5 ms·

A Fight to Fix Symplectic Geometry

60 points by KKKKkkkk1 10y ago

imode 10y ago
title should be "The Fight to Fix Symplectic Geometry".
- bnjmn 10y ago
  Yes! Wayyy too clickbaity without the niche adjective.
- dang 10y ago
  Belated thanks—a moderator saw your suggestion and fixed the title earlier today.
thanatropism 10y ago
I'm doing my thesis in numerics for symplectic flow maps, and was really excited to see the popular press to somewhat cover the world I'm living in. I also spent days on end joking about how the new generation of alt-medicine gurus will be "symplectic holistic healers" and how I want a piece of that. Out with quantum theory, in with symplectic physics!
- balsam 10y ago
  Hi, I'm just getting started out in this area (background: quantum physics). Are there review papers or slides you would recommend? For the numerics. I'm somewhat familiar with Wehrheim and da Silva, but didn't realize that people were going into numerics already.
  - thanatropism 10y ago
    On numerics: there are entire textbooks at a relatively elementary level (no differential geometry) with titles like "Simulating Hamiltonian dynamics". At the end of the day, verifying that a discrete map (approximating a Hamiltonian system) is symplectomorphic is a matter of linear algebra. On theory: there are a couple of good lectures by Dusa McDuff on YouTube; she also has a short tutorial-like PDF somewhere googleable.
auggierose 10y ago
Very interesting article, I also recommend reading the comments, as they help to illuminate the nature of the "fight". What seems to have happened there is this: A genius used his intuition to solve a foundational issue, but did not produce an adequate proof that he actually did. 16 years later others come in and make sure that he does. Now there is some controversial about who gets credit for the proof and for the solution of the foundational issue. There shouldn't be such a controversial. Credit is not a zero-sum game. If somebody now goes ahead and produces a formal machine-checked framework for symplectic geometry, that should deserve credit as well: Obviously not for solving the original problem, but for checking beyond any doubt that the produced mathematics is solid. Of course ideas are the life-blood of mathematics, but then again ideas are the life-blood of any creative human endeavour. What is special about mathematics are the proofs. So I don't see any problem in giving substantial credit to people who try to make proofs most rigorous.
- Certhas 10y ago
  I don't know which comments you are referring to exactly, but I would like to add that the comments by Motl should be taken very, very, very carefully. He is a climate change denier who believes women are over-represented in STEM. In fact his blog post on this article is titled: "How feminism helped to poison and politicize symplectic geometry". Not a reliable source of judgement by any stretch of the imagination. Edit: If you start reading the comments make sure you read to the end, where actual mathematicians are commenting.
  - auggierose 10y ago
    I am referring to all the comments.
  - deleted 10y ago
    [deleted]
- heymijo 10y ago
  Thanks for additional context. I don't agree with this statement though. "Credit is not a zero-sum game." History sadly says differently. It seems many discoveries are made concurrently, yet only one becomes synonymous with the discovery. And wouldn't you know it, as I went to Google a few examples (evolution and DNA were on my mind), I found Stigler's Law of Eponymy[0]. The derivation makes sense for our context here: "Historical acclaim for discoveries is often assigned to persons of note who bring attention to an idea that is not yet widely known, whether or not that person was its original inventor." [0]https://en.wikipedia.org/wiki/Stigler's_law_of_eponymy https://en.wikipedia.org/wiki/Stigler's_law_of_eponymy
mlmlmasd 10y ago
Has infinite mathematics actually been useful for anything at all? I mean uncountable infinities, higher cardinalities, etc - not limits, and other 'pseudo-infinite' abstractions. I have never encountered these things applied to anything other than infinite mathematics for infinite mathematics' sake. Seems like a waste of time.
- joshuata 10y ago
  Many of the ideas from infinite mathematics interact with finite mathematics in very interesting ways. For example, the traditional integral is the Riemann integral which cannot handle sets with too many (uncountably many) discontinuities. The Lebesgue integral loosens those restrictions by using measure to guarantee there aren't too many discontinuties. The Lebesgue integral, in turn, is used to integrate over function spaces, allowing for techniques like fourier transform and technologies such as mp3 encoding.
  - gugagore 10y ago
    Why does encoding of discrete-time signals have to do with Lebesgue integrals?

60 points by KKKKkkkk1 10y ago

imode 10y ago

title should be "The Fight to Fix Symplectic Geometry".

bnjmn 10y ago
Yes! Wayyy too clickbaity without the niche adjective.
dang 10y ago
Belated thanks—a moderator saw your suggestion and fixed the title earlier today.

thanatropism 10y ago

I'm doing my thesis in numerics for symplectic flow maps, and was really excited to see the popular press to somewhat cover the world I'm living in. I also spent days on end joking about how the new generation of alt-medicine gurus will be "symplectic holistic healers" and how I want a piece of that. Out with quantum theory, in with symplectic physics!

balsam 10y ago
Hi, I'm just getting started out in this area (background: quantum physics). Are there review papers or slides you would recommend? For the numerics. I'm somewhat familiar with Wehrheim and da Silva, but didn't realize that people were going into numerics already.
- thanatropism 10y ago
  On numerics: there are entire textbooks at a relatively elementary level (no differential geometry) with titles like "Simulating Hamiltonian dynamics". At the end of the day, verifying that a discrete map (approximating a Hamiltonian system) is symplectomorphic is a matter of linear algebra. On theory: there are a couple of good lectures by Dusa McDuff on YouTube; she also has a short tutorial-like PDF somewhere googleable.

auggierose 10y ago

Very interesting article, I also recommend reading the comments, as they help to illuminate the nature of the "fight". What seems to have happened there is this: A genius used his intuition to solve a foundational issue, but did not produce an adequate proof that he actually did. 16 years later others come in and make sure that he does. Now there is some controversial about who gets credit for the proof and for the solution of the foundational issue. There shouldn't be such a controversial. Credit is not a zero-sum game. If somebody now goes ahead and produces a formal machine-checked framework for symplectic geometry, that should deserve credit as well: Obviously not for solving the original problem, but for checking beyond any doubt that the produced mathematics is solid. Of course ideas are the life-blood of mathematics, but then again ideas are the life-blood of any creative human endeavour. What is special about mathematics are the proofs. So I don't see any problem in giving substantial credit to people who try to make proofs most rigorous.

Certhas 10y ago
I don't know which comments you are referring to exactly, but I would like to add that the comments by Motl should be taken very, very, very carefully. He is a climate change denier who believes women are over-represented in STEM. In fact his blog post on this article is titled: "How feminism helped to poison and politicize symplectic geometry". Not a reliable source of judgement by any stretch of the imagination. Edit: If you start reading the comments make sure you read to the end, where actual mathematicians are commenting.
- auggierose 10y ago
  I am referring to all the comments.
- deleted 10y ago
  [deleted]
heymijo 10y ago
Thanks for additional context. I don't agree with this statement though. "Credit is not a zero-sum game." History sadly says differently. It seems many discoveries are made concurrently, yet only one becomes synonymous with the discovery. And wouldn't you know it, as I went to Google a few examples (evolution and DNA were on my mind), I found Stigler's Law of Eponymy[0]. The derivation makes sense for our context here: "Historical acclaim for discoveries is often assigned to persons of note who bring attention to an idea that is not yet widely known, whether or not that person was its original inventor." [0]https://en.wikipedia.org/wiki/Stigler's_law_of_eponymy https://en.wikipedia.org/wiki/Stigler's_law_of_eponymy

mlmlmasd 10y ago

Has infinite mathematics actually been useful for anything at all? I mean uncountable infinities, higher cardinalities, etc - not limits, and other 'pseudo-infinite' abstractions. I have never encountered these things applied to anything other than infinite mathematics for infinite mathematics' sake. Seems like a waste of time.

joshuata 10y ago
Many of the ideas from infinite mathematics interact with finite mathematics in very interesting ways. For example, the traditional integral is the Riemann integral which cannot handle sets with too many (uncountably many) discontinuities. The Lebesgue integral loosens those restrictions by using measure to guarantee there aren't too many discontinuties. The Lebesgue integral, in turn, is used to integrate over function spaces, allowing for techniques like fourier transform and technologies such as mp3 encoding.
- gugagore 10y ago
  Why does encoding of discrete-time signals have to do with Lebesgue integrals?