*David Colander and Roland Kupers*

- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691169132
- eISBN:
- 9781400850136
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691169132.003.0007
- Subject:
- Economics and Finance, History of Economic Thought

This chapter focuses on Stephen Wolfram, an early advocate of the importance of complexity science. He founded the Journal of Complex Systems back in 1987, and saw the transformational aspect of ...
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This chapter focuses on Stephen Wolfram, an early advocate of the importance of complexity science. He founded the Journal of Complex Systems back in 1987, and saw the transformational aspect of computer analysis long before it was generally understood. But his ego and his disdain for standard scientific conventions kept him and the complexity science he favored outside the mainstream scientific establishment that discourages such grandiose claims. In 2002, his self-published book A New Kind of Science was seen by the scientific community as the delusions of a former wunderkind. It is argued that Wolfram’s book represents the insights of a brilliant visionary about “a new tool of science”—computational tools that earlier scientists could hardly have imagined. These computational tools provide not only new tools for analysis, but also a new vision of how to frame thinking about complex processes. It is the blending of the computational tools and the vision that makes up complexity science.Less

This chapter focuses on Stephen Wolfram, an early advocate of the importance of complexity science. He founded the *Journal of Complex Systems* back in 1987, and saw the transformational aspect of computer analysis long before it was generally understood. But his ego and his disdain for standard scientific conventions kept him and the complexity science he favored outside the mainstream scientific establishment that discourages such grandiose claims. In 2002, his self-published book *A New Kind of Science* was seen by the scientific community as the delusions of a former wunderkind. It is argued that Wolfram’s book represents the insights of a brilliant visionary about “a new tool of science”—computational tools that earlier scientists could hardly have imagined. These computational tools provide not only new tools for analysis, but also a new vision of how to frame thinking about complex processes. It is the blending of the computational tools and the vision that makes up complexity science.

*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0006
- Subject:
- Mathematics, Educational Mathematics

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined ...
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This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.Less

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0032
- Subject:
- Mathematics, Educational Mathematics

This chapter explores some applications of equivariant cohomology. Since its introduction in the Fifties, equivariant cohomology has found applications in topology, symplectic geometry, K-theory, and ...
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This chapter explores some applications of equivariant cohomology. Since its introduction in the Fifties, equivariant cohomology has found applications in topology, symplectic geometry, K-theory, and physics, among other fields. Its greatest utility may be in converting an integral on a manifold to a finite sum. Since many problems in mathematics can be expressed in terms of integrals, the equivariant localization formula provides a powerful computational tool. The chapter then discusses a few of the applications of the equivariant localization formula. In order to use the equivariant localization formula to compute the integral of an invariant form, the form must have an equivariantly closed extension.Less

This chapter explores some applications of equivariant cohomology. Since its introduction in the Fifties, equivariant cohomology has found applications in topology, symplectic geometry, *K*-theory, and physics, among other fields. Its greatest utility may be in converting an integral on a manifold to a finite sum. Since many problems in mathematics can be expressed in terms of integrals, the equivariant localization formula provides a powerful computational tool. The chapter then discusses a few of the applications of the equivariant localization formula. In order to use the equivariant localization formula to compute the integral of an invariant form, the form must have an equivariantly closed extension.