Mathematicians Explore Mirror Link Between Two Geometric Worlds | Hacker News Reader

Mathematicians Explore Mirror Link Between Two Geometric Worlds

quantamagazine.org

74 pointsnsoonhui21 comments

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 names of the major branches you're unifying without having to read half the article. Holy moley, everything is written to waste our attention on suspense and it's driving me wild. Where's the dry but laymen-approachable math and physics publication at? Oh yeah that doesn't sell time in the face of ads.

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.

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??

I remember many years ago an extremely bright and enthusiastic mathmo who was under the mistaken impression I was nearly as bright as he was explained some result in involving topology and mapping spaces. It got really quite involved and abstract and then he pulls a rabbit out of the hat and says "And this thing has the same order as the monster and no-one knows why." In the years that have followed we've just discovered more and more of these impossible co-incidences and tantalising hints that large numbers of research areas are in fact different representations of the same thing.

The cutting edge discoveries in analysis at the end of the 19th century became the fundamentals of the subject taught to UK undergraduates within a 100 years. This kind of thing looks to me like it could revolutionize our understanding yet again.

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.

> 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.

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.

I would like to read more on this, do you have any further references?

" algebraic geometry is like plotting the equation of a circle" this description and definition is as helpful and accurate as saying a circle is approximated by a square. Pop math writers need to stop likening something as complicated as algebraic geometry to high school algebra.

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.

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.

Math is useless, says Hacker News commenter. More news at 11.

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.

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.

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