In 1900, David Hilbert, in a lecture entitled "Problems of Mathematics", proposed 23 public problems [18]. The 15th question, which focuses on the 19th-century theory of counting geometry and intersection, is entitled "Establishing a Rigorous Foundation for Schubert's Counting Calculus".
The Springer Encyclopedia of Mathematics [28] recalls that "clarifying the Schubert calculus is an important topic in algebraic geometry of the 20th century". The purpose of this paper is to trace the origins of modern mathematical geometry by taking the stories of geometry pioneers as a glimpse and the answer to question 15 as the context.
Written by | Duan Haibao (Researcher, Academy of Mathematics and Systems Science, Chinese Academy of Sciences), Zhao Xuezhi (Professor, School of Mathematical Sciences, Capital Normal University)
01 A Brief History of Intersection Theory
In the 3rd century B.C., the Greek geometer Apollonius (262-190 B.C.) proved the following results in his article "Tangencies". is the first example of counting geometry:
Theorem 1.1: For 3 circles in a general position in a plane, exactly 8 circles are tangent to them.
Schematic diagram of Apollonius' theorem
The first proof of this theorem has been lost to the dust of history, and posterity can only learn from a account by Pappus in the 4th century AD. During the Renaissance, numerous geometricians worked on the search for proof of this theorem, among which Viete, Adriaan van Roomen, Joseph Diaz Gergonne, and Newton were successful. The illustration above is the cover story of the 2016 Algebraic Geometry course 3264 and Related Stories [15] published by the University of Cambridge, where the number 3264 refers to the number of conic curves tangent to 5 conic curves in the general position in the complex projection plane, obtained by Chasles in 1864 [2].
The discovery of the Descartes (1627) spatial coordinate system enabled geometricians (e.g., Maclaurin (1720), Euler (1748), Bezout (1764)) to use polynomial equations to characterize geometric objects that met specific geometric conditions. As a result, many counting geometry problems are formulated as follows:
Polynomial problem: For a system of polynomial equations where the number of independent variables on a complex field is equal to the number of equations:
Schematic diagram of the intersection theory
The problem of intersecting numbers can be attributed to the French mechanic and mathematician Jean-Victor Poncelet (1788-1867). He graduated from the École Polytechnique de Paris in 1811 and participated in Napoleon's war of aggression against Russia as a captain in the engineering corps. Abandoned on the battlefield during the Battle of Krasnoye, near Moscow, he was captured and imprisoned in the Saratow prisoner of war camp in Siberia. Ponslie survived the difficult years in the prisoner of war camp by studying geometry. Relying only on the foundation of drawing geometry taught by Monge in college, he independently discovered and established a systematic theory of high-dimensional projective geometry without knowing anything about projective geometry in the 17th century. He proposed and studied the invariant properties of the figure through the central projection; The concepts of "intercomposive" and "infinite" elements were introduced; The polarization theory of quadratic curves and surfaces is established, and the general duality principle is obtained. In addition, he studied the properties of graphs that are maintained when they change continuously within a certain range, and proposed "the principle of continuity", which is the prototype of "intersecting number homotopy invariance" in topology today and "Chow's moving lemma" in algebraic geometry. Ponsley organized his work during his time in the prisoner of war camps into "On the Projective Nature of Figures, 1816", which was the foundational work of modern projective geometry and the theory of intersection. However, Cauchy was dissatisfied with Ponsley's participation in the French Revolution and refused to publish the article on the grounds that it would lead to "serious errors".
In 1900, Poincaré creatively introduced the homology group H*(M) of manifold M in the classification of three-dimensional manifolds, which enabled people to apply the operation in the group to analyze the geometry of M. In the process of studying the intersection number problem, Lefschetze further established the theory of upper cohomology of manifold M H*(M) [21, 1926]. From the level of chain complexes, the latter is only a dual of the former, but compared with the cohomology theory, the upper cohomology has a prominent advantage: the diagonal mapping d:M→MxM induces a multiplication operation called "cup product" in the upper cohomology group
At this point, we review three historical assumptions for solving counting problems (or intersecting number problems) and how they relate to each other. A natural question is, which option is effectively computable? This is not only the core requirement of the problem of counting geometry, but also the driving force for the development of algebraic geometry in the 20th century.
02 Hilbert question 15
Hermann Schubert (1848-1911) received his doctorate from the University of Halle, Germany, in 1870. His doctoral dissertation, Theory of Eigennumbers[22], was on the topic of counting geometry. Previously, he has published related articles, proving that there are 16 spheres tangent to 4 spheres in general positions in space, which is a direct generalization of Apollonius' theorem in space cases.
In 1879, Schubert published the culmination of the 19th-century theory of intersection, The Calculus of Counting Geometry [23]:
Schubert and the Geometric Calculus of Counting
In this book, he develops Chasles' work on conic curves[2] and demonstrates the geometric charm of intersection theory through a series of examples. For example:
Example 2.1: There are exactly 4,407,296 conic curves tangent to 8 quadric surfaces in a general position in a given space;
Example 2.2: For 9 quadrics in a given space in a general position, there are exactly 666, 841, 088 quadrics tangent to them;
Example 2.3: Given 12 quadric surfaces in a general position in space, exactly 5,819,539,783,680 cubic solid curves are tangent to them.
Schubert's work was widely criticized because of its extensive application of the "principle of continuity" that Cauchy opposed. To avoid criticism, he renamed the principle "the principle of special position" in 1874 and "the principle of conservation of numbers" two years later. [19] Still under attack by Study and Kohn. The most pertinent comment comes from van der Waerden, who recalls in his literature [32] that Schubert's argument was so general that "no definition of the intersection number is given, no way to find it, no way to calculate it".
In question 15, Hilbert asks for "a rigorous foundation for Schubert's counting algorithm". At the same time, Hilbert affirmed the advantages of Schubert's method in predicting the solution of polynomial problems:
"The problem is to strictly prove the correctness of those geometric numbers in counting geometry on the premise of accurately defining their scope of application. Of particular interest is the geometric numbers that Schubert calculates in his book based on the so-called special position principle (or the principle of conservation of intersecting numbers).
Although today's algebra guarantees in principle the possibility of implementing the elimination method, the proof of those theorems in counting geometry places higher demands on algebra. This is because it requires that the number of the resulting equations and the repleons of their solutions be foreseen before the elimination method is applied to those particular equations (systems). ”
Hilbert with question 15
03 Basic Problems of Schubert's Calculus: The Eigennumber Problem
To get to the heart of Schubert's calculus, let's quote a table of counts from the original book [23]:
Table 1. Eigennumber equation for a spatial conic cross-section
The symbols ρ, μ, and ν represent three algebraic clusters in space formed by a fixed point, intersecting a fixed line, tangent and a conic section of a fixed plane. Schubert himself referred to the equations in the table as "eigennumber equations", while early researchers also called them "Schubert symbolic equations". In his work [22-24], Schubert repeatedly emphasized that the problem of eigennumbers is the main theoretical problem of counting geometry [20]. However, mathematicians have spent more than 60 years in order to arrive at a rigorous formulation of the "eigennumber problem", and this section reviews the relevant stories.
3.1意大利学派 (The Italian school)
The mathematical group that first studied problem 15 was the Italian school represented by Enriques and Severi. Their representative works are Severi's articles "The Principle of Conservation of (Eigen) Numbers" and "Fundamentals of Counting Geometry and the Theory of Eigennumbers" [26, 27]. According to Van der Walden [28], "they built admirable structures, but the logical basis was unstable, the concepts were poorly defined, and the evidence was insufficient".
The scene of the debate of the Italian school on the theory of intersection
3.2 哥廷根学派 (The Gottingen School)}
In 1930, Van der Walden published the article "Topological Foundations of Counting-Geometry Calculus" [29], which was an important milestone in the history of algebraic geometry. In this article, he first proposed the idea of answering question 15 within the framework of Lefshetz's theory of upper cohomology. In his article, he keenly noted:
a) Each Schubert sign equation should be a relation in a cohomology group on a projective manifold;
b) that the premise for solving the eigennumber problem is to determine an additive substrate for the cohomology group on the projective manifold;
c) The common goal of all counting problems, which is to calculate the intersection number of algebraic clusters in projective manifolds, successfully led the follow-up to problem 15.
Van der Walden and the Topological Foundations of Counting Geometry
3.3 布尔巴基学派 (Bourbaki)
Let G be a compactly connected Lie group and P be a parabolic subgroup of G. Through the adjoint representation of G to its Lie algebra, the homogeneous space G/P is realized as a flag manifold called the Lie group G. Below, we follow the ref. [1] by using W(G) to denote the Weyl group of Li group G, and using W(G; P) denotes the left companion set W(G)/W(P) of the subgroup W(P). It was first discovered by C. Ehresmann in 1934[14]
a) the parameter space of the geometric object that Schubert's calculus is concerned with is essentially some special case of the flag manifold G/P;
b) In the special case of the complex Grassmann manifold Gn,k(C), the classic Schubert symbol happens to be an additive base for the cohomology group on it.
With the deepening of research, the vague term "Schubert symbol" in the early literature was gradually replaced by rigorous geometric objects such as "Schubert cavity" or "Schubert cluster". In particular, Chevalley [3, 1958], Bernstein - Gel'fand - Gel'fand et al. [1, 1973] successively demonstrated that each flag manifold G/P has a cavity decomposition:
Amazingly, 50 years before the formal birth of the theory of cohomology, Schubert was already applying the theory and working on the geometric calculus of counts. As an example, we cite a passage from Coolidge [4]: "Schubert's fundamental problem was to linearly express the product of these symbols with other symbols. He had only partial success. ”
04 The birth of algebraic geometry
Van der Walden begins his article [30] by pointing out that "the central problem of Problem 15 is to give a definition (or formula) of intersection multiplicities, with the help of which we can effectively calculate the solutions of Schubert's counting geometry problems while maintaining the principle of conservation of intersecting numbers". With that, he began a plan to build the foundations of algebraic geometry. He published a series of articles in Mathematische Annalen entitled "ZurAl- gebraische Geometrie (#1~#20)" and in 1939 he published his famous book Introduction to Algeberaic Geometry, whose primary task was to find a rigorous definition of "intersecting multiplicity".
Andrei Weil.A is the soul of the Bourbaki school. In 1946, he published his landmark work Fundamentals of Algebraic Geometry [31], in which he systematically and completely defined "intersecting multiple numbers" for algebraic clusters on algebraic closed fields for the first time. Subsequently, based on Chevalli's discovery of Schubert's calculus theorem (Theorem 3.1), in the second edition of the book, he equated the solution to Hilbert's problem 15 with the problem of "determining the upper cohomology loop of all flag manifolds G/P" [31, p. 331]. Hereinafter referred to as the "Wey Problem".
Wey and Fundamentals of Algebraic Geometry
On the basis of Wey's work [31], the "beautiful formula" for the two dimensional complementary subclusters X, Y, and Serre J.P. in the irreducible algebraic cluster W yields the "beautiful formula" of the intersecting multiplicity [25, 1965]:
where A denotes the local loops O(X,W), a and b are the ideals of the algebraic clusters X and Y, respectively, and L is the length of the A mode. Subsequently, Fulton W, together with MacPherson R.D., generalized the formula to algebraic clusters with singularities [16]. Unfortunately, such formulas do not allow for effective calculations, especially for the number of features that concern question 15.
Problem 15 is a far-reaching problem in contemporary mathematics, which promoted the theory of counting geometry and intersection in the 19th century, and grew into the algebraic geometry established by the mathematicians Fan de Walden and André Wey in the 20th century [29-31], which made Schubert's calculus deeply integrated into the fields of differential geometry, algebraic topology, and Lie group representation theory, and profoundly influenced the development trajectory of these fields. All this is not only a powerful testimony of Hilbert's broad vision and foresight for the development of mathematics, but also an urgent requirement for exploring the effective algorithms of Schubert's calculus, especially the solutions to the eigennumber problem and the Wey problem.
In 1931, Wei-Liang Chow (1011-1995) received his Ph.D. from the University of Chicago. His admiration for Van der Walden's Algebra led him to study algebraic geometry at the University of Leipzig in Germany in 1933. In 1958, he announced the Chow Ring named after him at a seminar in C. Chevalley [4], which is a foundational platform for contemporary intersecting theory.
In his work on the construction of periphonic rings, Zhou Weiliang proved that there is a canonical isomorphism between all flag manifold circumferential rings A*(G/P) and upper cohomology rings H*(G/P). Taking advantage of this isomorphism, Duan Haiseal and Zhao Xuezhi developed differential topology, algebraic topology, and symbolic computing techniques in their series of work [6-11], and solved the "eigennumber problem" and "Weiyi problem" from both theoretical and computational perspectives. Accordingly, they have stated in the article [12, 13] that question 15 has been answered.
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