What is the moduli space of $ntimes n$ matrices with values on some field $k$ under conjugation?
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Maybe we can start with an easy example. I want to understand what is the moduli space of say $1times 1$ matrices under conjugation. So if such a matrix is $A$, the conjugation means $A = G^{-1}AG$ where say $Gin GL_n$. What is the moduli space then? And what is the corresponding moduli space for $2times 2$ such matrices? Let's say that $k = mathbb{C}$. What is the moduli space for $n times n$ such matrices? And do singularities appear?
This is the first question a speaker asked at a seminar last week. And I really want to understand not only what the moduli space of such matrices is but also when and why singularities appear.
linear-algebra algebraic-geometry moduli-space
asked Nov 2 '15 at 19:05
MarionMarion
825618
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show 2 more comments
$begingroup$
Maybe we can start with an easy example. I want to understand what is the moduli space of say $1times 1$ matrices under conjugation. So if such a matrix is $A$, the conjugation means $A = G^{-1}AG$ where say $Gin GL_n$. What is the moduli space then? And what is the corresponding moduli space for $2times 2$ such matrices? Let's say that $k = mathbb{C}$. What is the moduli space for $n times n$ such matrices? And do singularities appear?
This is the first question a speaker asked at a seminar last week. And I really want to understand not only what the moduli space of such matrices is but also when and why singularities appear.
linear-algebra algebraic-geometry moduli-space
asked Nov 2 '15 at 19:05
MarionMarion
825618
$endgroup$
1
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
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– Travis
Nov 2 '15 at 20:12
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Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
2
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
2
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You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
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– A.P.
Nov 2 '15 at 20:27
3
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Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53
|
show 2 more comments
$begingroup$
Maybe we can start with an easy example. I want to understand what is the moduli space of say $1times 1$ matrices under conjugation. So if such a matrix is $A$, the conjugation means $A = G^{-1}AG$ where say $Gin GL_n$. What is the moduli space then? And what is the corresponding moduli space for $2times 2$ such matrices? Let's say that $k = mathbb{C}$. What is the moduli space for $n times n$ such matrices? And do singularities appear?
This is the first question a speaker asked at a seminar last week. And I really want to understand not only what the moduli space of such matrices is but also when and why singularities appear.
linear-algebra algebraic-geometry moduli-space
asked Nov 2 '15 at 19:05
MarionMarion
825618
$endgroup$
Maybe we can start with an easy example. I want to understand what is the moduli space of say $1times 1$ matrices under conjugation. So if such a matrix is $A$, the conjugation means $A = G^{-1}AG$ where say $Gin GL_n$. What is the moduli space then? And what is the corresponding moduli space for $2times 2$ such matrices? Let's say that $k = mathbb{C}$. What is the moduli space for $n times n$ such matrices? And do singularities appear?
This is the first question a speaker asked at a seminar last week. And I really want to understand not only what the moduli space of such matrices is but also when and why singularities appear.
linear-algebra algebraic-geometry moduli-space
linear-algebra algebraic-geometry moduli-space
asked Nov 2 '15 at 19:05
MarionMarion
825618
asked Nov 2 '15 at 19:05
MarionMarion
825618
asked Nov 2 '15 at 19:05
MarionMarion
825618
asked Nov 2 '15 at 19:05
MarionMarion
825618
asked Nov 2 '15 at 19:05
MarionMarion
825618
825618
1
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
$endgroup$
– Travis
Nov 2 '15 at 20:12
$begingroup$
Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
2
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
2
$begingroup$
You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
$endgroup$
– A.P.
Nov 2 '15 at 20:27
3
$begingroup$
Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53
|
show 2 more comments
1
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
$endgroup$
– Travis
Nov 2 '15 at 20:12
$begingroup$
Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
2
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
2
$begingroup$
You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
$endgroup$
– A.P.
Nov 2 '15 at 20:27
3
$begingroup$
Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53
1
1
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
$endgroup$
– Travis
Nov 2 '15 at 20:12
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
$endgroup$
– Travis
Nov 2 '15 at 20:12
$begingroup$
Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
$begingroup$
Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
2
2
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
2
2
$begingroup$
You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
$endgroup$
– A.P.
Nov 2 '15 at 20:27
$begingroup$
You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
$endgroup$
– A.P.
Nov 2 '15 at 20:27
3
3
$begingroup$
Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53
$begingroup$
Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53
|
show 2 more comments
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1
$begingroup$
Do you mean, What is the structure of the space $M_n(k) / sim$, where $A sim B$ if $A$ and $B$ are similar?
$endgroup$
– Travis
Nov 2 '15 at 20:12
$begingroup$
Was that a rethorical question? If so, you could try asking the speaker him/herself...
$endgroup$
– A.P.
Nov 2 '15 at 20:14
2
$begingroup$
No, I'm asking OP to verify that this is precisely what they mean by moduli space here. (Or is interpreting what the speaker meant by this term part of OP's question?)
$endgroup$
– Travis
Nov 2 '15 at 20:19
2
$begingroup$
You made me smile @Travis. My question was directed at the OP, regarding the intent behind the speaker's question. :)
$endgroup$
– A.P.
Nov 2 '15 at 20:27
3
$begingroup$
Mumford has a beautiful expository paper about this. dam.brown.edu/people/mumford/alg_geom/papers/…
$endgroup$
– Fredrik Meyer
Nov 2 '15 at 21:53