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The Hodge conjecture put forward by the British mathematician William Vallance Douglas Hodge in 1950 is undoubtedly the most difficult to understand of all the millennium puzzles. This is a highly professional question that only a few professional mathematicians can really understand. Here is Hodge's conjecture:

Every (certain type) harmonic differential form on a nonsingular projective algebraic family is a rational combination of cohomology classes of algebraic closed chains.

Do you find that you don't understand every technical term in this sentence? In the article about Birch and Swinnerton-Dell conjecture, I can also associate that conjecture with simple geometry, that is, the area of triangles.

There is not even a simple analogy for Hodge's conjecture. The Hodge conjecture illustrates most clearly that the nature of modern mathematics makes it almost impossible for ordinary people to understand most of it.

Over the past century, mathematicians have established new abstractions on top of old abstractions, and the objects being considered have become more abstract than new ones. Take Hodge's conjecture as an example, calculus plays a major role here, but it is not carried out on real numbers, or even in the plural, as many high school students have learned. This is calculus in a more general and abstract context.

For ordinary people, the difficulty of this question is the most interesting part of it. Having said that, I still want to try to explain what Hodge conjectured.

In the 17th century, the French philosopher Descartes algebraized the geometry, put the geometry in the Cartesian coordinate system, and then established their mathematical equations. Geometry studied by algebra is usually called algebraic geometry, also known as Cartesian geometry.

During the 19th century, mathematicians took the Cartesian method a step forward. Instead of using algebra as a tool to study geometric objects, they start with algebraic equations and define the solutions of these equations as "geometric" objects. But most of the equations do not correspond to the geometric objects we are familiar with. So it doesn't make sense to call them "geometric objects". In this way, the objects generated from algebraic equations are called "algebraic clusters" by mathematicians.

When defining algebraic clusters, mathematicians do not consider only one algebraic equation, but a system of equations (finite). In a system of equations consisting of two equations, each equation defines a geometric figure, then the cluster defined by the system of equations will be a common part of the two graphs.

Therefore, algebraic clusters are a generalization of geometric objects. Any geometric object is an algebraic cluster, but there are many algebraic clusters that cannot be visible. However, it is not because a particular algebraic cluster cannot be visualized that we cannot study it.

Now, we can look at a technical term ∶ in Hodge's conjecture, a family of nonsingular projective algebras, which is simply a smooth multidimensional "surface" generated by the solution of an algebraic equation. It's like a sphere by solving algebraic equations.

And get a smooth two-dimensional surface.

This conjecture makes an assertion against the "harmonic differential form" on the "surface". A harmonic differential form is a solution of a very important partial differential equation (called Laplace equation). It arises not only from physics, but also from the study of complex functions.

Calculus studied in college is usually on a two-dimensional plane. But with a little effort, it can be extended to other surfaces, such as spheres. With a little more effort, calculus can be extended to a variety of more general clusters. Hodge conjecture involves calculus which is extended to a family of nonsingular projective algebras. It makes an assertion about a certain type of abstract object, which we call H object. If we start with a certain type of cluster and do some calculus on it, we will produce H object.

When we use calculus to define an object, the defined object is not necessarily "geometric" in any sense. Hodge guessed that the H object was an exception to that sentence. Although they may not be geometric objects themselves, they can be built from geometric objects in a fairly simple way. In the term of this conjecture, the H object is a rational combination of cohomology classes of algebraically closed chains. That is to say, any H object can be constructed from geometric objects in a purely algebraic way.

So, you can think of Hodge's conjecture as ∶.

By using calculus on clusters, we have created a class of objects (H objects) that not only dash our hopes of visualizing them, but also make it impossible for us to describe them in an algebraic way. However, these objects can be constructed in an algebraic way from "objects that can be described by algebra".

The function of Hodge conjecture is to provide experts with a powerful mathematical structure that can be used to analyze H objects. This is very important in many modern mathematics. Mathematicians are constantly looking for new structures on objects, or looking for connections from one field to another, so that they can transform methods from one field to another. apply it to another field.

Now, we have an overall understanding of Hodge's conjecture. Here is another way to understand the problem.

We can put forward the Hodge conjecture from the integral along the generalized path on the algebraic cluster. Since the value of this integral can be kept constant by deforming the path, you can assume that the integral is defined on the path class. Hodge conjecture suggests that if some of these integrals are zero, then there is a path in this path class that can be described by a polynomial equation.

Let's first blow the significance of Hodge's conjecture: the proof of Hodge's conjecture will establish a basic relationship between algebraic geometry, analysis and topology.

Until now, Hodge's conjecture is still a conjecture. In 1991, the American Mathematical Society published a book documenting some of the research on Hodge's conjecture and listing 71 papers published between 1950 and 1996. These papers are only one aspect of this conjecture, the so-called Hodge conjecture on the Abelian cluster.

The following is a statement of Hodge's conjecture at the beginning of the American Mathematical Society's book.

William Hodge knows next to nothing about such a good mathematician as Hodge. He was born in Edinburgh, Scotland in 1903. He finished his studies first in Edinburgh and then in Cambridge. At the age of 33, he was appointed professor at Cambridge University in 1936 until he retired in 1970.

He is a leading figure in the development of connections between geometry, analysis, and topology. Mathematicians still remember him mainly because (except for his conjecture) his harmonic integral theory.

He was elected to the Royal Society of London in 1938 and was awarded the Royal Medal in 1957 for his outstanding contributions to algebraic geometry. From 1947 to 1949, he served as president of the Mathematical Society of London, where he won the Bellick Prize in 1952. In 1974, the Royal Society rewarded him again, this time with the Copley Medal for his groundbreaking work in algebraic geometry, especially his theory of harmonic integrals. Hodge died in 1975 at the age of 72.

For most of his career, Hodge devoted himself to the development of algebraic geometry theory, one of which is now called "Hodge theory". His conjecture is derived from algebraic geometry. Hodge announced this conjecture in his speech at the International Congress of mathematicians held in Cambridge, England in 1950.

When plural encountered the mathematics of fluid during the Renaissance, mathematicians talked about an incredible thing ∶ introduced a number in algebra, its square is-1. This number is represented by I and becomes the basis of the plural.

Although it is difficult for human beings to accept that the square of a number is negative, complex numbers have a set of effective arithmetic operations, just like the usual real arithmetic operations, and we can also solve polynomial equations containing complex numbers. The way to overcome the counterintuitive plural is to realize that they can be drawn as "points" on an ordinary two-dimensional plane.

In real numbers, we can correspond each real number r to its opposite number-r. Drawn on a straight line ("real line"), each number is paired by a point on the other side of the origin and at the same distance from the origin. This specific pairing plays an important role in the arithmetic operation of real numbers.

A complex number can be drawn as a point on a complex plane. For these numbers, the similar pairing between x+iy and-x-iy is a reflection about the origin. But there is another kind of pairing of plural numbers, which plays an important role in the arithmetic operation of plural numbers. The second kind of pairing is to correspond each complex x+iy to its conjugate complex x-iy. Complex conjugate pairing is about the reflection of the real axis (that is, the x axis) on the complex plane.

By the 19th century, the basic theory of complex number has been successfully studied, and complex number is generally regarded as the standard number system of mainstream mathematics. Moreover, mathematicians began to develop a theory of extending calculus to complex functions, resulting in complex analysis.

The two main characters in the early study of complex analysis are Riemann and Cauchy. They associate complex functions with physics. They started thinking like this, ∶.

If f (z) is a complex-valued function of a complex variable z, then we can write the value of f (z) as f (z) = u (z) + iv (z), where u (z) and v (z) are real numbers. This gives two new functions u and v, both of which are real-valued functions of the complex variable z.

The two mathematicians found that if the complex function f has a well-defined derivative (in modern terms, if the function f is analytic), then its real part u and imaginary part v must satisfy two partial differential equations:

These equations are familiar to physicists. They are Laplace equations, which play an important role in gravity theory, electromagnetic theory and fluid mechanics. A solution of the Laplace equation is called a harmonic function. The discovery of the close relationship between the calculus of complex functions and the Laplace equation has led to great progress in mathematical physics.

A major development in the theory of complex functions is the invention of Riemannian surfaces. There are some functions that are friendly to real numbers, but when independent variables or function values are allowed to be complex, the result does not look like a normal function at all, because more than one function value can be derived from an argument. Square root functions and logarithms are two examples.

For real numbers, any positive real number has two square roots, but because one of them is positive and the other is negative, as long as the positive root is specified, the problem can be eliminated. But when the root is plural, there is no natural and effective way to choose between the two roots. Riemann suggested that the best way to deal with these "multi-valued functions" (they are not real functions at all) is to think of them as single-valued functions (that is, real functions) defined on a multi-layered surface.

Riemannian surface has a more complex topological structure than complex plane. One way to look at them is to think of them as a spiral ladder configuration of a complex plane.

In the early 20th century, mathematicians extended the idea of Riemannian surface to a highly abstract concept-complex manifold, that is, a multi-dimensional simulation of Riemannian surface with a complex topology. Such a manifold has a structure that ensures that the concept of complex analytic functions is meaningful. In particular, it is possible to define the so-called differential form, that is, the product of extending the differential df of the function f in ordinary (real) calculus to the multi-dimensional case.

Some differential forms can be divided into different types with some common key characteristics, so they are called cohomology classes. These cohomology classes are exactly what Hodge conjecture says.

To understand the concept of cohomology class requires a series of advanced professional mathematical knowledge. Here is a very brief summary of ∶

First of all, we need to know that there is a specific operation in differential form, called external derivative. The external differential itself is a kind of differential.

A differential form is appropriate if it is the outer derivative of another differential form.

If the external derivative of a differential form is zero, the differential form is said to be closed.

If the difference between two closed differential forms is appropriate, they are said to be cohomology.

Therefore, the elements of the cohomology class are closed differential forms. Appropriateness is the "similarity" property shared by elements in the same cohomology class. Note that the definition of the cohomology class depends heavily on the concept from calculus.

Cohomology classes define useful topological invariants, which capture important aspects of basic complex manifolds. By obtaining the concept of cohomology class (in closed differential form), we can return to the concepts of algebraic geometry and algebraic clusters. A complex algebraic family is a multi-dimensional "surface" defined by the complex solution of a system of algebraic equations.

If the solution of the system of equations defining a complex algebraic family depends only on the ratio of the relevant numbers, mathematicians call the complex algebraic family projective.

If a cluster is smooth as a "surface", they call the cluster nonsingular.

Therefore, a nonsingular projective complex algebraic family is a special type of complex manifold.

Hodge realized that he could apply the methods from analysis to these algebraic manifolds. In particular, he realized that the rational cohomology class of differential form generated by a family of nonsingular projective complex algebras can be regarded as the solution of Laplace equation.

Hodge's observations make it possible to write such a class as a sum of some special components, which are called harmonic forms. They are the solutions of the Laplace equation which can be defined by p complex variables and Q conjugate complex variables. Moreover, every (p-dimensional) algebraic cohomology class gives a (p _ (p)) form.

In his report to the International Congress of mathematicians in 1950, Hodge proposed that for nonsingular projective complex algebraic families, the last property mentioned above may completely characterize the algebraic cohomology class. In other words, each harmonic form is a rational combination of closed algebraic forms (that is, it can be constructed in an algebraic-- that is, without calculus-- methods).

This is how Hodge's conjecture was born. But is this conjecture correct? Nobody knows.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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