Is differential topology a dying field? [closed]Is the field of q-series 'dead'?Can the Math 2.0 Forum's Closure be prevented?
Is differential topology a dying field? [closed]
Is the field of q-series 'dead'?Can the Math 2.0 Forum's Closure be prevented?
$begingroup$
I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.
I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!
soft-question differential-topology
edited May 30 at 23:04
Wojowu
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
$endgroup$
closed as primarily opinion-based by Timothy Chow, Mike Miller, Deane Yang, LSpice, Todd Trimble♦ May 30 at 21:40
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 18 more comments
$begingroup$
I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.
I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!
soft-question differential-topology
edited May 30 at 23:04
Wojowu
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
$endgroup$
closed as primarily opinion-based by Timothy Chow, Mike Miller, Deane Yang, LSpice, Todd Trimble♦ May 30 at 21:40
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
14
$begingroup$
"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
$endgroup$
– Francesco Polizzi
May 30 at 12:04
4
$begingroup$
I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
$endgroup$
– Todd Trimble♦
May 30 at 12:04
40
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
$endgroup$
– Mark Grant
May 30 at 12:17
5
$begingroup$
@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
$endgroup$
– user140765
May 30 at 13:15
11
$begingroup$
+1 "we can just call each other wrong and stop there"
$endgroup$
– Nik Weaver
May 30 at 14:36
|
show 18 more comments
$begingroup$
I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.
I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!
soft-question differential-topology
edited May 30 at 23:04
Wojowu
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
$endgroup$
I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.
I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!
soft-question differential-topology
soft-question differential-topology
edited May 30 at 23:04
Wojowu
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
edited May 30 at 23:04
Wojowu
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
edited May 30 at 23:04
Wojowu
8,03613462
edited May 30 at 23:04
Wojowu
8,03613462
edited May 30 at 23:04
Wojowu
8,03613462
8,03613462
asked May 30 at 11:42
James BaxterJames Baxter
38518
asked May 30 at 11:42
James BaxterJames Baxter
38518
asked May 30 at 11:42
James BaxterJames Baxter
38518
38518
closed as primarily opinion-based by Timothy Chow, Mike Miller, Deane Yang, LSpice, Todd Trimble♦ May 30 at 21:40
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as primarily opinion-based by Timothy Chow, Mike Miller, Deane Yang, LSpice, Todd Trimble♦ May 30 at 21:40
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
14
$begingroup$
"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
$endgroup$
– Francesco Polizzi
May 30 at 12:04
4
$begingroup$
I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
$endgroup$
– Todd Trimble♦
May 30 at 12:04
40
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
$endgroup$
– Mark Grant
May 30 at 12:17
5
$begingroup$
@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
$endgroup$
– user140765
May 30 at 13:15
11
$begingroup$
+1 "we can just call each other wrong and stop there"
$endgroup$
– Nik Weaver
May 30 at 14:36
|
show 18 more comments
14
$begingroup$
"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
$endgroup$
– Francesco Polizzi
May 30 at 12:04
4
$begingroup$
I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
$endgroup$
– Todd Trimble♦
May 30 at 12:04
40
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
$endgroup$
– Mark Grant
May 30 at 12:17
5
$begingroup$
@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
$endgroup$
– user140765
May 30 at 13:15
11
$begingroup$
+1 "we can just call each other wrong and stop there"
$endgroup$
– Nik Weaver
May 30 at 14:36
14
14
$begingroup$
"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
$endgroup$
– Francesco Polizzi
May 30 at 12:04
$begingroup$
"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
$endgroup$
– Francesco Polizzi
May 30 at 12:04
4
4
$begingroup$
I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
$endgroup$
– Todd Trimble♦
May 30 at 12:04
$begingroup$
I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
$endgroup$
– Todd Trimble♦
May 30 at 12:04
40
40
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
$endgroup$
– Mark Grant
May 30 at 12:17
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
$endgroup$
– Mark Grant
May 30 at 12:17
5
5
$begingroup$
@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
$endgroup$
– user140765
May 30 at 13:15
$begingroup$
@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
$endgroup$
– user140765
May 30 at 13:15
11
11
$begingroup$
+1 "we can just call each other wrong and stop there"
$endgroup$
– Nik Weaver
May 30 at 14:36
$begingroup$
+1 "we can just call each other wrong and stop there"
$endgroup$
– Nik Weaver
May 30 at 14:36
|
show 18 more comments
2 Answers
2
active
oldest
votes
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
$endgroup$
add a comment |
$begingroup$
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.
There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.
The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.
Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.
Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:
Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, hatA(X) = 1$, and $hatA(X, T_C) = 0$?
Here $hatA$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.
Shmuel Weinberger has championed the study of decidability
questions in differential topology.
The Novikov conjectures would imply that $mathcalL$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.
There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
$endgroup$
1
$begingroup$
Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
$endgroup$
– Aleksandar Milivojevic
Jun 1 at 18:40
1
$begingroup$
@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
$endgroup$
– Ian Agol
Jun 4 at 15:24
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
$endgroup$
add a comment |
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
$endgroup$
add a comment |
$begingroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
$endgroup$
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
answered May 30 at 15:14
Mark GrantMark Grant
23.5k664142
23.5k664142
add a comment |
add a comment |
$begingroup$
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.
There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.
The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.
Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.
Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:
Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, hatA(X) = 1$, and $hatA(X, T_C) = 0$?
Here $hatA$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.
Shmuel Weinberger has championed the study of decidability
questions in differential topology.
The Novikov conjectures would imply that $mathcalL$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.
There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
$endgroup$
1
$begingroup$
Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
$endgroup$
– Aleksandar Milivojevic
Jun 1 at 18:40
1
$begingroup$
@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
$endgroup$
– Ian Agol
Jun 4 at 15:24
add a comment |
$begingroup$
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.
There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.
The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.
Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.
Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:
Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, hatA(X) = 1$, and $hatA(X, T_C) = 0$?
Here $hatA$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.
Shmuel Weinberger has championed the study of decidability
questions in differential topology.
The Novikov conjectures would imply that $mathcalL$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.
There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
$endgroup$
1
$begingroup$
Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
$endgroup$
– Aleksandar Milivojevic
Jun 1 at 18:40
1
$begingroup$
@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
$endgroup$
– Ian Agol
Jun 4 at 15:24
add a comment |
$begingroup$
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.
There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.
The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.
Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.
Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:
Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, hatA(X) = 1$, and $hatA(X, T_C) = 0$?
Here $hatA$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.
Shmuel Weinberger has championed the study of decidability
questions in differential topology.
The Novikov conjectures would imply that $mathcalL$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.
There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
$endgroup$
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.
There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.
The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.
Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.
Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:
Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, hatA(X) = 1$, and $hatA(X, T_C) = 0$?
Here $hatA$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.
Shmuel Weinberger has championed the study of decidability
questions in differential topology.
The Novikov conjectures would imply that $mathcalL$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.
There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
edited May 30 at 20:00
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
answered May 30 at 18:52
Ian AgolIan Agol
51k3133252
51k3133252
1
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Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
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– Aleksandar Milivojevic
Jun 1 at 18:40
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@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
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– Ian Agol
Jun 4 at 15:24
add a comment |
1
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Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
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– Aleksandar Milivojevic
Jun 1 at 18:40
1
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@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
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– Ian Agol
Jun 4 at 15:24
1
1
$begingroup$
Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
$endgroup$
– Aleksandar Milivojevic
Jun 1 at 18:40
$begingroup$
Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something.
$endgroup$
– Aleksandar Milivojevic
Jun 1 at 18:40
1
1
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@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
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– Ian Agol
Jun 4 at 15:24
$begingroup$
@AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference.
$endgroup$
– Ian Agol
Jun 4 at 15:24
add a comment |
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"Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field?
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– Francesco Polizzi
May 30 at 12:04
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I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford.
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– Todd Trimble♦
May 30 at 12:04
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Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
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– Mark Grant
May 30 at 12:17
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@NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something?
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– user140765
May 30 at 13:15
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+1 "we can just call each other wrong and stop there"
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– Nik Weaver
May 30 at 14:36