200 years later, Euclid listed some facts recognized by people as definitions and axioms, and used these definitions and axioms to study the properties of various figures by means of formal logic, thus establishing a set of geometric argumentation methods to demonstrate propositions and obtain theorems from axioms and definitions, forming a strict logical system-geometry.
After thousands of years of changes, people's research on shape has become more and more complicated, and at this time Hodge conjecture came into being.
The background of the birth of Hodge conjecture
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Century 70
Before 1980s, geometry and algebra had made great progress, but they were two independent disciplines. Descartes compared the geometric method and algebraic method at that time. He advocated simplifying geometric problems into algebraic problems, calculating and proving them by algebraic methods, and finally solving geometric problems. According to this idea, he founded what we now call analytic geometry.
Descartes' mathematical thought proves that if you abstract further, geometry is actually the same as algebra, and geometry can be transformed into algebraic equations, and algebraic equations can also be transformed into geometric figures.
If you want to know where a line passes through a specific circle, you can draw this shape geometrically, or just compare equations algebraically. Both methods will give the same answer.
/kloc-In the 9th century, mathematicians tried to popularize Descartes' method. They started with some algebraic equations and defined the solutions of these equations as "geometric" objects. Objects generated by algebraic equations in this way are called "algebraic clusters".
Algebraic clusters on spherical projective spaces
Therefore, algebraic clusters are the generalization of geometric images. Any geometric correspondence is an algebraic cluster, but many algebraic clusters cannot be visualized. However, just because a particular algebraic cluster cannot be visualized, you can't do (algebraic) geometry on it. Yes, but this is geometry without graphics.
After that, mathematicians soon discovered that more complex equations, even equations, can work together in various dimensions to produce amazing shapes.
Mathematicians have found a very practical way to obtain more complex shapes. The basic idea is to what extent we can form the shape of a given object by gluing together simple geometric building blocks with increasing sizes. This technology is very easy to use, so it can be popularized in many different ways.
Mathematicians hope that through this method, they can expand it step by step in different ways, and finally establish a set of powerful algebraic equations or/and geometric tools, so as to classify various complex objects into some specific simple geometric objects and their combinations. This makes mathematicians make great progress in classifying all kinds of objects encountered in research.
Unfortunately, in this extension, the geometric starting point of the program becomes blurred. In this extension process, the geometric starting point becomes blurred-simple geometric objects are combined from this; What is a combined program/sequence? Therefore, without any geometric explanation, it is necessary to add some "non-geometric" basic modules.
Based on this dilemma, in 1958, Professor Hodge, a British mathematician and chairman of the 3rd International Mathematical Congress, proposed that any Hodge class can be expressed as a rational linear (geometric component) combination of algebraic closed-chain classes on nonsingular complex projective algebraic cluster space.
What does this sentence mean? "Nonsingular projective algebraic cluster" refers to the "surface" of a smooth multidimensional object generated by the solution of an algebraic equation. Simply put, any geometric figure of any shape, no matter how complicated (as long as you can think of it), can be composed of a bunch of simple geometric figures.
The significance of Hodge conjecture
Since the birth of Galois Group Theory, modern mathematics tends to extract abstract understanding of the essence of things.
For more than a hundred years, mathematicians have been building deeper abstractions on the basis of abstractions, and each abstraction is farther away from our daily experience world. Taking group theory as an example, our general "addition, subtraction, multiplication and division" is abstracted into four algorithms.
Hodge conjecture is a difficult problem born under the extremely abstract system of modern mathematics. As a highly professional problem, the object it deals with is far from human intuition, so that it is difficult to judge whether the conjecture itself is right or wrong, and even the expression of the problem itself is seeking to establish a real * * * knowledge.
In other words, whether the expression of this question is rigorous and reasonable is still controversial in the field of mathematics. Some people even say that it should be more accurate to call irrelevant guesses.
The proof of Hodge conjecture will establish a basic connection between algebraic geometry, analysis and topology.
Progress in proving Hodge's conjecture
The American Mathematical Society published a book on the research progress of Hodge conjecture. At the beginning of its preface, there is a statement about Hodge's conjecture, which is described by this book as a "popular version" of this conjecture:
This book has been published twice. The second edition was published in 1999, and * * * has 368 pages, each of which is dense. Update according to known conditions. It lists 7 1 papers published from 1950 to 1996. These papers are only about one aspect of this conjecture, that is, the so-called Hodge conjecture on Abelian clusters. In the preface, the author of this book admits that even with this appendix, this comprehensive report is still incomplete and readers should refer to other materials.
That is to say, from 1958, the research progress of Hodge conjecture was almost zero, and the only breakthrough that was proved was solved by American mathematician Lefschetz in 1925 before Hodge conjecture was put forward. He proved an example of Hodge's conjecture.
Complex geometric figure
Compared with the famous Poincare conjecture and Goldbach conjecture. Hodge conjecture is the most difficult mathematical problem in the world. For more than half a century, mathematicians are still at a loss.
At present, two post-80s mathematicians in China, Xun Zhiwei and Zhang Wei, who graduated from Peking University Institute of Mathematical Sciences, have proved the higher-order Gan-Gross-Prasad conjecture in the function domain. The formulas discovered and proved by Zhang Wei and Yun Zhiwei are related to three of the seven Millennium problems (Hodge conjecture, Riemann hypothesis and BSD conjecture). In an interview with CCTV, Yun Zhiwei said: "Our equation is the connection between number theory and geometry. Geometry is related to Hodge conjecture in algebraic geometry, and number theory is related to Riemann zeta function in Riemann hypothesis. This equation itself can be regarded as some extensions under the framework of BSD conjecture. "
Yun Zhiwei, Zhang Wei
I hope that China mathematicians can make a breakthrough on this unsolved problem for thousands of years, and let mathematicians know where Hodge's conjecture will lead.