Does there exist some relations between Functional Analysis and Algebraic Topology?
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As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dimensional Euclidean space. But when we come across some infinite dimensional spaces, such as Banach spaces, do the tools in Algebraic Topology also take effect?
Moreover, are there some books discussing such relation? My learning background is listed following:basic algebra(group, ring, field, polynomial); Rudin's real & complex analysis and functional analysis; general topology(Munkres level).
Any viewpoint will be appreciated.
functional-analysis algebraic-topology
edited Dec 4 '18 at 9:00
Kei
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
$endgroup$
add a comment |
$begingroup$
As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dimensional Euclidean space. But when we come across some infinite dimensional spaces, such as Banach spaces, do the tools in Algebraic Topology also take effect?
Moreover, are there some books discussing such relation? My learning background is listed following:basic algebra(group, ring, field, polynomial); Rudin's real & complex analysis and functional analysis; general topology(Munkres level).
Any viewpoint will be appreciated.
functional-analysis algebraic-topology
edited Dec 4 '18 at 9:00
Kei
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
$endgroup$
add a comment |
$begingroup$
As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dimensional Euclidean space. But when we come across some infinite dimensional spaces, such as Banach spaces, do the tools in Algebraic Topology also take effect?
Moreover, are there some books discussing such relation? My learning background is listed following:basic algebra(group, ring, field, polynomial); Rudin's real & complex analysis and functional analysis; general topology(Munkres level).
Any viewpoint will be appreciated.
functional-analysis algebraic-topology
edited Dec 4 '18 at 9:00
Kei
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
$endgroup$
As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dimensional Euclidean space. But when we come across some infinite dimensional spaces, such as Banach spaces, do the tools in Algebraic Topology also take effect?
Moreover, are there some books discussing such relation? My learning background is listed following:basic algebra(group, ring, field, polynomial); Rudin's real & complex analysis and functional analysis; general topology(Munkres level).
Any viewpoint will be appreciated.
functional-analysis algebraic-topology
functional-analysis algebraic-topology
edited Dec 4 '18 at 9:00
Kei
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
edited Dec 4 '18 at 9:00
Kei
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
edited Dec 4 '18 at 9:00
Kei
347
edited Dec 4 '18 at 9:00
Kei
347
edited Dec 4 '18 at 9:00
Kei
347
347
asked Aug 15 '13 at 13:33
mathonmathon
536513
asked Aug 15 '13 at 13:33
mathonmathon
536513
asked Aug 15 '13 at 13:33
mathonmathon
536513
536513
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
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Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
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add a comment |
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See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
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$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
add a comment |
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Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too
http://en.wikipedia.org/wiki/Von_Neumann_algebra
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
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add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
$endgroup$
add a comment |
$begingroup$
Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
$endgroup$
add a comment |
$begingroup$
Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
$endgroup$
Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
edited Aug 15 '13 at 14:15
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
answered Aug 15 '13 at 13:53
Prahlad VaidyanathanPrahlad Vaidyanathan
26.1k12152
26.1k12152
add a comment |
add a comment |
$begingroup$
See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
$endgroup$
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
add a comment |
$begingroup$
See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
$endgroup$
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
add a comment |
$begingroup$
See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
$endgroup$
See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
answered Aug 15 '13 at 13:43
njguliyevnjguliyev
13.3k12041
13.3k12041
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
add a comment |
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
$begingroup$
That's a vigorous field!But unfortunately,I haven't any backgroud on Differential Geometry,not to mention Atiyah-Singer index theorem.
$endgroup$
– mathon
Aug 15 '13 at 13:50
add a comment |
$begingroup$
Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too
http://en.wikipedia.org/wiki/Von_Neumann_algebra
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
$endgroup$
add a comment |
$begingroup$
Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too
http://en.wikipedia.org/wiki/Von_Neumann_algebra
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
$endgroup$
add a comment |
$begingroup$
Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too
http://en.wikipedia.org/wiki/Von_Neumann_algebra
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
$endgroup$
Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too
http://en.wikipedia.org/wiki/Von_Neumann_algebra
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
answered Aug 15 '13 at 14:12
AutolatryAutolatry
2,68911015
2,68911015
add a comment |
add a comment |
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