Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general,...
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I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or perturbation.
I know there are way ways to treat these problems numerically, e.g. the immersed boundary method for free boundary problems, and my limited impressions of general relativity suggest that there is some rigorous, differential description of how mass deforms space-time dynamically.
I guess the model situation I'm thinking of is some sphere (or torus; insert smooth surface here) whose boundary can stretch due to some interaction with an external vector field (e.g. electric, fluid flow) or local perturbation (e.g. chemical reaction). Have there been attempts to abstract these scientifically motivated problems to some level of rigorous analysis?
differential-geometry soft-question smooth-manifolds
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
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add a comment |
$begingroup$
I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or perturbation.
I know there are way ways to treat these problems numerically, e.g. the immersed boundary method for free boundary problems, and my limited impressions of general relativity suggest that there is some rigorous, differential description of how mass deforms space-time dynamically.
I guess the model situation I'm thinking of is some sphere (or torus; insert smooth surface here) whose boundary can stretch due to some interaction with an external vector field (e.g. electric, fluid flow) or local perturbation (e.g. chemical reaction). Have there been attempts to abstract these scientifically motivated problems to some level of rigorous analysis?
differential-geometry soft-question smooth-manifolds
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
$endgroup$
2
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
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– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
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– CasaBonita
Apr 28 '16 at 16:02
1
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
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– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42
add a comment |
$begingroup$
I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or perturbation.
I know there are way ways to treat these problems numerically, e.g. the immersed boundary method for free boundary problems, and my limited impressions of general relativity suggest that there is some rigorous, differential description of how mass deforms space-time dynamically.
I guess the model situation I'm thinking of is some sphere (or torus; insert smooth surface here) whose boundary can stretch due to some interaction with an external vector field (e.g. electric, fluid flow) or local perturbation (e.g. chemical reaction). Have there been attempts to abstract these scientifically motivated problems to some level of rigorous analysis?
differential-geometry soft-question smooth-manifolds
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
$endgroup$
I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or perturbation.
I know there are way ways to treat these problems numerically, e.g. the immersed boundary method for free boundary problems, and my limited impressions of general relativity suggest that there is some rigorous, differential description of how mass deforms space-time dynamically.
I guess the model situation I'm thinking of is some sphere (or torus; insert smooth surface here) whose boundary can stretch due to some interaction with an external vector field (e.g. electric, fluid flow) or local perturbation (e.g. chemical reaction). Have there been attempts to abstract these scientifically motivated problems to some level of rigorous analysis?
differential-geometry soft-question smooth-manifolds
differential-geometry soft-question smooth-manifolds
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
asked Apr 28 '16 at 15:55
CasaBonitaCasaBonita
1206
1206
2
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
$endgroup$
– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
$endgroup$
– CasaBonita
Apr 28 '16 at 16:02
1
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
$endgroup$
– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42
add a comment |
2
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
$endgroup$
– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
$endgroup$
– CasaBonita
Apr 28 '16 at 16:02
1
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
$endgroup$
– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42
2
2
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
$endgroup$
– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
$endgroup$
– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
$endgroup$
– CasaBonita
Apr 28 '16 at 16:02
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
$endgroup$
– CasaBonita
Apr 28 '16 at 16:02
1
1
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
$endgroup$
– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
$endgroup$
– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42
add a comment |
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2
$begingroup$
The whole field of geometric flows studies this kind of thing - particularly of the "evolving to a steady-state configuration" variety.
$endgroup$
– Anthony Carapetis
Apr 28 '16 at 15:58
$begingroup$
Let's assume I have a background in undergraduate differential geometry and some first year graduate analysis. What do I need to do to get involved in this field? Is there a canonical text or comprehensive review of geometric flows?
$endgroup$
– CasaBonita
Apr 28 '16 at 16:02
1
$begingroup$
The field is very broad and deep. You need a better background than the one you currently have. To get some feel for it, take a look at the Wikipedia article in the mean curvature flow. My suggestion is to get a better background in Riemannian Geometry (say, do Carmo) and nonlinear PDEs (say, Evans) before trying geometric flows.
$endgroup$
– Moishe Kohan
May 2 '16 at 3:35
$begingroup$
@studiosus didn't expect to be able to jump into it immediately. I just wanted a heading, so to speak. Thanks for the advice!
$endgroup$
– CasaBonita
May 2 '16 at 3:42