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Mathematicians Explore Mirror Link Between Two Geometric Worlds

quantamagazine.org

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Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#11

Post by goldenkey »

spekcular wrote: post #9
goldenkey wrote: post #8

Earlier quoted context omitted.

All of it stems from the complexity of formal systems. It seems that all the hard problems in these various subjects are just different masks for the same fundamental hardnesses in properties of programs. https://en.wikipedia.org/wiki/Langlands_program

Wait, what? Are you claiming that mirror symmetry is "secretly" Langlands? This is not my understanding, and I'd appreciate a citation if you are indeed making that claim. Moreover, it does not seem to me that, for example, the Clay Millennium problems are all glosses on the same underlying problem, so I'm a bit confused by your statement.

I was just linking to Langlands as another example of duality of topics in mathematics. Just like NP-complete problems are equivalent, the hardest problems in mathematics are most likely also part of some complexity class and equivalent to eachother. Take what we see here and in Langlands and generalize it to information, computation, formal languages, model theory, in general.

There is probably a unified theory of "problems" that glues all the hardness together.

Just my opinion..not a claim.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#12

Post by Y_Y »

boxfire wrote: post #10

Okay so this is one thing that boils me about quanta magazine. The authors obviously have enough background in mathematics to know the nature of the field, yet they do these very long winded, vague intros, which serve to mystify the subject and build it up. That is a major disservice to the knowledge itself. Sure you can't drop the mathematical definitions in the front, but your thesis paragraph can say least say the…

And then about ten paragraphs in the reveal that a torus is like a donut, and now it's time for symplectic geometry. I think that it's a noble effort, but you can't go from zero to graduate level geometry in a little article like that.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#13

Post by Y_Y »

moomin wrote: post #5
rightbyte wrote: post #2

It most be nice to work with stuff that is in practice unfalsifiable paycheck after paycheck. Relieves stress! Research fields like these has never convinced me of its valour really. I read the article as fan fiction.

It's only unfalsifiable in the sense that it's proven correct.

Le mot juste.

(Of course this is all subject to some wishy-washy ZFC "theory" which nobody has ever been able to prove.)

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#14

Post by hrhth »

Y_Y wrote: post #13
moomin wrote: post #5

Earlier quoted context omitted.

It's only unfalsifiable in the sense that it's proven correct.

Le mot juste. (Of course this is all subject to some wishy-washy ZFC "theory" which nobody has ever been able to prove.)

Theory in math is a suit of axioms and theorems and as such you don't prove it. The best you can do is to prove a statement. Also, ZFC is irrelevant to people who do not do set theory.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#15

Post by hrhth »

rightbyte wrote: post #7
jacobolus wrote: post #3

Earlier quoted context omitted.

Research fields like algebraic geometry and symplectic geometry? Or pure mathematics in general? Pure mathematics relies on written proofs, not falsification by experimental evidence (though finding a counterexample is one powerful way to disprove a mathematical conjecture).

I was thinking of the physical interpretation not mainly the math behind it, string theory: “You might have this small space that you can’t see or measure directly, but some aspects of the geometry of that space might influence real-world physics,” said Mark Gross, a mathematician at the University of Cambridge. I mean modern research math can be silly, but it is (hopefully) not bogus.

Math only has to be consistent. It doesn't have to have any relation to physics or "reality". I can start with 1 + 1 = 3(favorite example of "common sense" people when they point out something inconsistent) and other "nonsensical" statements and derive a bunch of results to make up a new theory within math. It might have jack-all to do with physics as we know it, but far as math is concerned, so long as the derived results are correct, then the derived theory is valid. The question is whether or not this new theory is interesting. Often times - not.

Some physicists who express their dislike of string theory are not making a statement about math. Roughly they are saying the theory is more math-y rather than it's physics-y.

It's similar to how some people dislike "soupy lasagna". They are not making a statement about soup. They are not saying soup is bad. They mean they like their lasagna be more like lasagna and less like soup.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#16

Post by voldacar »

boxfire wrote: post #10

Okay so this is one thing that boils me about quanta magazine. The authors obviously have enough background in mathematics to know the nature of the field, yet they do these very long winded, vague intros, which serve to mystify the subject and build it up. That is a major disservice to the knowledge itself. Sure you can't drop the mathematical definitions in the front, but your thesis paragraph can say least say the…

Huge agree. I knew it was quanta just from the headline. For the life of me I will never understand who their target audience is. People who claim to like math but cant actually be trusted to learn any??

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#17

Post by Y_Y »

hrhth wrote: post #14
Y_Y wrote: post #13

Earlier quoted context omitted.

Le mot juste. (Of course this is all subject to some wishy-washy ZFC "theory" which nobody has ever been able to prove.)

Theory in math is a suit of axioms and theorems and as such you don't prove it. The best you can do is to prove a statement. Also, ZFC is irrelevant to people who do not do set theory.

I'd argue that proving theorems is what makes them theorems. Then again, I assumed what I wrote above was clearly a joke, but it seems it wasn't clear enough.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#18

Post by hrhth »

Y_Y wrote: post #17
hrhth wrote: post #14

Earlier quoted context omitted.

Theory in math is a suit of axioms and theorems and as such you don't prove it. The best you can do is to prove a statement. Also, ZFC is irrelevant to people who do not do set theory.

I'd argue that proving theorems is what makes them theorems. Then again, I assumed what I wrote above was clearly a joke, but it seems it wasn't clear enough.

A correct statement is a result. An important result is a theorem. Indeed, I must've missed the joke. They often fly over my head.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#19

Post by q-big »

spekcular wrote: post #9
goldenkey wrote: post #8

Earlier quoted context omitted.

All of it stems from the complexity of formal systems. It seems that all the hard problems in these various subjects are just different masks for the same fundamental hardnesses in properties of programs. https://en.wikipedia.org/wiki/Langlands_program

Wait, what? Are you claiming that mirror symmetry is "secretly" Langlands? This is not my understanding, and I'd appreciate a citation if you are indeed making that claim. Moreover, it does not seem to me that, for example, the Clay Millennium problems are all glosses on the same underlying problem, so I'm a bit confused by your statement.

> Wait, what? Are you claiming that mirror symmetry is "secretly" Langlands?

There is a geometric analogue to the Langlands correspondence called "geometric Langlands correspondence":

> https://en.wikipedia.org/wiki/Geometric_Langlands_correspond...

There is a relationship beetween the geometric Langlands correspondence and S-duality

> https://en.wikipedia.org/wiki/S-duality

a duality that is considered in string theory. Another duality that is considered in string theory is T-duality:

> https://en.wikipedia.org/wiki/T-duality

The latter duality is strongly related to mirror symmetry.

Re: Mathematicians Explore Mirror Link Between Two Geometric Worlds

#20

Post by spekcular »

q-big wrote: post #19
spekcular wrote: post #9

Earlier quoted context omitted.

Wait, what? Are you claiming that mirror symmetry is "secretly" Langlands? This is not my understanding, and I'd appreciate a citation if you are indeed making that claim. Moreover, it does not seem to me that, for example, the Clay Millennium problems are all glosses on the same underlying problem, so I'm a bit confused by your statement.

> Wait, what? Are you claiming that mirror symmetry is "secretly" Langlands? There is a geometric analogue to the Langlands correspondence called "geometric Langlands correspondence": > https://en.wikipedia.org/wiki/Geometric_Langlands_correspond... There is a relationship beetween the geometric Langlands correspondence and S-duality > https://en.wikipedia.org/wiki/S-duality a duality that is considered in string the…

Sure, but the fact that one can play "six degrees of Kevin Bacon" with Wikipedia and get from Langlands to mirror symmetry is not evidence one problem is actually the other in disguise. Already at the first link, it's unclear that resolving the usual Langlands program would say anything about geometric Langlands.
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