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In [[mathematics]], the '''Pontryagin classes''', named for [[Lev Pontryagin]], are certain [[characteristic class]]es. The Pontryagin class lies in [[cohomology group]]s with degree a multiple of four. It applies to real [[vector bundle]]s.
{{other persons}}
'''Edward Nelson''' (born May 4, 1932, in [[Decatur, Georgia]]) is a professor in the Mathematics Department at [[Princeton University]].    He is known for his work on [[mathematical physics]] and [[mathematical logic]].  In mathematical logic, he is noted especially for his [[internal set theory]], and his controversial views on [[ultrafinitism]] and the [[consistency]] of [[Peano arithmetic|arithmetic]].


== Definition ==
==Career==
Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class ''p<sub>k</sub>''(''E'') is defined as
:''p<sub>k</sub>''(''E'') = ''p<sub>k</sub>''(''E'', '''Z''') = (−1)<sup>''k''</sup> ''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') ∈ ''H''<sup>4''k''</sup>(''M'', '''Z'''),
where:
*''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') denotes the 2''k''-th [[Chern class]] of the [[complexification]] ''E'' ⊗ '''C''' = ''E'' ⊕ ''iE'' of ''E'',
*''H''<sup>4''k''</sup>(''M'', '''Z''') is the 4''k''-[[cohomology]] group of ''M'' with [[integer]] coefficients.


The rational Pontryagin class ''p<sub>k</sub>''(''E'', '''Q''') is defined to be the image of ''p<sub>k</sub>''(''E'') in ''H''<sup>4''k''</sup>(''M'', '''Q'''), the 4''k''-[[cohomology]] group of ''M'' with [[Rational number|rational]] coefficients.
Nelson received his Ph.D. in 1955 from the [[University of Chicago]], where he worked with [[Irving Segal]].
He was a member of the [[Institute for Advanced Study]] from 1956–1959.  He has held a position at [[Princeton University]] from 1959 to the present, attaining the rank of professor there in 1964.


== Properties ==
==Early work==
The '''total Pontryagin class'''
:<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\mathbf{Z}),</math>
is (modulo 2-torsion) multiplicative with respect to
[[Glossary of differential geometry and topology#W|Whitney sum]] of vector bundles, i.e.,
:<math>2p(E\oplus F)=2p(E)\smile p(F)</math>
for two vector bundles ''E'' and ''F'' over ''M''.  In terms of the individual Pontryagin classes ''p<sub>k</sub>'',
:<math>2p_1(E\oplus F)=2p_1(E)+2p_1(F),</math>
:<math>2p_2(E\oplus F)=2p_2(E)+2p_1(E)\smile p_1(F)+2p_2(F)</math>
and so on.


The vanishing of the Pontryagin classes and [[Stiefel-Whitney class]]es of a vector bundle does not guarantee that the vector bundle is trivial.  For example, up to [[Vector bundle#Vector bundle morphisms|vector bundle isomorphism]], there is a unique nontrivial rank 10 vector bundle ''E''<sub>10</sub> over the [[N-sphere|9-sphere]].  (The [[clutching function]] for ''E''<sub>10</sub> arises from the [[Orthogonal group#Homotopy groups|stable homotopy group]] π<sub>8</sub>(O(10)) = '''Z'''/2'''Z'''.)  The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the [[Stiefel-Whitney class]] ''w''<sub>9</sub> of ''E''<sub>10</sub> vanishes by the [[Stiefel-Whitney class#Relations over the Steenrod algebra|Wu formula]] ''w''<sub>9</sub> = ''w''<sub>1</sub>''w''<sub>8</sub> + Sq<sup>1</sup>(''w''<sub>8</sub>).  Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of ''E''<sub>10</sub> with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}
Nelson has made contributions to the theory of infinite dimensional [[group representation]]s, the mathematical treatment of [[quantum field theory]], the use of [[stochastic process]]es in [[quantum mechanics]], and the reformulation of [[probability theory]] in terms of [[non-standard analysis]].


Given a 2''k''-dimensional vector bundle ''E'' we have
For many years he worked on [[mathematical physics]] and probability theory, and still has a residual interest in these fields, particularly in possible extensions of stochastic mechanics to [[field theory (physics)|field theory]].
:<math>p_k(E)=e(E)\smile e(E),</math>
where ''e''(''E'') denotes the [[Euler class]] of ''E'', and <math>\smile</math> denotes the [[cup product]] of cohomology classes.  


=== Pontryagin classes and curvature ===
In 1950, Nelson formulated a popular variant of the [[four color problem]]. What is the chromatic number, denoted <math>\chi</math>, of the plane? In more detail, what is the smallest number of colors sufficient for coloring the points of the Euclidean plane in such a way that no two points of the same color are unit distance apart?<ref>p.23, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1</ref> We know by simple arguments that 4&nbsp;≤&nbsp;''&chi;''&nbsp;≤&nbsp;7. The problem was introduced to a wide mathematical audience by [[Martin Gardner]] in his October 1960 [[Mathematical Games]] column. The chromatic number problem, also now known as the [[Hadwiger–Nelson problem]], was also a favorite of [[Paul Erdős]], who mentioned it frequently in his problems lectures.
As was shown by [[Shiing-Shen Chern]] and [[André Weil]] around 1948, the rational Pontryagin classes
:<math>p_k(E,\mathbf{Q})\in H^{4k}(M,\mathbf{Q})</math>
can be presented as differential forms which depend polynomially on the [[curvature form]] of a vector bundle. This [[Chern–Weil theory]] revealed a major connection between algebraic topology and global differential geometry.


For a [[vector bundle]] ''E'' over a ''n''-dimensional [[differentiable manifold]] ''M'' equipped with a [[connection form|connection]], the total Pontryagin class is expressed as
==Work on foundations==
:<math>p=\left[1-\frac{{\rm Tr}(\Omega ^2)}{8 \pi ^2}+\frac{{\rm Tr}(\Omega ^2)^2-2 {\rm Tr}(\Omega ^4)}{128 \pi ^4}-\frac{{\rm Tr}(\Omega ^2)^3-6 {\rm Tr}(\Omega ^2) {\rm Tr}(\Omega ^4)+8 {\rm Tr}(\Omega ^6)}{3072 \pi ^6}+\cdots\right]\in H^*_{dR}(M),</math>
In recent years he has been working on mathematical logic and the foundations of mathematics. One of his goals is to extend IST ([[Internal Set Theory]]—a version of a portion of [[Abraham Robinson]]'s [[non-standard analysis]]) in a natural way to include external functions and sets, in a way that provides an external function with specified properties unless there is a finitary obstacle to its existence. Other work centers on fragments of arithmetic, studying the divide between those theories interpretable in [[Robinson arithmetic|Raphael Robinson's Arithmetic]] and those that are not; [[computational complexity theory|computational complexity]], including the problem of whether [[P = NP problem|P is equal to NP or not]]; and automated proof checking.


where Ω denotes the [[curvature form]], and ''H*''<sub>dR</sub>(''M'') denotes the [[de Rham cohomology]] groups.{{Citation needed|date=July 2009}}
In September 2011, Nelson announced that his he had proved that [[Peano arithmetic]] was logically inconsistent. An error was found in the proof, and he retracted the claim.


=== Pontryagin classes of a manifold ===
==Notes==
The '''Pontryagin classes of a smooth manifold''' are defined to be the Pontryagin classes of its [[tangent bundle]].
{{reflist}}


[[Sergei Novikov (mathematician)|Novikov]] proved in 1966 that if manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes ''p<sub>k</sub>''(''M'', '''Q''') in ''H''<sup>4''k''</sup>(''M'', '''Q''') are the same.
==References==
* [http://math.princeton.edu/~nelson/cv.pdf Curriculum Vitae]


If the dimension is at least five, there are at most finitely many different smooth manifolds with given [[Homotopy#Homotopy equivalence of spaces|homotopy type]] and Pontryagin classes.
==See also==
*[[Influence of non-standard analysis]]


== Pontryagin numbers ==
==External links==
'''Pontryagin numbers''' are certain [[topological invariant]]s of a smooth [[manifold]]. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a [[manifold]] as follows:
*[http://www.math.princeton.edu/~nelson/ Edward Nelson's Homepage]
 
Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers
:''k''<sub>1</sub>, ''k''<sub>2</sub>, ..., ''k<sub>m</sub>'' such that ''k''<sub>1</sub>+''k''<sub>2</sub>+...+''k<sub>m</sub>'' =''n''.
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
where ''p<sub>k</sub>'' denotes the ''k''-th Pontryagin class and [''M''] the [[fundamental class]] of ''M''.
 
=== Properties ===
#Pontryagin numbers are oriented [[cobordism]] invariant; and together with [[Stiefel-Whitney number]]s they determine an oriented manifold's oriented cobordism class.
#Pontryagin numbers of closed Riemannian manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
#Such invariants as [[Signature (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers.
 
== Generalizations ==
There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure.


== See also ==
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
*[[Chern–Simons form]]
| NAME              = Nelson, Edward
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION =
| DATE OF BIRTH    = May 4, 1932
| PLACE OF BIRTH    =
| DATE OF DEATH    =
| PLACE OF DEATH    =
}}
{{DEFAULTSORT:Nelson, Edward}}
[[Category:1932 births]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:American mathematicians]]
[[Category:American logicians]]
[[Category:Set theorists]]
[[Category:University of Chicago alumni]]
[[Category:Princeton University faculty]]
[[Category:Living people]]
[[Category:Mathematical physicists]]


== References ==
*{{cite book
  |author= [[John Milnor|Milnor John W.]]
  |author2=Stasheff, James D. |authorlink2=Jim Stasheff
  |title= Characteristic classes
  |work= Annals of Mathematics Studies
  |issue=76
  |publisher=Princeton University Press / University of Tokyo Press
  |location=Princeton, New Jersey; Tokyo
  |year= 1974
  |isbn= 0-691-08122-0}}
* {{Cite journal | last=Hatcher | first=Allen | author-link=Allen Hatcher  | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | ref=harv | postscript=<!--None--> | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}}
==External links==
* {{springer|title=Pontryagin class|id=p/p073750}}


[[Category:Characteristic classes]]
[[de:Edward Nelson]]
[[Category:Differential topology]]
[[fr:Edward Nelson (mathématicien)]]
[[ht:Edward Nelson]]

Revision as of 18:14, 11 August 2014

Template:Other persons Edward Nelson (born May 4, 1932, in Decatur, Georgia) is a professor in the Mathematics Department at Princeton University. He is known for his work on mathematical physics and mathematical logic. In mathematical logic, he is noted especially for his internal set theory, and his controversial views on ultrafinitism and the consistency of arithmetic.

Career

Nelson received his Ph.D. in 1955 from the University of Chicago, where he worked with Irving Segal. He was a member of the Institute for Advanced Study from 1956–1959. He has held a position at Princeton University from 1959 to the present, attaining the rank of professor there in 1964.

Early work

Nelson has made contributions to the theory of infinite dimensional group representations, the mathematical treatment of quantum field theory, the use of stochastic processes in quantum mechanics, and the reformulation of probability theory in terms of non-standard analysis.

For many years he worked on mathematical physics and probability theory, and still has a residual interest in these fields, particularly in possible extensions of stochastic mechanics to field theory.

In 1950, Nelson formulated a popular variant of the four color problem. What is the chromatic number, denoted , of the plane? In more detail, what is the smallest number of colors sufficient for coloring the points of the Euclidean plane in such a way that no two points of the same color are unit distance apart?[1] We know by simple arguments that 4 ≤ χ ≤ 7. The problem was introduced to a wide mathematical audience by Martin Gardner in his October 1960 Mathematical Games column. The chromatic number problem, also now known as the Hadwiger–Nelson problem, was also a favorite of Paul Erdős, who mentioned it frequently in his problems lectures.

Work on foundations

In recent years he has been working on mathematical logic and the foundations of mathematics. One of his goals is to extend IST (Internal Set Theory—a version of a portion of Abraham Robinson's non-standard analysis) in a natural way to include external functions and sets, in a way that provides an external function with specified properties unless there is a finitary obstacle to its existence. Other work centers on fragments of arithmetic, studying the divide between those theories interpretable in Raphael Robinson's Arithmetic and those that are not; computational complexity, including the problem of whether P is equal to NP or not; and automated proof checking.

In September 2011, Nelson announced that his he had proved that Peano arithmetic was logically inconsistent. An error was found in the proof, and he retracted the claim.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

See also

External links

Template:Persondata


de:Edward Nelson fr:Edward Nelson (mathématicien) ht:Edward Nelson

  1. ↑ p.23, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1
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