Let's discuss Mathematics

[AdrienIer]

I assume it's a typo ? Technically the answer is actually -1 because the question doesn't state it's the lowest common positive factor, but that's even more ridiculous.

 

[Kyriakos]

I assume it's a typo ? Technically the answer is actually -1 because the question doesn't state it's the lowest common positive factor, but that's even more ridiculous.

Now wouldn't it be really something if this was a test question and indeed they wanted -1? ^^

 

[Samson]

Technically the answer is actually -1 because the question doesn't state it's the lowest common positive factor, but that's even more ridiculous.

Does that mean that technically the lowest common multiple of any two numbers is minus infinity (or aleph null or whatever)?

 

[AdrienIer]

The common definition of "lowest common multiple" states that it's the lowest common positive multiple, but if you don't take that part into account I would say there is no lowest common multiple. You define a multiple of A as a number B where there exists another integer K for which A times K = B. Therefore minus infinity is not a multiple of whatever number you chose.

 

[Kyriakos]

Is "proof by intimidation" the best method? ^^

 

[AdrienIer]

Any proof that seems complicated enough to not be read in its entirety by the reader is a proof by intimidation.

 

[Kyriakos]

Any proof that seems complicated enough to not be read in its entirety by the reader is a proof by intimidation.

But proof by intimidation is typically a stronger form of proof by authority.

 

[Kyriakos]

I have another geometry question (this time it is one about sole unequal sides' comparison to their corresponding angles).

It seems to me that the following (which is generally the schoolbook's proof) is a bit too overkill (the actual proof is of all cases, but also bypasses working on the specific triangle with m)

Isn't there a way to work on the triangle with angles m1,m2 to be compared, without relying on the far more general theorem (theorem 1 as I named it here)? (as always, angles of triangle =2 right angles is not part of the progression of theorems to be used in a proof; and goes without saying that this implies that the theorem equating the external angle to the two non-suplementary to it internal ones isn't allowed).
If it was an allowed part, one could just do something like this:

And argue that as b>c, a1>a2, and all three angles=2L, obviously m2<m1. But it is not allowed, so got any ideas?

 

[Kyriakos]

Hm, if the question is verbatim stated as "draw 12 circles, so that each of them is tangential to exactly 5 of them", wouldn't the following be a perfectly sufficient construction? (I know that others, including very different ones, exist; just asking about this one)

Inferred that in this construction each of the two identical groups of 6 circles have 1 common point.

Book only had one construction of the many possible ones, and it made an impression on me that they went with one which would only be forced under a very specific limitation (not in the question). In the spoiler:

Spoiler :

This is - I think the only possibility? - an arrangement where each point of tangency is unique:

 

[AdrienIer]

Yep

 

[Kyriakos]

Yep

What about the question in the spoiler? Is that the only arrangement where each point of tangency is unique?

 

[AdrienIer]

I have no idea

 

[Kyriakos]

I have no idea

I am shocked Due to my type of personality I'd take it as implying that I am starting to ask decent/non-simple questions ^^

Well, it is an interesting question, and I would like to know. It's also imo potentially far easier/faster to use such constructions to establish relations than (the analogue in complication) with algebra. Provided, of course, that the question is translatable to such in the first place.

 

[AdrienIer]

The thing is, non-coordinate geometry is always much harder. Also "being the only possibility" in geometry is particularly hard to prove.
In such situations, either you have the answer or you don't. It's much rarer to have an idea without knowing how it goes with problems like these, in algebra you usually have basic methods to move forward before declaring that you don't know something.

 

[danjuno]

Any proof that seems complicated enough to not be read in its entirety by the reader is a proof by intimidation.

Greatest math quote I've ever read!

 

[Samson]

I do not really understand it, but this sounds pretty cool.

The breakthrough proof bringing mathematics closer to a grand unified theory

One of the biggest stories in science is quietly playing out in the world of abstract mathematics. Over the course of last year, researchers fulfilled a decades-old dream when they unveiled a proof of the geometric Langlands conjecture — a key piece of a group of interconnected problems called the Langlands programme. The proof — a gargantuan effort — validates the intricate and far-reaching Langlands programme, which is often hailed as the grand unified theory of mathematics but remains largely unproven. Yet the work’s true impact might lie not in what it settles, but in the new avenues of inquiry it reveals.

“It’s a huge triumph. But rather than closing a door, this proof throws open a dozen others,” says David Ben-Zvi at the University of Texas at Austin, who was not involved with the work.

Proving the geometric Langlands conjecture has long been considered one of the deepest and most enigmatic pursuits in modern mathematics. Ultimately, it took a team of nine mathematicians to crack the problem, in a series of five papers spanning almost 1,000 pages [note: all five are referenced as preprints on arXiv. I am not sure what is going on there]. The group was led by Dennis Gaitsgory at the Max Planck Institute for Mathematics in Bonn, Germany, and Sam Raskin at Yale University in New Haven, Connecticut, who completed his PhD with Gaitsgory in 2014.

The magnitude of their accomplishment was quickly recognized by the mathematical community: in April, Gaitsgory received the US$3-million Breakthrough Prize in Mathematics, and Raskin was awarded a New Horizons prize for promising early-career mathematicians. Like many landmark results in mathematics, the proof promises to forge bridges between different areas, allowing the tools of one domain to tackle intractable problems in another. All told, it’s a heady time for researchers in these fields.

“It gives us the strongest evidence yet that something we’ve believed in for decades is true,” says Ben-Zvi. “Now we can finally ask: what does it really mean?”

Spoiler Rest of article :

The hole story

The Langlands programme traces its origins back 60 years, to the work of a young Canadian mathematician named Robert Langlands, who set out his vision in a handwritten letter to the leading mathematician André Weil. Over the decades, the programme attracted increasing attention from mathematicians, who marvelled at how all-encompassing it was. It was that feature that led Edward Frenkel at the University of California, Berkeley, who has made key contributions to the geometric side, to call it the grand unified theory of mathematics.

Langlands’ aim was to connect two very separate major branches of mathematics — number theory (the study of integers) and harmonic analysis (the study of how complicated signals or functions break down into simple waves). A special case of the Langlands programme is the epic proof that Andrew Wiles published, in 1995, of Fermat’s last theorem — that no three positive integers a, b and c satisfy the equation an + bn = cn if n is an integer greater than 2.

Robert Langlands discussed his ideas in a letter to André Weil in 1967. On the cover page to the letter (left), Langlands says, “If you are willing to read it as pure speculation I would appreciate that; if not — I am sure you have a waste basket handy.”Credit: Institute for Advanced Study (Princeton, N.J.) Shelby White and Leon Levy Archives Center

The geometric Langlands conjecture was first developed in the 1980s by Vladimir Drinfeld, then at the B. Verkin Institute for Low Temperature Physics and Engineering in Kharkiv, Ukraine. Like the original or arithmetic form of the Langlands conjecture, the geometric conjecture also makes a type of connection: it suggests a correspondence between two different sets of mathematical objects. Although the fields linked by the arithmetic form of Langlands are separate mathematical ‘worlds’, the differences between the two sides of the geometric conjecture are not so pronounced. Both concern properties of Riemann surfaces, which are ‘complex manifolds’ — structures with coordinates that are complex numbers (with real and imaginary parts). These manifolds can take the form of spheres, doughnuts or pretzel-like shapes with two or more holes.

Many mathematicians strongly suspect that the ‘closeness’ of the two sides means the proof of the geometric Langlands conjecture could eventually offer some traction for furthering the arithmetic version, in which the relationships are more mysterious. “To truly understand the Langlands correspondence, we have to realize that the ‘two worlds’ in it are not that different — rather, they are two facets of one and the same world,” says Frenkel. “Seeing this unity requires a new vision, a new understanding. We are still far from it in the original formulation. But the fact that, for Riemann surfaces, the two worlds sort of coalesce means that we are getting closer to finding this secret unity underlying the whole programme,” he adds.

One side of the geometric Langlands conjecture concerns a characteristic called a fundamental group. In basic terms, the fundamental group of a Riemann surface describes all the distinct ways in which loops can be tied around it. With a doughnut, for example, a loop can run horizontally around the outer edge or vertically through the hole and around the outside. The geometric Langlands deals with the ‘representation’ of a surface’s fundamental group, which expresses the group’s properties as matrices (grids of numbers).

The other side of the geometric Langlands programme has to do with special kinds of ‘sheaves’. These tools of algebraic geometry are rules that allot ‘vector spaces’ (where vectors — arrows — can be added and multiplied) to points on a manifold in much the same way as a function describing a gravitational field, say, can assign numbers for the strength of the field to points in standard 3D space.

Bridgework in progress

Work on bridging this divide began back in the 1990s. Using earlier work on Kac–Moody algebras, which ‘translate’ between representations and sheaves, Drinfeld and Alexander Beilinson, both now at the University of Chicago, Illinois, described how to build the right kind of sheaves to make the connection. Their paper (see go.nature.com/4ndp5ev), nearly 400 pages long, has never been formally published. Gaitsgory, together with Dima Arinkin at the University of Wisconsin–Madison, made this relationship more precise in 20126; then, working alone, Gaitsgory followed up with a step-by-step outline of how the geometric Langlands might be proved7.

“The conjecture as such sounds pretty baroque — and not just to outsiders,” says Ben-Zvi. “I think people are much more excited about the proof of geometric Langlands now than they would have been a decade ago, because we understand better why it’s the right kind of question to ask, and why it might be useful for things in number theory.”

One of the most immediate consequences of the new proof is the boost it provides to research on ‘local’ versions of the different Langlands conjectures, which ‘zoom in’ on particular objects in the ‘global’ settings. In the case of the geometric Langlands programme, for example, the local version is concerned with the properties of objects associated with discs around points on a Riemann surface — rather than the whole manifold, which is the domain of the ‘global’ version.

Peter Scholze, at the Max Planck Institute for Mathematics, has been instrumental in forging connections between the local and global Langlands programmes. But initially, even he was daunted by the geometric side.

“To tell the truth,” Scholze says, “until around 2014, the geometric Langlands programme looked incomprehensible to me.” That changed when Laurent Fargues at the Institute of Mathematics of Jussieu in Paris proposed a reimagining of the local arithmetic Langlands conjectures in geometrical terms8. Working together, Scholze and Fargues spent seven years showing that this strategy could help to make progress on proving a version of the local arithmetic Langlands conjecture9 concerning the p-adic numbers, which involve the primes and their powers. They connected it to the global geometric version that the team led by Gaitsgory and Raskin later proved.

The papers by Scholze and Fargues built what Scholze describes as a “wormhole” between the two areas, allowing methods and structures from the global geometric Langlands programme to be imported into the local arithmetic context. “So I’m really happy about the proof,” Scholze says. “I think it’s a tremendous achievement and am mining it for parts.”

Quantum connection

According to some researchers, one of the most surprising bridges that the geometric Langlands programme has built is to theoretical physics. Since the 1970s, physicists have explored a quantum analogue of a classical symmetry: that swapping electric and magnetic fields in Maxwell’s equations, which describe how the two fields interact, leaves the equations unchanged. This elegant symmetry underpins a broader idea in quantum field theory, known as S-duality.

In 2007, Edward Witten at the Institute for Advanced Study (IAS) in Princeton, New Jersey, and Anton Kapustin at the California Institute of Technology in Pasadena were able to show10 that S-duality in certain four-dimensional gauge theories — a class of theories that includes the standard model of particle physics — possesses the same symmetry that appears in the geometric Langlands correspondence. “Seemingly esoteric notions of the geometric Langlands program,” the pair wrote, “arise naturally from the physics.”

Although their theories include hypothetical particles, called superpartners, that have never been observed, their insight suggests that geometric Langlands is not just a rarefied idea in pure mathematics; instead, it can be seen as a shadow of a deep symmetry in quantum physics. “I do think it is fascinating that the Langlands programme has this counterpart in quantum field theory,” says Witten. “And I think this might eventually be important in the mathematical development of the Langlands programme.”

Among the first to take that possibility seriously was Minhyong Kim, director of the International Centre for Mathematical Sciences in Edinburgh, UK. “Even simple-sounding problems in number theory — like Fermat’s last theorem — are hard,” he says. One way to make headway is by using ideas from physics, like those in Witten and Kapustin’s work, as a sort of metaphor for number-theoretic problems, such as the arithmetic Langlands conjecture. Kim is working on making these metaphors more rigorous. “I take various constructions in quantum field theory and try to cook up precise number-theoretic analogues,” he says.

 

[a pen-dragon]

[note: all five are referenced as preprints on arXiv. I am not sure what is going on there]

I guess that means they have not completed peer-review, a part of the scientific publishing process, basically some people unrelated to the authors check for mistakes. I have no idea how long this typically takes in maths, but I imagine it could be quite a long while with such papers, due to their probable complexity.

If one wants to fully understand a proof one needs to also check every small step, as any error, however tiny, may invalidate the entire proof.

Today typically a paper is uploaded to arxiv when it is done, basically at the same time as it is submitted to a journal for publishing. This means that for recent papers it is typical that they are not yet published.

 

[Samson]

I guess that means they have not completed peer-review, a part of the scientific publishing process, basically some people unrelated to the authors check for mistakes. I have no idea how long this typically takes in maths, but I imagine it could be quite a long while with such papers, due to their probable complexity.

If one wants to fully understand a proof one needs to also check every small step, as any error, however tiny, may invalidate the entire proof.

Today typically a paper is uploaded to arxiv when it is done, basically at the same time as it is submitted to a journal for publishing. This means that for recent papers it is typical that they are not yet published.

Yeah, I am aware of how that works, but I have not heard of someone winning anything for a paper that has not been accepted for publication, let alone $3 million.