Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What is Symplectic Geometry? (2016) [pdf]

ams.org

Post reply on HN
Print view View original

21–30 of 30 posts

Re: What is Symplectic Geometry? (2016) [pdf]

#21

Post by ajkjk »

danharaj wrote: post #14
ajkjk wrote: post #8

Earlier quoted context omitted.

Yeah, but I'm interested in understanding things the other direction: if there is not some major class of problems that are best described by symplectic manifolds, then why care about Darboux's theorem at all? If there are, why isn't that front-and-center? This article conspicuously avoids motivating symplectic geometry _at all_, which is so frustrating. It mentions connections to subjects, but it doesn't mention why…

Is "any hamiltonian dynamics on any phase space of a physical system" not a major enough class of problems for you? This is one way mathematicians study physics. Are you asking for applications of symplectic geometry? If so, here, I've been reading a cluster of papers in this area for a few weeks now: https://www.math.colostate.edu/~clayton/research/papers/fram...

No, I'm plenty aware of the applications. My complaint is that this article hardly mentioned them! What's the point of an article explaining a theory without motivating why it exist?

Re: What is Symplectic Geometry? (2016) [pdf]

#23

Post by sidek »

Ragib_Zaman wrote: post #19

Quite interestingly, Symplectic Geometry is currently under review/investigation for some of the foundational papers in the field having serious gaps and outright errors after closer inspection. These concerns were always spoken of in hush hush tones and only in recent times have people stated their concerns publically. Some of the original authors refuse to retract their papers despite being assured their academic p…

As a result, lots of recent work is being done in the algebraic setting (rather than analytic), where the foundations are on much firmer footing.

Being algebraic symplectic is a much stronger condition than analytic symplectic, but is still interesting enough (and, for geometry related to linear algebra problems, as is often relevant in CS, is not a very strong restriction at all.)

Re: What is Symplectic Geometry? (2016) [pdf]

#24

Post by tobmlt »

pjbk wrote: post #2

For those into physics, I wholeheartedly recommend Marsden and Ratiu's book, "Introduction to Mechanics and Symmetry", which deals mainly with the different formulations of physics applied to symplectic and associated geometries.

Thanks for the recommendation! I didn't know this one. I try and lap up everything I can by Marsden, (though more often through the lens of applied researchers: e.g. Desbrun, Hirani, Crane, and others -- much involving computer graphics and/or discrete differential geometry applied to physical simulation. In short, I better say that I am not familiar with the scope of Marsden's work. I am sure much of it is beyond me, but gosh darned it, the exterior calculus is beautiful and these guys write brilliantly readable stuff for an engineer.

Even as a hydrodynamics software guy, I found the computer graphics research community to be the easiest entry-point for, especially, the topology and modern differential geometry. It's especially nice when they do a simulation paper with a high end geometric/analytic approach.

This might be a good place to go in order to have a start at, say, Arnold's ``topological methods in hydrodynamics'' or anything TQFT-esque.

Re: What is Symplectic Geometry? (2016) [pdf]

#25

Post by mikhailfranco »

ajkjk wrote: post #21
danharaj wrote: post #14

Earlier quoted context omitted.

Is "any hamiltonian dynamics on any phase space of a physical system" not a major enough class of problems for you? This is one way mathematicians study physics. Are you asking for applications of symplectic geometry? If so, here, I've been reading a cluster of papers in this area for a few weeks now: https://www.math.colostate.edu/~clayton/research/papers/fram...

No, I'm plenty aware of the applications. My complaint is that this article hardly mentioned them! What's the point of an article explaining a theory without motivating why it exist?

Yes, too many mathematicians write about areas of mathematics that could be useful to others, without giving any motivation, or any hand-waving explanations, or the critical distinctions that make it a useful approach.

There is a barrier of communication that (most) mathematicians don't want to take any time to overcome. Often, the opening is enticing, then it's straight into lemmas, formal language, and citing of famous results by name. Their slight inclination to explain it to others dissipates in the first paragraph, then it's off to the races to impress their peers.

This is easy to see on Wikipedia - most mathematics articles are utterly useless to any non-mathematician who wants to get a general appreciation of an approach to see if it could shed some light on their problem.

In almost all cases, it is better to read domain generalists coming the other way into the higher mathematics.

Of course, one shining exception is John Baez:

http://math.ucr.edu/home/baez/twf.html

http://math.ucr.edu/home/baez/symplectic.html

Re: What is Symplectic Geometry? (2016) [pdf]

#26

Post by kkylin »

danharaj wrote: post #7
ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

> (In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.) In fact, every symplectic manifold locally looks like this (Darboux's theorem).

Don't disagree with the broader point you're making, but nitpick: every symplectic manifold looks locally like R^{2n} with the canonical symplectic structure. Not necessarily so globally.

Re: What is Symplectic Geometry? (2016) [pdf]

#27

Post by kkylin »

turnersr wrote: post #9
ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

What does "paired-off coordinates, like (x,v) in mechanics" mean? What is x and v here? Thanks!

Generalized coordinates and their "conjugate" momenta. These are the basic dynamical variables used to describe mechanical systems in Hamiltonian mechanics:

https://en.wikipedia.org/wiki/Hamiltonian_mechanics

For certain systems (e.g., particles in euclidean space interacting via conservative forces dependent only on position), the momenta coincide with velocity, but mathematically they are different objects: velocities are vectors, and momenta are covectors. They transform differently under coordinate transformations.

I wish I knew a concise self-contained exposition off the top of my head, but I don't. You can probably find something on-line. I think there's likely a discussion in Structure and Interpretation of Classical Mechanics (Sussman & Wisdom):

https://mitpress.mit.edu/sites/default/files/titles/content/...

Try looking up "cotangent bundles" (mathematical name for the type of spaces appropriate for Hamiltonian mechanics, formulated in terms of generlized coordinates and momenta) and "tangent bundles" (more appropriate for Lagrangian mechanics, formulated in terms of generalized coordinates and velocities).

Re: What is Symplectic Geometry? (2016) [pdf]

#28

Post by akimball »

tomrod wrote: post #22

This seems to have crossover with topological data analysis.

In particular, for geometrizing semantics. Montague grammar is a tarpit, and pragmatic utility of inference on distributed representations has been abundantly demonstrated in the past decade. Symplectic structure is one of a small class of structures which capture and relate essential features of natural semantics in a metric (read, tractable) representation. This offers a tantalizing prospect for bridging the gap between computation and cognition.

Re: What is Symplectic Geometry? (2016) [pdf]

#29

Post by danharaj »

ajkjk wrote: post #21
danharaj wrote: post #14

Earlier quoted context omitted.

Is "any hamiltonian dynamics on any phase space of a physical system" not a major enough class of problems for you? This is one way mathematicians study physics. Are you asking for applications of symplectic geometry? If so, here, I've been reading a cluster of papers in this area for a few weeks now: https://www.math.colostate.edu/~clayton/research/papers/fram...

No, I'm plenty aware of the applications. My complaint is that this article hardly mentioned them! What's the point of an article explaining a theory without motivating why it exist?

The blunt reason is that this article was written for mathematics grad students who already know that symplectic manifolds are a hot topic via diffusion.

Re: What is Symplectic Geometry? (2016) [pdf]

#30

Post by tomrod »

akimball wrote: post #28
tomrod wrote: post #22

This seems to have crossover with topological data analysis.

In particular, for geometrizing semantics. Montague grammar is a tarpit, and pragmatic utility of inference on distributed representations has been abundantly demonstrated in the past decade. Symplectic structure is one of a small class of structures which capture and relate essential features of natural semantics in a metric (read, tractable) representation. This offers a tantalizing prospect for bridging the gap be…

Hear here!
Post reply on HN