Teaching eight mathematics courses (quasi-)simultaneously is just a matter of organization. At least that is what I currently try to prove. Meanwhile I learn a lot about analytic number theory and Erdös’s discrepancy conjecture.
Warning to teaching staff: summing infinite sequences of positive (>0) integers is difficult, but possible. For example is not a mistake and students must get full marks. Let me show you why.
A couple of months ago I was hoping to ‘cheat’ a proof of the Erdös discrepancy conjecture by using a variant of an idea of Fürstenberg’s proof on the infinitude of primes. Remember, Fürstenberg considered the integers with a new topology. Its open sets are for and .
This topology is metrizable. There are a couple of hand-waving arguments how this metric could look like. However, as far as I am aware of, there is so far no neat description in the literature. A couple of days ago, R. Lovas and I. Mezö have published a fairly straightforward proof that with induces the above topology.
Since the sequence converges to 0 in this topology. The partial sums of satisfy and thus .
R. Lovas and I. Mezö have collected more such observations in their note. What they did not mention explicitly, but what I consider interesting is that with the above metric, the integers become an ultrametric space. Without loss of generality we assume . Then are all divisors of m and n and thus they are divisors of m+n. Therefore . The strong triangle inequality now follows .
Last week several hundred thousands of young people from all over the world gathered in Duisburg (Germany) to celebrate this years Love Parade. 21 of them are now dead. Their families and the public are mourning. What remains are questions. Questions on why this could happen and who is responsible.
Since this is a blog about science I am not so much interested in the mayor, who refuses to step back since he wants to ‘clear things up’. I am also not interested in the organizer who claimed 1.400.000 participants and, maybe after he realized that this will get him into trouble with his insurance, counted again to just find 250.000. Police and fire services also played some role which is not of my concern.
I am interested in the security concept which was certified by a professor from a near university. No names here since this is an ongoing case and it is hard to get the facts. Sure is that the organizer planned to get more than 250.000 people through a 40 meter wide tunnel to the location. This tunnel was also planned to be the only exit from the location. At first glance that sounds crazy and … at a second glance that still sounds crazy. How could someone certify such a security concept? Our scientist in question has a Ph. D. in theoretical physics and his wiki entry contains some name dropping in form of nobel prize winning collaborators. If he signs such a concept then maybe because of some ‘deep’ insights stemming from his research on ‘transport and traffic’. However, this is buried in some proprietory journals to which I have no access (as an unaffiliated random guy).
That leaves me with the publicly available information. That is a TV interview right after the catastrophe with the event still going on. My rough translation (out of memory) of the decisive passage ‘… the behaviour of panicking people is hard to model …’ followed by some disgusting allocation of responsibility. In a further interview our scientist denies responsibility at all since he was not allowed to see the whole concept. This assertion is shockingly unmasking.
My conclusion: As long as results are not publicly available, as long as concepts are signed without being fully seen and as long as no responsibility for mistakes is taken over it is very hard to differentiate between a charlatan and a scientist.
Because of the omnipresence of this case in all media science has lost a lot of reputation in germany.
Its Polymath time. My resources, especially time and most importantly skill, are limited and therefore I have to restrict myself a little. Let me just sketch where I set the boundaries, what I want to try and what a possible (successful) outcome might look like. Some acquaintance with Tim Gowers’s proof of Roth’s theorem and its (hoped for) connection to EDP is necessary.
What is the idea?
Let me first collect some observations:
- ‘Translation’ acts as a group on APs and is periodic on APs with common difference.
- ROI starts with some representation of the translation group in terms of rank one projections using exponentials.
- An elementary formula from Fourier Analysis describing the interaction of translation and exponentials is used to express the exponentials in terms of Fourier coefficients of some characteristic function and its translates.
- The result is an ‘efficient’ representation, i.e. it allows to deduce unbounded discrepancy.
Translation as described above lives on the domain of the Fourier coefficients. We do not lose information if we consider it for the corresponding Fourier series. On all reasonable spaces translations form a strongly continuous (semi-) group of linear operators. For periodic strongly continuous groups we have representations of the group. If we work with rotation groups on with we even have a tensorial representation in terms of exponentials. By the way, if we choose our space carefully, exponentials are at least approximative eigenvectors of translations (rotations).
The idea now is to get information on discrepancy on (the domain of the Fourier coefficients) by studying ROI on various spaces of Fourier series, like or .
If I am not able to translate the ROI idea to the infinite section directly I will also try to use the so called ‘Complex Inversion Formula’, expressing the integral over the group in terms of some inverse Laplace transform. This can be seen as an infinite dimensional version of Perron’s formula. However that would be a ‘last try’ since it is not connected to Polymath anymore.
What is the goal?
The result I am aiming for looks roughly as follows:
Let be a Banach (or better a Hilbert) space and let be a strongly continuous (maybe only semi-)group of linear operators (translations, rotations, periodic?) with generator . If the resolvent satisfies some conditions (including norm estimates) then ‘something’ has unbounded ‘discrepancy’.
That sounds incredibly naive, since Tim Gowers’s proof needed some clever estimates and preyed upon cancellations for different common differences . My hope is to hide these technicalities in an estimate on the resolvent. Such estimates like e.g. in the Hille-Yosida theorem are usually harder to obtain than to state. That is good news.
So far, that would be the first part. In a way this is just translating the main idea of the proof into some other language. The hard part would then be to apply the general theorem to other situations (maybe even EDP). Here the resolvent has to be estimated and technicalities enter. However, I do not want to think that far ahead.
So far that was a hot summer. With all written and oral exams finished and with bearable temperatures returning I plan to spend some free time to think about the Erdös Discrepancy Conjecture. Is it wise to post this plan? I think so. It puts some pressure on me and this is good.
The idea is that we have probably not made maximal use of Tim Gowers’s proof of Roth’s theorem. Behind all the numbers and estimates there could be some abstract/infinitary content. If this is the case, then it might be easier to generalize or at least to spot the limits of the method.
Intuition or wishful thinking? Maybe the later, but I want to know for sure.
Tim Gowers has published a new proof of Roth’s theorem (link to Polymath5) as part of the project. He extends previous polymath work of Kevin Obryant (and himself) to reach Roth’s exact bounds. All this in a very elegant and accessible way. The draft is a really good start to join the project and watch or contribute to a trend-setting and outstanding form of mathematics in the net.
S. Abramsky and J. Zvesper have uploaded
In section 4 ‘The Lawvere Fixpoint Lemma’ contains a ‘positive’ version of Cantor’s theorem.