let f(x) be a polynomial with non-negative integer coefficients such that f(10) is prime. If all coefficients are ā¤49598666989151226098104244512918, then f(x) is irreducible over Z[x]. Moreover, they proved that this bound is also sharp. In other words, coefficients larger than 49598666989151226098104244512918 do not guarantee irreducibility.
I made a Mƶbius transformation visualizer together with Taketo SanoĀ in a hackathon(http://hackday.jp/), and we got GOLD prize!
Parabolic Mƶbius transformation has a fixed point, and other Mƶbius transformations have two fixed points. (cf.http://hyrodium.tumblr.com/post/138314686744)
Bring two fixed points of a Mƶbius transformation closer to a point, then the transformation changes into a parabolic Mƶbius transformation.
The Leiden Declaration is not just for mathematicians
Recently some leading mathematicians have been studying the use, impact, and risks of using artificial intelligence in mathematical research and institutions.
They have now published the Leiden Declaration to articulate their concerns and to make recommendations.
I learned about this today from the New York Times, and you can learn a lot by reading their article. But you can learn more by reading the Declaration.
I feel that the Declaration is directly relevant not only to the mathematical community, but also, in reality, to all of us who regard thinking as an integral part of what we do. And that, in the end, is really all of us.
I strongly suggest that you study the Declaration and adopt its recommendations.
Here is my own take on the recommendations...
Provide disclosure and references for your own uses of artificial intelligence.
Invest all the work needed to rigorously vet any result that you have obtained through artificial intelligence.
Shoot down any crap work that you have identified. Don't make it personal, but don't be gentle.
Uphold the highest values of your chosen field of work. Both science and the arts are things that we do just out of curiosity, or to express ourselves, or because we just have to. In other words, these activities are ends in themselves. They are among the highest and most wonderful things that we, as a species, have done. Forcefully combat efforts to prioritize commercial, political, and military uses of artificial intelligence over more important reasons to use it.
Do everything you can to ensure that everyone who could benefit from using artificial intelligence, can use it. In a democratic world, anyone with a smartphone or computer should be able to find, afford, and use all the compute they need.
I have become convinced, through my own use of artificial intelligence for music composition and software development, that it has enormous, world-changing potential to amplify the scope, speed, and even depth of what scientists, artists, and indeed all of us can do.
Don't let selfish interests hoard or abuse this power.
Among the potential threats that the Leiden Declaration authors articulate are accuracy and reliability: Journal editors are already complaining about a flood of plausible seeming A.I.- generated papers and proofs that have turned out to be incorrect, and in ways that are difficult for mathematicians to discern.
Perhaps most pointedly, the authors raise the question of whether the many A.I. companies tackling mathematics ā major players such as OpenAI, Google DeepMind and Anthropic, or start-ups such as Harmonic, Math, Inc. and Axiom Math ā are keeping the fieldās best interests in mind. āTechnology companiesā involvement in research,ā they write, āraises the risk that research questions are prioritized and incentivized because of their amenability to A.I. methods and models, rather than their deeper significance to understanding.ā In turn, they point out, this disadvantages researchers who choose not to use the technology, and those who do not have access to it.
and...
OCHIGAMEĀ Mathematics is a rich form of cultural expression with an ancient history, and I am not worried that any technology will ever render it obsolete. Its most precious aspects, such as the collective quest to understand beautifully intricate ideas, and to explore the limits of the human imagination, cannot ever be automated. What I am worried about is that a handful of corporations are mobilizing their vast financial resources to impose an impoverished view of mathematics so forcefully ā at a moment when scientific research is already under political attack ā that they may well end up destroying the social institutions that allow mathematics to flourish. What could be futile about resisting that?
it's crazy how much the field of computing has been able to do with just semiconductors. imagine what would happen if they managed to get full conductors working. twice as much!
I promised some maths so here comes. The overarching theme for these posts is a framework called deformation/rigidity (you have to give it to us, we have pretty cool names) introduced by Sorin Popa in the early 2000's. One crucial aspect here is the notion of an s-malleable deformation. I'm trying to keep things intuitive for now so I won't include yet the actual definition but some intuition below:
Naive definition: One starts with an action of a group G on a von Neumann algebra, say (A, Ļ). The commutative counterpart would be an action of a group by (potentially discontinuous) Borel automorphisms of [0,1] which preserve the Lebesgue measure - this is in fact a much more general example than one might expect. Then an s-malleable deformation of this action consist of an action of G on a larger algebra (š, Ļ) which contains two copies of A, sitting 'orthogonal' to each other inside š and which can be 'deformed' into one-another by a continuous path of automorphisms that commute with the G-action. Furthermore we ask that this path is symmetric in time.
There will ideally be a number of posts explaining why anyone would care and what this is useful for but what I want to draw attention to is this idea of 'giving yourself space' to work by putting your object inside a larger object. It's really ubiquitous in maths I feel but somehow I get excited every time I run into it even if I've seen it so many times already (maybe because it inevitably goes wrong whenever I try to use it myself).
So really I just though I'd draw attention to this kind of phenomena:
The Lebesgue integral and DCT. At the end of the day, every time I want to compute an integral I do it by reducing to a Riemann integral so clearly the point of the Lebesgue integral is not that it's computable, but that by developing this fancier language you can prove things which are true for the Riemann integral itself but which you couldn't prove before - e.g. without secretly using the Lebesgue measure or Fatou's Lemma try proving that if a sequence of Riemann integrable functions on [0,1] is unformly bounded and converges pointwise to 0 then the integrals converge to 0. You need 0 knowledge about the Lebesgue measure to state it but good luck proving it without the language of Lebesgue (*I am not saying it's impossible but rather non-trivial I think)
Very often in number theory/diophantine equations. Just one example which I've seen recently: say a is a positive integer and consider n=a^3-3a+1. Prove all prime factors of n are either 1 (mod 9) or -1 (mod 9). Clearly you'll want to work in š½_p but really you'll have to move to a degree-2 extension of š½_p in order to write a=t+t^(-1) which will generally not have roots in the base field. (It's a cute exercise to finish the problem for here)
Furstenberg's topological proof for the infinity of primes *but also* his proof of Van der Waerden's Theorem (any way you partition the naturals into finitely many subsets one of them contains arithmetic progressions of arbitrarily long lengths) using the Poincare/Birkhoff multiple recurrence theorem from ergodic theory.
Bernstein's p^Ī±q^Ī² theorem using rep theory but also a bit of Galois theory.
Even the basic Cauchy's Theorem from group theory is just a really cute counting argument, I wasn't aware combinatorics was invited to this party??
I'll leave you with two more examples. They both can be done with func-anal tools but you wouldn't expect it. It's quite fun thinking about them:
Let G be a discrete group acting on a set I. Prove that the following are equivalent: 1. all orbits are infinite and 2. whenever A and B are finite subsets of I there is some g in G such that gA is disjoint from B. As an aside, if you endow I with the counting measure this is saying that the action is ergodic iff it is weakly mixing but this is rather irrelevant for the proof.
(This one you can also do with martingales I think but also there is a functional-analytic proof) Suppose on each lattice point (m,n) of the plane you write a real number a_{m,n}. You know that the bi-infinite family (a_{m,n})_{m,n} is bounded and that every number is the (arithmetic) mean of its four neighbours. Show that in fact all the a_{m,n} must be equal.
I'm attending an "elementary geometry" lecture taught by a category theorist this semester and I've never felt so much like I was living through the chinese room hypothetical in real life
Always in awe at how much more my advisor knows than me, he comes up with such wonderfully clever tricks, which are probably adapted versions of something he saw like 15 years ago. I can see why he likes to repeat that saying "the devil is cunning because he's old, not because of his devilry".
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