Mixing
moduli space of linear partial differential equations?
Clash Royale CLAN TAG#URR8PPP
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is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities...?
This is in connection with a quote from someone on the web that i saw long time ago, at that time i had contacted the author but they chose not to answer.
The quote:
"In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety , where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety."
Any pointers/refs on any of the points made in the quote would be gratefully recieved...
ag.algebraic-geometry ap.analysis-of-pdes moduli-spaces linear-pde
asked 3 hours ago
david mercurio
663
add a comment |Â
up vote
4
down vote
favorite
is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities...?
This is in connection with a quote from someone on the web that i saw long time ago, at that time i had contacted the author but they chose not to answer.
The quote:
"In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety , where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety."
Any pointers/refs on any of the points made in the quote would be gratefully recieved...
ag.algebraic-geometry ap.analysis-of-pdes moduli-spaces linear-pde
asked 3 hours ago
david mercurio
663
It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
The original quote is here.
– j.c.
56 mins ago
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities...?
This is in connection with a quote from someone on the web that i saw long time ago, at that time i had contacted the author but they chose not to answer.
The quote:
"In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety , where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety."
Any pointers/refs on any of the points made in the quote would be gratefully recieved...
ag.algebraic-geometry ap.analysis-of-pdes moduli-spaces linear-pde
asked 3 hours ago
david mercurio
663
is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities...?
This is in connection with a quote from someone on the web that i saw long time ago, at that time i had contacted the author but they chose not to answer.
The quote:
"In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety , where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety."
Any pointers/refs on any of the points made in the quote would be gratefully recieved...
ag.algebraic-geometry ap.analysis-of-pdes moduli-spaces linear-pde
ag.algebraic-geometry ap.analysis-of-pdes moduli-spaces linear-pde
asked 3 hours ago
david mercurio
663
asked 3 hours ago
david mercurio
663
edited 3 hours ago
asked 3 hours ago
david mercurio
663
asked 3 hours ago
david mercurio
663
asked 3 hours ago
david mercurio
663
663
It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
The original quote is here.
– j.c.
56 mins ago
add a comment |Â
It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
The original quote is here.
– j.c.
56 mins ago
It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
The original quote is here.
– j.c.
56 mins ago
The original quote is here.
– j.c.
56 mins ago
add a comment |Â
1 Answer
1
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Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $mathbbRn$ can be written as
$$
Pu = sum_alpha a^alphapartial_alpha u,
$$
where each coefficient $a^alpha$ is a smooth function, $alpha = (alpha_1, dots, alpha_n)$ and $partial_alpha = (partial_1)^alpha_1cdots(partial_n)^alpha_n$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
sigma(x,xi) = sum_ = k a^alpha(x)xi_alpha,
$$
where $xi = (xi_1, dots, xi_n) in mathbbR^n$ and $xi_alpha = (xi_1)^alpha_1cdots(xi_n)^alpha_n$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $mathbbRP^n-1$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
answered 1 hour ago
Deane Yang
19.6k560138
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $mathbbRn$ can be written as
$$
Pu = sum_alpha a^alphapartial_alpha u,
$$
where each coefficient $a^alpha$ is a smooth function, $alpha = (alpha_1, dots, alpha_n)$ and $partial_alpha = (partial_1)^alpha_1cdots(partial_n)^alpha_n$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
sigma(x,xi) = sum_ = k a^alpha(x)xi_alpha,
$$
where $xi = (xi_1, dots, xi_n) in mathbbR^n$ and $xi_alpha = (xi_1)^alpha_1cdots(xi_n)^alpha_n$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $mathbbRP^n-1$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
answered 1 hour ago
Deane Yang
19.6k560138
add a comment |Â
up vote
4
down vote
Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $mathbbRn$ can be written as
$$
Pu = sum_alpha a^alphapartial_alpha u,
$$
where each coefficient $a^alpha$ is a smooth function, $alpha = (alpha_1, dots, alpha_n)$ and $partial_alpha = (partial_1)^alpha_1cdots(partial_n)^alpha_n$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
sigma(x,xi) = sum_ = k a^alpha(x)xi_alpha,
$$
where $xi = (xi_1, dots, xi_n) in mathbbR^n$ and $xi_alpha = (xi_1)^alpha_1cdots(xi_n)^alpha_n$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $mathbbRP^n-1$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
answered 1 hour ago
Deane Yang
19.6k560138
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $mathbbRn$ can be written as
$$
Pu = sum_alpha a^alphapartial_alpha u,
$$
where each coefficient $a^alpha$ is a smooth function, $alpha = (alpha_1, dots, alpha_n)$ and $partial_alpha = (partial_1)^alpha_1cdots(partial_n)^alpha_n$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
sigma(x,xi) = sum_ = k a^alpha(x)xi_alpha,
$$
where $xi = (xi_1, dots, xi_n) in mathbbR^n$ and $xi_alpha = (xi_1)^alpha_1cdots(xi_n)^alpha_n$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $mathbbRP^n-1$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
answered 1 hour ago
Deane Yang
19.6k560138
Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $mathbbRn$ can be written as
$$
Pu = sum_alpha a^alphapartial_alpha u,
$$
where each coefficient $a^alpha$ is a smooth function, $alpha = (alpha_1, dots, alpha_n)$ and $partial_alpha = (partial_1)^alpha_1cdots(partial_n)^alpha_n$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
sigma(x,xi) = sum_ = k a^alpha(x)xi_alpha,
$$
where $xi = (xi_1, dots, xi_n) in mathbbR^n$ and $xi_alpha = (xi_1)^alpha_1cdots(xi_n)^alpha_n$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $mathbbRP^n-1$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
answered 1 hour ago
Deane Yang
19.6k560138
answered 1 hour ago
Deane Yang
19.6k560138
answered 1 hour ago
Deane Yang
19.6k560138
answered 1 hour ago
Deane Yang
19.6k560138
19.6k560138
add a comment |Â
add a comment |Â
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It is not finite dimensional, but I suppose that won't stop some daring people.
– Ben McKay
1 hour ago
The original quote is here.
– j.c.
56 mins ago