Intersection of Algebraic Geometry and Algebraic Topology
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What mathematical areas lie at the intersection of algebraic geometry and algebraic topology? I'm aware of certain ones such as derived algebraic geometry and motivic homotopy theory, which all involve a fair amount of higher category theory. Are there any areas at the intersection that focus more on the algebraic geometry/algebraic topology themselves rather than more on the category theory? Do certain aspects of derived geometry satisfy this criteria, such as the recent work on derived symplectic geometry [PTVV]? For instance, reading some books such as Lurie's HTT and Higher Algebra seems more focused on category theory rather than homotopy theory itself.
algebraic-geometry algebraic-topology soft-question
asked Mar 16 '16 at 1:11
user238194user238194
37618
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add a comment |
$begingroup$
What mathematical areas lie at the intersection of algebraic geometry and algebraic topology? I'm aware of certain ones such as derived algebraic geometry and motivic homotopy theory, which all involve a fair amount of higher category theory. Are there any areas at the intersection that focus more on the algebraic geometry/algebraic topology themselves rather than more on the category theory? Do certain aspects of derived geometry satisfy this criteria, such as the recent work on derived symplectic geometry [PTVV]? For instance, reading some books such as Lurie's HTT and Higher Algebra seems more focused on category theory rather than homotopy theory itself.
algebraic-geometry algebraic-topology soft-question
asked Mar 16 '16 at 1:11
user238194user238194
37618
$endgroup$
$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
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– Charlie
Mar 16 '16 at 1:19
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This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
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– Nicolas Bourbaki
Mar 16 '16 at 1:25
1
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One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
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– Mohan
Mar 16 '16 at 2:51
2
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"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
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– John Brevik
Mar 16 '16 at 3:20
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Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
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– Fredrik Meyer
Mar 16 '16 at 9:16
add a comment |
$begingroup$
What mathematical areas lie at the intersection of algebraic geometry and algebraic topology? I'm aware of certain ones such as derived algebraic geometry and motivic homotopy theory, which all involve a fair amount of higher category theory. Are there any areas at the intersection that focus more on the algebraic geometry/algebraic topology themselves rather than more on the category theory? Do certain aspects of derived geometry satisfy this criteria, such as the recent work on derived symplectic geometry [PTVV]? For instance, reading some books such as Lurie's HTT and Higher Algebra seems more focused on category theory rather than homotopy theory itself.
algebraic-geometry algebraic-topology soft-question
asked Mar 16 '16 at 1:11
user238194user238194
37618
$endgroup$
What mathematical areas lie at the intersection of algebraic geometry and algebraic topology? I'm aware of certain ones such as derived algebraic geometry and motivic homotopy theory, which all involve a fair amount of higher category theory. Are there any areas at the intersection that focus more on the algebraic geometry/algebraic topology themselves rather than more on the category theory? Do certain aspects of derived geometry satisfy this criteria, such as the recent work on derived symplectic geometry [PTVV]? For instance, reading some books such as Lurie's HTT and Higher Algebra seems more focused on category theory rather than homotopy theory itself.
algebraic-geometry algebraic-topology soft-question
algebraic-geometry algebraic-topology soft-question
asked Mar 16 '16 at 1:11
user238194user238194
37618
asked Mar 16 '16 at 1:11
user238194user238194
37618
edited Mar 16 '16 at 1:29
user238194
asked Mar 16 '16 at 1:11
user238194user238194
37618
asked Mar 16 '16 at 1:11
user238194user238194
37618
asked Mar 16 '16 at 1:11
user238194user238194
37618
37618
$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
$endgroup$
– Charlie
Mar 16 '16 at 1:19
$begingroup$
This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
$endgroup$
– Nicolas Bourbaki
Mar 16 '16 at 1:25
1
$begingroup$
One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
$endgroup$
– Mohan
Mar 16 '16 at 2:51
2
$begingroup$
"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
$endgroup$
– John Brevik
Mar 16 '16 at 3:20
$begingroup$
Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
$endgroup$
– Fredrik Meyer
Mar 16 '16 at 9:16
add a comment |
$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
$endgroup$
– Charlie
Mar 16 '16 at 1:19
$begingroup$
This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
$endgroup$
– Nicolas Bourbaki
Mar 16 '16 at 1:25
1
$begingroup$
One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
$endgroup$
– Mohan
Mar 16 '16 at 2:51
2
$begingroup$
"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
$endgroup$
– John Brevik
Mar 16 '16 at 3:20
$begingroup$
Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
$endgroup$
– Fredrik Meyer
Mar 16 '16 at 9:16
$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
$endgroup$
– Charlie
Mar 16 '16 at 1:19
$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
$endgroup$
– Charlie
Mar 16 '16 at 1:19
$begingroup$
This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
$endgroup$
– Nicolas Bourbaki
Mar 16 '16 at 1:25
$begingroup$
This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
$endgroup$
– Nicolas Bourbaki
Mar 16 '16 at 1:25
1
1
$begingroup$
One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
$endgroup$
– Mohan
Mar 16 '16 at 2:51
$begingroup$
One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
$endgroup$
– Mohan
Mar 16 '16 at 2:51
2
2
$begingroup$
"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
$endgroup$
– John Brevik
Mar 16 '16 at 3:20
$begingroup$
"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
$endgroup$
– John Brevik
Mar 16 '16 at 3:20
$begingroup$
Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
$endgroup$
– Fredrik Meyer
Mar 16 '16 at 9:16
$begingroup$
Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
$endgroup$
– Fredrik Meyer
Mar 16 '16 at 9:16
add a comment |
1 Answer
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You can study about arithmetic topology for knots and links and this field in intersection of algebraic topology and algebraic geometry. You can talk about this field with alias miler. She is at Harvard university.
answered Jan 8 at 17:28
husseinhussein
786
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$begingroup$
You can study about arithmetic topology for knots and links and this field in intersection of algebraic topology and algebraic geometry. You can talk about this field with alias miler. She is at Harvard university.
answered Jan 8 at 17:28
husseinhussein
786
$endgroup$
add a comment |
$begingroup$
You can study about arithmetic topology for knots and links and this field in intersection of algebraic topology and algebraic geometry. You can talk about this field with alias miler. She is at Harvard university.
answered Jan 8 at 17:28
husseinhussein
786
$endgroup$
add a comment |
$begingroup$
You can study about arithmetic topology for knots and links and this field in intersection of algebraic topology and algebraic geometry. You can talk about this field with alias miler. She is at Harvard university.
answered Jan 8 at 17:28
husseinhussein
786
$endgroup$
You can study about arithmetic topology for knots and links and this field in intersection of algebraic topology and algebraic geometry. You can talk about this field with alias miler. She is at Harvard university.
answered Jan 8 at 17:28
husseinhussein
786
answered Jan 8 at 17:28
husseinhussein
786
answered Jan 8 at 17:28
husseinhussein
786
answered Jan 8 at 17:28
husseinhussein
786
786
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$begingroup$
this is a very naive response, but one specific object to look at is the étale fundamental group of a scheme and associated/similar things, since it is the algebro-geometric analogue of the usual $pi_1$.
$endgroup$
– Charlie
Mar 16 '16 at 1:19
$begingroup$
This is probably not a very good response. But what I think of is cohomology. Sometimes computing cohomology in algebraic geometry and algebraic topology are equivalent, even though they are defined quite differently. Perhaps, a problem of intersection between these two subjects, is to study some cohomology bridge among these two theories?
$endgroup$
– Nicolas Bourbaki
Mar 16 '16 at 1:25
1
$begingroup$
One of the important directions is towards Shafarevich's conjecture. Have a look at Kollar's book.
$endgroup$
– Mohan
Mar 16 '16 at 2:51
2
$begingroup$
"It was my lot to plant the harpoon of algebraic topology into the belly of the whale of algebraic geometry." -- Lefschetz. (Cf. Noether-Lefschetz theory.)
$endgroup$
– John Brevik
Mar 16 '16 at 3:20
$begingroup$
Motivic homotopy theory combines scheme theory with homotopy theory in order to define motives.
$endgroup$
– Fredrik Meyer
Mar 16 '16 at 9:16