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How to explain to an engineer what algebraic geometry is?
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This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.
Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.
So, say the engineer is happy with the start
"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."
An engineer certainly understands this.
"...with complex coefficients..."
Here she's starting to feel a bit intimidated: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.
Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.
Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?
Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?
ag.algebraic-geometry soft-question big-list mathematics-education
edited 12 hours ago
Martin Sleziak
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
 |Â
show 4 more comments
up vote
11
down vote
favorite
This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.
Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.
So, say the engineer is happy with the start
"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."
An engineer certainly understands this.
"...with complex coefficients..."
Here she's starting to feel a bit intimidated: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.
Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.
Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?
Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?
ag.algebraic-geometry soft-question big-list mathematics-education
edited 12 hours ago
Martin Sleziak
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
12
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
4
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
2
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
7
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
7
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago
 |Â
show 4 more comments
up vote
11
down vote
favorite
up vote
11
down vote
favorite
This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.
Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.
So, say the engineer is happy with the start
"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."
An engineer certainly understands this.
"...with complex coefficients..."
Here she's starting to feel a bit intimidated: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.
Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.
Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?
Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?
ag.algebraic-geometry soft-question big-list mathematics-education
edited 12 hours ago
Martin Sleziak
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.
Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.
So, say the engineer is happy with the start
"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."
An engineer certainly understands this.
"...with complex coefficients..."
Here she's starting to feel a bit intimidated: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.
Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.
Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?
Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?
ag.algebraic-geometry soft-question big-list mathematics-education
ag.algebraic-geometry soft-question big-list mathematics-education
edited 12 hours ago
Martin Sleziak
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
edited 12 hours ago
Martin Sleziak
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
edited 12 hours ago
Martin Sleziak
2,78432028
edited 12 hours ago
Martin Sleziak
2,78432028
edited 12 hours ago
Martin Sleziak
2,78432028
2,78432028
asked 20 hours ago
Qfwfq
9,771877164
asked 20 hours ago
Qfwfq
9,771877164
asked 20 hours ago
Qfwfq
9,771877164
9,771877164
12
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
4
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
2
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
7
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
7
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago
 |Â
show 4 more comments
12
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
4
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
2
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
7
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
7
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago
12
12
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
4
4
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
2
2
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
7
7
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
7
7
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago
 |Â
show 4 more comments
4 Answers
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up vote
7
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Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...
answered 7 hours ago
Donu Arapura
24.5k264120
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up vote
6
down vote
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
answered 10 hours ago
Ben McKay
13.7k22757
add a comment |Â
up vote
4
down vote
In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.
Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case:
You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.
answered 3 hours ago
Steven Landsburg
15.8k365114
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up vote
0
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I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality†part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is tooâ€Â
This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could go up to a quadratic extension (!) to “see what happens†and let them play around
I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.
I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena
answered 25 mins ago
Rohit Chatterjee
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...
answered 7 hours ago
Donu Arapura
24.5k264120
add a comment |Â
up vote
7
down vote
Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...
answered 7 hours ago
Donu Arapura
24.5k264120
add a comment |Â
up vote
7
down vote
up vote
7
down vote
Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...
answered 7 hours ago
Donu Arapura
24.5k264120
Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...
answered 7 hours ago
Donu Arapura
24.5k264120
edited 6 hours ago
answered 7 hours ago
Donu Arapura
24.5k264120
answered 7 hours ago
Donu Arapura
24.5k264120
answered 7 hours ago
Donu Arapura
24.5k264120
24.5k264120
add a comment |Â
add a comment |Â
up vote
6
down vote
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
answered 10 hours ago
Ben McKay
13.7k22757
add a comment |Â
up vote
6
down vote
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
answered 10 hours ago
Ben McKay
13.7k22757
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
answered 10 hours ago
Ben McKay
13.7k22757
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.
answered 10 hours ago
Ben McKay
13.7k22757
answered 10 hours ago
Ben McKay
13.7k22757
answered 10 hours ago
Ben McKay
13.7k22757
answered 10 hours ago
Ben McKay
13.7k22757
13.7k22757
add a comment |Â
add a comment |Â
up vote
4
down vote
In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.
Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case:
You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.
answered 3 hours ago
Steven Landsburg
15.8k365114
add a comment |Â
up vote
4
down vote
In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.
Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case:
You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.
answered 3 hours ago
Steven Landsburg
15.8k365114
add a comment |Â
up vote
4
down vote
up vote
4
down vote
In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.
Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case:
You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.
answered 3 hours ago
Steven Landsburg
15.8k365114
In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.
Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.
I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case:
You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.
answered 3 hours ago
Steven Landsburg
15.8k365114
answered 3 hours ago
Steven Landsburg
15.8k365114
answered 3 hours ago
Steven Landsburg
15.8k365114
answered 3 hours ago
Steven Landsburg
15.8k365114
15.8k365114
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I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality†part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is tooâ€Â
This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could go up to a quadratic extension (!) to “see what happens†and let them play around
I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.
I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena
answered 25 mins ago
Rohit Chatterjee
1012
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up vote
0
down vote
I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality†part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is tooâ€Â
This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could go up to a quadratic extension (!) to “see what happens†and let them play around
I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.
I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena
answered 25 mins ago
Rohit Chatterjee
1012
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality†part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is tooâ€Â
This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could go up to a quadratic extension (!) to “see what happens†and let them play around
I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.
I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena
answered 25 mins ago
Rohit Chatterjee
1012
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality†part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is tooâ€Â
This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.
Then you could go up to a quadratic extension (!) to “see what happens†and let them play around
I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.
I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena
answered 25 mins ago
Rohit Chatterjee
1012
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 25 mins ago
Rohit Chatterjee
1012
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 25 mins ago
Rohit Chatterjee
1012
answered 25 mins ago
Rohit Chatterjee
1012
1012
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Rohit Chatterjee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
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12
To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse
– Kevin Casto
20 hours ago
4
One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry.
– François Brunault
20 hours ago
2
Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22.
– Gerhard Paseman
19 hours ago
7
Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry.
– Zach Teitler
19 hours ago
7
@ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math.
– Somatic Custard
18 hours ago