should be 3g-3, although Riemann failed to strictly prove its existence.
Riemann also proved Riemann inequality by analytical method: l(D)≥d(D)-g+ 1, where D is the divisor on a given Riemann surface. Then his student G. Roche added a term to this inequality to make it an equation. This equation is the original form of the famous Riemann-Roche theorem of F. Hirzebruch and A. Grotendick (see algebraic function field). Algebraic geometry-content After Riemann, German mathematician M Nott and others obtained many profound properties of algebraic curves by geometric methods. Nott also studied the properties of algebraic surfaces. His achievements laid the foundation for the work of Italian school in the future.
From the end of 19, the Italian school represented by G. Castelnovo, F. enriquez, F. Seville and H. Poincare, (C.-)? French school represented by piccard and Lefschetz. They have done a lot of very important work in the classification of low-dimensional algebraic clusters in complex fields, especially the classification theory of algebraic surfaces, which is considered as one of the most beautiful theories in algebraic geometry. However, due to the lack of strict theoretical basis in the early research of algebraic geometry, there are many loopholes and mistakes in these works, some of which have not been remedied until now.
One of the most important advances in algebraic geometry since the 20th century is to establish its theoretical basis in the most general situation. In 1930s, O. Za Riski and B L Van de Walden first introduced the method of commutative algebra into the study of algebraic geometry. On this basis, in the 1940s, awei established the algebraic geometry theory on abstract fields by using the method of abstract algebra, and then successfully proved that the zeta function on the algebraic curve has properties similar to Riemann conjecture when k is a finite field by reconstructing the algebraic correspondence theory on abstract fields of Italian school. In the mid-1950s, the French mathematician J.P. Searle established the algebraic cluster theory on the concept of layers and the cohesion theory of cohesive layers, which laid the foundation for Grotendick to establish the probability theory later. The establishment of probability theory makes the research of algebraic geometry enter a new stage. The concept of probability is a generalization of algebraic clusters, which allows the coordinates of points to be selected in any commutative ring with identity elements and allows nilpotent elements to exist in the structural layer.
Another important significance of probability theory is to unify the arithmetic of algebraic geometry and algebraic number field in the same language, which makes it possible to apply a large number of concepts, methods and results in algebraic geometry to the study of algebraic number theory. Two typical examples of this application are as follows: ① P. Deligne extended Wei Yi's theorem on zeta function to any algebraic cluster over a finite field in 1973, and proved that the famous Wei Yi conjecture used Grotendick's probability theory. ② The G pseudo-proposition proves the Mo Deer conjecture in 1983. A direct corollary of this result is that the Fermat equation x+y= 1 has only a limited number of non-zero values when n≥4, thus making a major breakthrough in the study of Fermat conjecture.
On the other hand, great progress has been made in transcendental methods in algebraic geometry over complex fields since the 20th century, such as G.-W. Drumm's analytic cohomology theory, the application of W.V.D D. Hodge's harmonic integral theory, the deformation theory of squares and D. C. Spencer, and some important works of P. Griffith.
Zhou Weiliang made many important contributions to the development of algebraic geometry in the early 20th century. His concepts of circle, circle cluster and circle coordinate have played an important role in the development of many fields of algebraic geometry. He also proved the famous Zhou Theorem: If a compact complex analytic manifold is projective, it must be an algebraic cluster.
In the late 20th century, great progress has been made in the classification theory of low-dimensional algebraic clusters in classical complex fields. In the classification of algebraic curves, thanks to the work of D.B. Mountford and others, people now have a deep understanding of the algebraic curve parameter cluster Mg. In 1960s, Mountford applied Grotendick's probability theory to the classical invariants theory, thus establishing the geometric invariants theory and using it to prove the existence and quasi-projectivity of Mg. As we all know, Mg is an irreducible algebraic cluster. When g≥24, it is general. At present, we have begun to understand the properties of subalgebraic clusters of Mg.
The classification theory of algebraic surfaces has also made great progress. For example, in the mid-1960 s, Xiao Pingbangyan thoroughly understood the classification and properties of elliptic surfaces; In 1976, Qiu Chengtong and Yoichi Miyaoka simultaneously proved an important inequality of general algebraic surfaces: с娝 ≤3с2, where с娝 and с2 are Chen Shu of surfaces. At the same time, the classification of three-dimensional or high-dimensional algebraic clusters has begun to attract more and more interest.
Algebraic geometry has extensive relations with many branches of mathematics. In addition to the number theory mentioned above, there are analytic geometry, differential geometry, commutative algebra, algebraic groups, K theory, topology and so on. The development of algebraic geometry and these disciplines play a mutually promoting role. At the same time, as a theoretical discipline, the application prospect of algebraic geometry has also begun to attract people's attention, one of which is the application of algebraic geometry in cybernetics.
In recent years, people have widely used algebraic geometry tools in the latest superstring theory of modern particle physics, which indicates that ancient algebraic geometry will play an important role in the development of modern physics.