What does duality mean in math?

[geordief]

That would depend on what depth of understanding you wanted to achieve. If you merely wanted to make some sort of sense out of the somewhat mystical mantra "gravity is curvature", then no. But the more details you skate over, the more you have take on trust - what is often called "hand waving"

Sadly I have gone as far as I am able with "following the math" (just too hard for me) and have to rely on others who have followed that path to "give reports" on what they have learned and what seems salient to them.

As you suggest "gravity is curvature" is a very intriguing observation and seemingly equally intriguing to the learned and the unlearned alike.

I imagine to understand anything ,familiarity with the phenomenon is required (in order to predict what happens next?) and the phenomenon of curved space can seem far removed from everyday experience apart from the predictive mathematical models or the extreme physical phenomena of black holes or optical lensing. (amongst others I am sure)

I am surprised at your "then no" as I did think that parallel transport seemed fairly intrinsic to the model.(perhaps I misinterpreted what you were saying?)

 

[Wizard of Whatever]

There is a sort of duality when there is more than one valid solution to a given problem.

 

[QuarkHead]

I am surprised at your "then no" as I did think that parallel transport seemed fairly intrinsic to the model.(perhaps I misinterpreted what you were saying?)

I don't think you did misunderstand, at least I hope not. I f you would like a quick and dirty intro to gravity and curvature I am willing to give it a try inanother thresd, but for now.......

I am surprise quantum chemists are not reasonably familiar with at keast one example of duality - dual vectors, sometimes called co-vectors, or heaven help us, covariant tensors of rank 1. They were introduced in quantum mechanics around 1930 by Paul Dirac ( he used the truly appalling notation "ket= |v>" for vectors and "bra= <v'|"for their duals

 

[geordief]

don't think you did misunderstand, at least I hope not. I f you would like a quick and dirty intro to gravity and curvature I am willing to give it a try inanother thresd, but for now.......

Not for now,thanks.I have been interested in this subject for a long time but I just tend to pick up the odd snippet from time to time by peaking into ongoing threads on the subject when I find one.

Sadly my ability to follow a subject in a disciplined way is not up to learning stuff like this from scratch and I am really just an interested observer when the opportunity presents.

 

[exchemist]

I don't think you did misunderstand, at least I hope not. I f you would like a quick and dirty intro to gravity and curvature I am willing to give it a try inanother thresd, but for now.......

I am surprise quantum chemists are not reasonably familiar with at keast one example of duality - dual vectors, sometimes called co-vectors, or heaven help us, covariant tensors of rank 1. They were introduced in quantum mechanics around 1930 by Paul Dirac ( he used the truly appalling notation "ket= |v>" for vectors and "bra= <v'|"for their duals

Just read this.

Sure we used bracket notation a lot. Just didn’t use the term duals. We talked mostly of complex conjugates - and integrals thereof. We didn’t really get much into the vector space mathematics of it. Probably cheating.

 

[QuarkHead]

Just read this.

Bored on a Friday night? Couldn't you find a pub with a decent fight to be had?

Just didn’t use the term duals. We talked mostly of complex conjugates

That's an intersting point. Quantum physics (and I assume chemistry) considers the state of a system to be a complex valued vector (But why complex?). Mathematics asserts that the prototypical complex vector space is $$\mathbb{C}$$ itself, with any complex number being a vector..
Mathematics also asserts that given a vector space $$V$$, its dual is the linear function $$V^*:V \to \mathbb{R}$$

Since we may regard the complex conjugate as the map $$^*:\mathbb{C}\to \mathbb{R}$$ ( since for any $$z=x+iy$$ the product of sums $$zz^* =(x+iy)(x-iy)=x^2+y^2 \in \mathbb{R}$$) the dual coincides the the conplex conjugate.

In general, though, all of the above is false

We didn’t really get much into the vector space mathematics of it. Probably cheating.

you and your physics chums alike

 

[Wizard of Whatever]

There are simple examples of types of duality. One, plus and minus numbers. Another real and imaginary numbers. Another could be primes and not primes in the integers.

 

[QuarkHead]

There are simple examples of types of duality. One, plus and minus numbers. Another real and imaginary numbers. Another could be primes and not primes in the integers.

Um...I think you may need to refresh your understanding here, specifically on the difference between partitiion and duality

 

[Wizard of Whatever]

Um...I think you may need to refresh your understanding here, specifically on the difference between partitiion and duality

May not meet the specific definition of duality, but fits the general concept.