Algebraic geometry and blockchain
analytic geometry includes plane analytic geometry and solid analytic geometry. Plane analytic geometry establishes one-to-one correspondence between point and real number pair, and one-to-one correspondence between curve and equation through plane rectangular coordinate system, and uses algebraic method to study geometric problems, or uses geometric method to study algebraic problems. Since the 17th century, e to the development of navigation, astronomy, mechanics, military and proction, as well as the rapid development of elementary geometry and elementary algebra, analytic geometry has been established and widely used in various branches of mathematics. Before the founding of analytic geometry, geometry and algebra were two independent branches. The establishment of analytic geometry for the first time truly realizes the combination of geometric method and algebraic method, and unifies shape and number, which is a major breakthrough in the history of mathematical development. Descartes is the first decisive step in the development of variable mathematics. The establishment of analytic geometry plays an inestimable role in the birth of calculus.
After the appearance of analytic geometry, the idea of studying geometry with algebraic method has developed into another branch of geometry, which is algebraic geometry. The objects of algebraic geometry are plane algebraic curves, space algebraic curves and algebraic surfaces
the rise of algebraic geometry is mainly e to the solution of general polynomial equations, and the research on the space composed of the solutions of such equations, that is, the so-called algebraic cluster, is carried out. The starting point of analytic geometry is to introce a coordinate system to represent the position of a point. Similarly, for any kind of algebraic family, coordinates can also be introced. Therefore, the coordinate method has become a powerful tool for the study of algebraic geometry
the study of algebraic geometry began in the first half of the 19th century with the study of cubic or higher plane curves. For example, Abel found the double periodicity of elliptic function in the study of elliptic integral, which laid the foundation of elliptic curve theory< In 1857, Riemann introced and developed the theory of algebraic functions, 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 introced the concept of so-called Riemann surface. Using this concept, Riemann defined the genus, which is one of the most important numerical invariants of algebraic curves. This is also the first absolute invariant in the history of algebraic geometry. It is the first time to consider the parameter cluster of Bi rational equivalence classes of all Riemannian surfaces with the same genus g, and it is found that the dimension of this parameter cluster should be 3g-3, although Riemannian has not been able to prove its existence strictly
after Riemann, German mathematician nott et al. Obtained many profound properties of algebraic curves by geometric methods. Nott also studied the properties of algebraic surfaces. His achievements laid a foundation for the later work of the Italian school
since the end of the 19th century, the Italian school represented by castelnovo, Enriquez and severi and the French school represented by Poincare, Pika and Lefschetz have emerged. They have done a lot of important work on the classification of low dimensional algebraic families over complex fields, especially the classification theory of algebraic surfaces, which is considered to be one of the most beautiful theories in algebraic geometry. However, e to the lack of a strict theoretical basis in the early study of algebraic geometry, there are many loopholes and errors in these works, some of which have not been made up until now
one of the most important developments in algebraic geometry since the 20th century is the establishment of its theoretical basis in the most general case. In the 1930s, zarinski and van der Walden first introced the method of commutative algebra in the study of algebraic geometry. On this basis, wye established the algebraic geometry theory on abstract field by using the method of abstract algebra in the 1940s. Then in the mid-1950s, the French mathematician Searle established the theory of algebraic cluster on the concept of layer and the cohomology theory of condensed layer, which laid the foundation for grotendick to establish the theory of probability, His lecture notes "Fundamentals of algebraic geometry" (EGA, SGA, FGA) became the Bible in this field. The establishment of probability theory makes the study of algebraic geometry enter a new stage. The concept of probability type is a generalization of algebraic family. It allows the coordinates of points to be selected from any commutative ring with identity elements, and allows nilpotent elements to exist in the structure layer< In recent years, algebraic geometry tools have been widely used in the latest superstring theory of modern particle physics, which indicates that abstract algebraic geometry will play an important role in the development of modern physics
analytic geometry is the algebra of geometric problems and the geometry of algebraic problems
combination of number and shape.
A. Grothendieck
(1)
Alexandre Grothendieck was born on March 28, 1928 in a Jewish family in Berlin, Germany. His father was killed by the Nazis in World War II. After the end of the war, Grothendieck went to France to study mathematics. He successively studied from Dieudonne, a master of Bourbaki school, and Laurent Schwartz, a famous master of functional analysis. In his 20s, Grothendieck became an authority on the theory of topological vector space, which was very popular at that time. But since 1957, Grothendieck's research has mainly turned to algebraic geometry and homology algebra. In 1959, he became the president of the newly established Institute of higher Sciences in Paris. His work develops the homology method and layer theory of algebraic geometry of Leray and Serre to a new height. His scheme theory laid the foundation of modern algebraic geometry. Because of many of his pioneering work, algebraic geometry, an ancient branch of mathematics, radiates new vitality, and eventually leads to the complete proof of Weil's conjecture by designe, which is considered to be one of the most significant achievements of pure mathematics in the 20th century. Under the leadership of Grothendieck, the Institute of higher studies in Paris was recognized as the world center for algebraic geometry ring that period. He also won the fields prize, the highest international prize in Mathematics in 1966. Perhaps because of his wartime experience as a young man, Grothendieck is a radical pacifist who can give up his mathematical research for the sake of war. During the Vietnam War, he taught category theory to local scholars in Hanoi forest. In 1970, at the age of 42, at the peak of his research, he gave up mathematics completely and left the Institute of higher studies in Paris. He later taught at Montpellier University in France until he retired at the age of 60. He also said he would go to the Pyrenees in southwest Europe to be a hermit Buddhist. On his 60th birthday in 1988, Grothendieck unexpectedly declined the Royal Swedish Academy of Sciences' award of $250000 to him. The reason is that he thinks the money should be spent on young and promising mathematicians. Although Grothendieck has been far away from the academic circle for a long time, he is still recognized as one of the greatest and most influential mathematicians in modern times. The profound theoretical system of modern algebraic geometry created by him has brought about great changes, which can be felt in almost all the core branches of mathematics
if you open any textbook or monograph of modern algebraic geometry, you will often see such nouns as gross. Topology, gross. Cohomology, gross. Ring, etc. At this time, I always think of Grothendieck,
this great mathematician whom we admire most, maybe he is living in a small town in Europe, but the great wealth he left to mankind will undoubtedly be recorded in history forever
(2)
"for these" pure "mathematicians, the material world is only an illusion, and only the spiritual world is eternal. They only need a pencil and a few pieces of white paper to carve out a brilliant world in the ivory tower of pure mathematics with their own smart mind. " The 1960s was a time of considerable uneasiness. The idols of young students at this time were Mao Zedong and Che Guevara. They will wear red sleeve hoops, carry the image of Guevara, and go to the street to confront the armed police. At this time, it seems that university professors are not very obedient because they have more contact with students. For example, s Smale, an American mathematician and winner of the fields prize in 1966, has publicly attacked the hegemonic policy of the United States and the Soviet Union many times. Because of this, he was "looked after" by the CIA. In 1966, ring the International Congress of mathematicians in Moscow, the KGB simply "invited" him to a car for a period of time. But compared with Grothendieck, Smale's behavior is not too unusual
Bourbaki is a school founded by a group of young French mathematicians in the 1930s. Its first members graated from Ecole normal sup é Rieure), including A. Weil, H. Cartan and J. Dieudonn é、 C. Chevalley, J. delsarte et al. Grothendieck joined the school in its heyday. In addition to the masters of the older generation, the Bourbaki school at that time also had such talented young people as L. Schwartz and J. - P. Serre. Here, Grothendieck came into contact with the frontier of mathematics, and then grew into a new generation of mathematicians. Grothendieck first studied functional analysis, and he profoundly changed the face of this subject. Dieudonn é It is said that the work of Grothendieck, like that of S. Banach, leaves the strongest mark in functional analysis. However, the most important work of Grothendieck is algebraic geometry. Algebraic geometry studies the graphs represented by the solutions of algebraic equations (Systems). It has been nearly 400 years since R Descartes invented analytic geometry. In the 1930s, O. Zariski and B. L. van der Waerden introced commutative algebra into algebraic geometry. In the mid-1940s, Weil completely established algebraic geometry on the basis of abstract algebra, and put forward the famous Weil conjecture. Later, Kodaira, F. Hirzebruch, J. - P. Serre and others made great breakthroughs in this discipline. In 1950s and 1960s, Grothendieck revolutionized algebraic geometry thoroughly, published more than ten books, and established a set of grand and complete "probability theory". Grothendieck's work can be called the peak of algebraic geometry, and his work is known as "Grothendieck Bible". Grothendieck's theory is valuable. On the basis of probability theory, mathematicians have made one amazing achievement after another: Grothendieck gave the first algebraic proof of the famous Riemann Roch theorem
it also led to the following events:
in 1973, P. delegate proved Weil's conjecture (won the 1978 fields prize)
in 1983, G. falings proved the Mordell conjecture (won the 1986 fields prize)
in 1995, A. wiles proved Taniyama Shimura conjecture, and then solved Fermat's theorem with a history of more than 350 years; S last Therem) (1996 fields special award)
these achievements represent the highest level of contemporary mathematics, and can be brilliant for thousands of years
in the 20th century, there were many talents and fields awards in algebraic geometry, but there was only one God, Grothendieck. His series of monographs EGA is recognized as the Bible of algebraic geometry
Grothendieck is a complete anarchist and pacifist. He often makes his political propaganda to those who come to him to ask math questions. In the 1960s, he was employed as a professor of the Institute des Hautes etudes Sciences in France, but when he found out that this institution was funded by NATO, he resigned and went back to his hometown to work in agriculture. In 1970, l Pontrjagin, the blind mathematician of the Soviet Union, gave a report on "differential games" at the international mathematician Congress, in which he talked about the problem of tracking aircraft with missiles. Grothendieck angrily stepped onto the stage and grabbed the microphone to protest that he mentioned military affairs at the
Mathematics Conference. G hardy once said: "real mathematics has no influence on war,... It is a" harmless and innocent "profession.". Maybe that's why Grothendieck chose mathematics. However, Grothendieck was graally disappointed to find that mathematics was often used in military affairs, such as algebraic geometry, which he studied, was used to compile ciphers, and most of the mathematical researches were directly or indirectly supported by the military. This is obviously contrary to his ideal. So in 1970, he left his favorite career in mathematics and turned to disarmament activities and farm management. In the 1980s, he simply disappeared in this dirty world. Only a few of his friends knew his address, but they kept their mouths shut. So far, Grothendieck is still at a loss. Hermits have existed since ancient times, but such as Grothendieck, it is rare that they do not love glory and retire after success.
the second is a circle with the origin as the center and a as the radius
there are two points in the first circle connecting point (4,3) and the origin and extending the intersection. The distance between these two points and the origin is the maximum and minimum value of a, and the value of a is within this range
You have to ask yourself,
Introction
algebraic geometry is a branch of mathematics, which combines abstract algebra, especially commutative algebra, with geometry. It can be regarded as the study of the solution set of algebraic equation system. Algebraic geometry focuses on algebraic clusters. An algebraic family is the locus of a point determined by one or more algebraic equations of a space coordinate. For example, algebraic families in three-dimensional space are algebraic curves and algebraic surfaces. Algebraic geometry studies the geometric properties of general algebraic curves and surfaces
algebraic geometry has extensive relations with many branches of mathematics, such as complex analysis, number theory, analytic geometry, differential geometry, commutative algebra, algebraic group, topology, etc. The development of algebraic geometry and these subjects promote each other.