Book contents
- Frontmatter
- Contents
- The Independent University of Moscow and Student Sessions at the IUM
- Mysterious mathematical trinities
- The principle of topological economy in algebraic geometry
- Rational curves, elliptic curves, and the Painlevé equation
- The orbit method and finite groups
- On the development of the theory of dynamical systems during the past quarter century
- Foundations of computational complexity theory
- The Schrödinger equation and symplectic geometry
- Rings and algebraic varieties
- Billiard table as a playground for a mathematician
- The Fibonacci numbers and simplicity of 2127 – 1
- On problems of computational complexity
- Values of the ζ-function
- Combinatorics of trees
- What is an operad?
- The orbit method beyond Lie groups. Infinite-dimensional groups
- The orbit method beyond Lie groups. Quantum groups
- Conformal mappings and the Whitham equations
- Projective differential geometry: old and new
- Haken's method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem
The principle of topological economy in algebraic geometry
Published online by Cambridge University Press: 18 December 2009
- Frontmatter
- Contents
- The Independent University of Moscow and Student Sessions at the IUM
- Mysterious mathematical trinities
- The principle of topological economy in algebraic geometry
- Rational curves, elliptic curves, and the Painlevé equation
- The orbit method and finite groups
- On the development of the theory of dynamical systems during the past quarter century
- Foundations of computational complexity theory
- The Schrödinger equation and symplectic geometry
- Rings and algebraic varieties
- Billiard table as a playground for a mathematician
- The Fibonacci numbers and simplicity of 2127 – 1
- On problems of computational complexity
- Values of the ζ-function
- Combinatorics of trees
- What is an operad?
- The orbit method beyond Lie groups. Infinite-dimensional groups
- The orbit method beyond Lie groups. Quantum groups
- Conformal mappings and the Whitham equations
- Projective differential geometry: old and new
- Haken's method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem
Summary
This part of the lecture is not related to the first part, so it can be understood independently. We start with an example.
Example 1. In ℂPn, consider two algebraic varieties X and Y of complementary dimensions. In general position, they intersect in finitely many points. Let [X] and [Y] be the homology classes realized by the varieties X and Y, and let [X] º [Y] be the intersection index of these classes (which is an integer). It is equal to the number of “positive” intersection points of X with Y minus the number of “negative” intersection points. Thus, the number #(X ∩ Y) of all intersection points is not smaller than the intersection index [X] º [Y] (and has the same parity). The Bézout Theorem asserts that #(X ∩ Y) is equal to the number [X] º [Y], i.e., there is no inequality! The point is that the orientation of complex manifolds is such that each intersection makes a contribution of +1, not –1, in the total intersection index. Negative intersections are “expensive,” they increase the number of intersection points of X with Y in comparison with the “topologically necessary” number. A propos, the same considerations imply that a polynomial of degree n has precisely n roots, not more.
This (well-known) and the following (newer) examples lead to a “principle of economy,” which, in its turn, can be used to state further conjectures. These conjectures can be verified in particular cases; sometimes, they can be proved and become theorems. But in most cases, they remain conjectures, i.e., assertions which we may try to disprove, for a long time.
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- Chapter
- Information
- Surveys in Modern Mathematics , pp. 13 - 23Publisher: Cambridge University PressPrint publication year: 2005