User jon paprocki - MathOverflow

Answer by Jon Paprocki for Colloquial catchy statements encoding serious mathematics

2012-07-04T18:19:36Z

I had to walk to school uphill both ways.

I've found that this is one of the better ways to try to explain the idea behind non-commutative geometry to a layperson.

Answer by Jon Paprocki for Interesting mathematical documentaries

2012-06-19T21:31:47Z

Science Lives, made by the Simons Foundation, has nine very long interviews that essentially amount to miniature documentaries about the lives of a number of 20th/21st century mathematicians and physicists.

Answer by Jon Paprocki for Mathematics of quasicrystals

2012-06-18T17:14:51Z

If you are interested in the non-commutative geometry side of things, there is an overview article, The Noncommutative Geometry of Aperiodic Solids (pdf link) by Jean Bellissard. He writes the paper building up from the most basic possible physical concepts and makes the use of noncommutative geometry to study quasicrystals seem quite natural, and it is done in a mathematically rigorous manner. Edit: I should emphasize that this paper is about the physical side of quasicrystals, written from a mathematical perspective. I wasn't sure if that was what you wanted, as opposed to just the mathematical study of quasicrystals without regard for any related physics.

Answer by Jon Paprocki for Examples of common false beliefs in mathematics.

2010-05-19T15:12:17Z

False belief: A function being continuous in some open interval implies that it is also differentiable in that interval:

Counterexample:

The Weierstrass function is an example of a function that is continuous everywhere but differentiable nowhere:

$f(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x)$

Where $a \in (0, 1)$, $b$ is a positive odd integer, and $ab > 1 + \frac{3\pi}{2}$. The function has fractal-like behavior, which leads to it not being differentiable. This notion is rather disheartening to most calculus students, though!

Another example that is maybe not a false belief so much as something that is very hard to believe at first is the Monty Hall problem. I remember spending most of a day in catatonic despair when I first learned of it...

Are there any mathematical objects that exist but have no concrete examples?

2011-04-22T16:37:44Z

I am curious as to whether there exists a mathematical object in any field that can be proven to exist but has no concrete examples? I.e., something completely non-constructive. The closest example I know of are ultrafilters, which only have one example that can be written down. MathOverflow user Harrison Brown mentioned to me that there are examples in Ramsey theory of objects that are proven to exist but have no known deterministic construction (but there might be), which is close to what I'm looking for. He also mentioned that the absolute Galois group of the rationals has only two elements that you can write down - the identity element and complex conjugation.

I am worried that this might be a terribly silly question, since typically there is a trivial example of an object, and a definition that specifically did not include the trivial case would be 'cheating' as far as I'm concerned. My motivation for this question is purely out of curiosity. Also, this is my first question on MO, so I probably need help with tags and such (I'm not terribly sure what this would belong to). I think that this should be a community wiki, but I do not have the reputation to make it so as far as I can tell.

Answer by Jon Paprocki for What are some examples of narrowly missed discoveries in the history of mathematics?

2010-06-21T16:01:32Z

In the book The Scientists by John Gribbin, he mentions that, in his search for the theory of general relativity, Einstein apparently wrote down a correct equation that would have led him to correctly discovering the rest of the equations for general relativity very quickly. But, he did not see the equation for what it was and ran down the wrong path for two entire years before coming back to the correct equation. Here's the quote from the book:

"Einstein himself is often presented as the prime example of someone who did great things alone, without the need for a community. This myth was fostered, perhaps even deliberately, by those who have conspired to shape our memory of him. Many of us were told a story of a man who invented general relativity out of his own head, as an act of pure individual creation, serene in his contemplation of the absolute as the First World War raged around him.

It is a wonderful story, and it has inspired generations of us to wander with unkempt hair and no socks around shrines like Princeton and Cambridge, imagining that if we focus our thoughts on the right question we could be the next great scientific icon. But this is far from what happened. Recently my partner and I were lucky enough to be shown pages from the actual notebook in which Einstein invented general relativity, while it was being prepared for publication by a group of historians working in Berlin. As working physicists it was clear to us right away what was happening: the man was confused and lost - very lost. But he was also a very good physicist (though not, of course, in the sense of the mythical saint who could perceive truth directly). In that notebook we could see a very good physicist exercising the same skills and strategies, the mastery of which made Richard Feynman such a great physicist. Einstein knew what to do when he was lost: open his notebook and attempt some calculation that might shed some light on the problem.

So we turned the pages with anticipation. But still he gets nowhere. What does a good physicist do then? He talks with his friends. All of a sudden a name is scrawled on the page: 'Grossman!!!' It seems that his friend has told Einstein about something called the curvature tensor. This is the mathematical structure that Einstein had been seeking, and is now understood to be the key to relativity theory.

Actually I was rather pleased to see that Einstein had not been able to invent the curvature tensor on his own. Some of the books from which I had learned relativity had seemed to imply that any competent student should be able to derive the curvature tensor given the principles Einstein was working with. At the time I had had my doubts, and it was reassuring to see that the only person who had ever actually faced the problem without being able to look up the answer had not been able to solve it. Einstein had to ask a friend who knew the right mathematics.

The textbooks go on to say that once one understand the curvature tensor, one is very close to Einstein's theory of gravity. The questions Einstein is asking should lead him to invent the theory in half a page. There are only two steps to take, and one can see from this notebook that Einstein has all the ingredients. But could he do it? Apparently not. He starts out promisingly, then he makes a mistake. To explain why his mistake is not a mistake he invents a very clever argument. With falling hearts, we, reading the notebook, recognize his argument as one that was held up to us as an example of how not to think about the problem. As good students of the subject we know that the agument being used by Einstein is not only wrong but absurd, but no one told us it was Einstein himself who invented it. By the end of the notebook he has convinced himself of the truth of a theory that we, with more experience of this kind of stuff than he or anyone could have had at the time, can see is not even mathematically consistent. Still, he convinced himself and several others of its promise, and for the next two years they pursued this wrong theory. Actually the right equation was written down, almost accidentally, on one page of the notebook we looked at it. But Einstein failed to recognize it for what it was, and only after following a false trail for two years did he find his way back to it. When he did, it was questions his good friends asked him that finally made him see where he had gone wrong."

Comment by Jon Paprocki

2012-08-08T23:44:19Z

interpretation in terms of noncommutative geometry can be found at physik.uni-regensburg.de/forschung/krey/papkre0

Comment by Jon Paprocki

2012-08-08T23:44:06Z

It is not a perfect analogy, but I think that the intuition built from this statement helps to understand something like the Aharanov-Bohm effect in quantum mechanics. In this case, the quantum phase of a particle moving from point A to point B depends on the path taken to get there. So if we wave our hands and replace 'phase' with 'altitude', then we could imagine that there are two different paths from A to B, one which is 'uphill' and one which is 'downhill'. And so you might have a notion of walking uphill to school both ways. A quick introduction to the Aharanov-Bohm effect and its

Comment by Jon Paprocki

2012-06-12T17:18:15Z

I believe a countably infinite number of lakes can actually be produced.

Comment by Jon Paprocki

2011-10-05T14:58:41Z

similarity to Gel`fand duality. So maybe if you can try explaining this to your friend that will help him out, since it sounds like he used to have a very similar attitude to mine.

Comment by Jon Paprocki

2011-10-05T14:58:08Z

Dori, I am embarrassed to admit that I also once 'hated polynomials', and even had the same attitude towards algebraic geometry because that was what it appeared to be about to me at first glance. But now I blame it on mostly being ignorant of the ubiquity of polynomials in mathematics. So like your friend, I was interested in non-commutative geometry and was really enamored with Gel`fand duality - and when I found out that one of the points of view of algebraic geometry is that rings are all viewed as being functions over their spectrum, my attitude immediately changed because of the

Comment by Jon Paprocki

2011-08-29T14:49:09Z

It appears to have resurfaced: matrix.cmi.ua.ac.be/fun

Comment by Jon Paprocki

2011-04-23T15:53:26Z

Very interesting! Definitely looks like something I will jack from the library.

Comment by Jon Paprocki

2011-04-23T15:48:48Z

Sorry for the vagueness! I think I've realized that I don't understand the question well enough to make it more concrete (a little ironic, given the nature of the question), so I don't think I'll try to rewrite it until I have a better grasp of what I'm asking.