Timeline for Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level
Current License: CC BY-SA 4.0
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| May 25, 2018 at 23:42 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| May 25, 2018 at 20:01 | comment | added | user13113 | @AlexandreEremenko: My impression of semi-algebraic geometry is that it both lacks the obstacles the modern formulation of algebraic geometry is meant to circumvent (e.g. implicit function theorem) and lacks the features that are very natural in the language of schemes. E.g. interescting $y=0$ and $y=x^2$ as schemes rather than point sets tells you something interesting about intersection multiplicities. But what could you hope to learn from intersecting $y=0$ and $y=\max(x^2, x^3)$ in a more sophisticated manner? | |
| May 25, 2018 at 18:31 | comment | added | Alexandre Eremenko | It is hard to tell what is "applied" and what is "applicable". Most of the examples that I know is alrebraic geometry applied in other fields of mathematics, like differential equations. | |
| May 25, 2018 at 18:29 | comment | added | Alexandre Eremenko | Yes, some sophisticated (real) algebraic geometry is used in robotics. People who do this are unfamiliar with the modern language. They use classical language. | |
| May 25, 2018 at 18:25 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| May 25, 2018 at 18:23 | comment | added | Paul Siegel | Are the examples that you have truly "applied" rather than possibly "applicable"? For instance, I've asked experts in robotics if they ever use any geometry in their work, and they usually respond that they remember hearing about it in graduate school but that the problems it solves are at best peripheral to the subject. They do not say the same about, say, statistics or linear algebra. | |
| May 25, 2018 at 18:19 | comment | added | No One | You don't have to prove that "classical theory won't be sufficient" for these papers at all (no one can)... please just add those papers in your answer... | |
| May 25, 2018 at 18:16 | comment | added | Alexandre Eremenko | I know such papers but cannot prove that "classical theory won't be sufficient". As I said, this was the author's choice. And I predict that you will not obtain an example which you want. Even if a paper is written in the modern language, no one can prove that this cannot be translated into a classical language. | |
| May 25, 2018 at 18:13 | comment | added | No One | I partially agree with you. But a my question indicates, I want a direction/approach in which "the classical theory of varieties won't be sufficient." Do you know any applied algebraic geometry paper written in modern language that "it is hard for people without this modern training to understand"? A few references would be greatly appreciated! | |
| May 25, 2018 at 18:07 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |