nklein software

Just a quick update on my previous post on gamifying the problem-solving process in high school algebra. I had a little time to spare on an airplane flight, of all things , so I decided to rework the mockup version of the algebra game into something a bit more structured, namely as 12 progressively difficult levels of solving a linear equation in one unknown. Requires either Java or Flash. Somewhat to my surprise, I found that one could create fairly challenging puzzles out of this simple algebra problem by carefully restricting the moves available at each level. Here is a ... [Link]

Ben Green and I have just uploaded to the arXiv our paper “New bounds for Szemeredi’s theorem, Ia: Progressions of length 4 in finite field geometries revisited“, submitted to Proc. Lond. Math. Soc.. This is both an erratum to, and a replacement for, our previous paper “New bounds for Szemeredi’s theorem. I. Progressions of length 4 in finite field geometries“. The main objective in both papers is to bound the quantity for a vector space over a finite field of characteristic greater than , where is defined as the cardinality of the largest subset of that does not contain an ... [Link]

Here is the final analysis question from 2003. 12C. State carefully the formula for integration by parts for functions of a real variable. Let be infinitely differentiable. Prove that for all and for all , . By considering the function at , or otherwise, prove that the series converges to . What is implied by “state carefully”? It probably means that more is required than just writing . What else can one put? The main thing is the conditions under which the formula is valid. So I think what is required is something like this. Let and be differentiable functions ... [Link]

I’ve been spending too much time recently thinking about among other things right properness of model categories. The ultimate goal is to build models of homotopy type theory in ∞,1 -toposes, but at the moment in this post I’m just trying to get a handle on about what right properness means, at an intuitive level. So this is going to be kind of rambly and philosophical and lacking in conclusions. The issue of right properness is an aspect of a more general question: how do properties of a model category — call it ℳ, say — reflect properties of the ... [Link]

These two links have homeomorphic exteriors. They’re both strongly invertible. Let’s quotient the first link by its strong inversion to get a tangle. Then we’ll isotop that tangle around and eventually take its double branched cover to get the second. Notice that this homeomorphism swaps the red and blue meridians and longitudes. [Link]

The coming Monday, Tuesday and Thursday, A. Wigderson will be in Zürich to give the yearly Pauli Lectures, with a general title of “The computational lens”. It seems to be the best possible time to share the following insight from one of my family’s Minecraft sessions… [Link]

Right now I am at the 17th NRW Topology Meeting. In a few minutes I will talk about Principal ∞-Bundles – Theory and Applications. By coincidence it turns out that the previous speaker, Ulrich Pennig discussed, in a nice talk, such an application: twisted 2-vector bundles. This is joint work of him and Brano Jurčo. They consider BDR 2-vector bundles which, by definition, are the objects classified by, roughly, the monoidal delooping of the monoidal category GL • Vect . Their starting point to consider twists of these structures is the discussion in Thomas Kragh’s Orientations and Connective Structures on ... [Link]

A natural way that one might hope to bring about a genuine change to the current subscription model where libraries pay through the nose for journals is that i we all put our papers on the arXiv and ii the libraries conclude, correctly, that the benefits from their very expensive subscriptions do not justify the costs. Bundling across subjects makes this a lot more difficult of course, but it seems that some institutions in Germany do not subscribe to the Freedom Collection see previous post for a definition , which makes it easier. And now there is an example. The ... [Link]

Over on Google+, David Roberts just told me the most exciting theorem I’ve heard all week. Every projective variety is the Grassmannian of a quiver representation! I suppose it’s just another indication of the ‘wildness’ of quiver representations once we leave the safe waters of Gabriel’s theorem. Let me explain…. To briefly recall: a quiver is a category Q freely generated by a finite directed graph. A quiver representation is a functor F:Q→FinVect. In other words, it’s just a finite-dimensional vector space for each vertex and a linear operator for each edge. A morphism of quiver representations is a natural ... [Link]

Here’s a basic example that comes up if you work with elliptic curves: Let be a prime which is . Let be the elliptic curve over a field of characteristic . Then has an endomorphism . It turns out that, in the group law on , we have . That is to say, plus copies of is trivial. I remember when I learned this trying to check it by hand, and being astonished at how out of reach the computation was. There are nice proofs using higher theory, but shouldn’t you just be able to write down an equation which ... [Link]

Elsevier has recently put out a new statement giving details of some changes it has made. In their own words, In February, we informed you of a series of important changes that we are making to how the Elsevier mathematics program will be run. In this letter, we would like to update you on where we currently stand, and inform you of some new initiatives we have undertaken based upon the feedback we have received from the community. I have known for some time that they were going to make an announcement of this kind, and that it would involve ... [Link]

Joy Christian is the author of numerous papers claiming to disprove Bell’s theorem. Yes, that Bell’s theorem: the famous result from the 1960s showing that no local hidden variable theory can reproduce all predictions of quantum mechanics for entangled states of two particles. Here a “local hidden variable theory” means—and has always meant—a theory where Alice gets some classical information x, Bob gets some other classical information y generally correlated with x , then Alice and Bob choose which respective experiments to perform, and finally Alice sees a measurement outcome that’s a function only of her choice and of x ... [Link]

This post belongs to a series that began here. Next up is a question about integration. 11B. Let be continuous. Define the integral . You are not asked to prove existence. Suppose that are real numbers such that for all . Stating clearly any properties of the integral that you require, show that . The function is continuous and non-negative. Show that . Now let be continuous on . By suitable choice of show that , and by making an appropriate change of variable, or otherwise, show that . The first part of this question is something I recommend practising ... [Link]

My previous post recommended a really nice talk by Jonathan Israel about human rights. Here is the link again. It is very recommended. Actually, I was in the audience and after the lecture, at the reception, I came to the lecturer to compliment him for the talk, except that I approached a different, rather similar looking, person. I am not going to make it a habit on this blog to recommend good videotaped lectures available on the Internet unless this new Genre will lead to a sky-rocketing rating , but, just one more time, let me recommend you a really ... [Link]

Here’s another one. 10F. State without proof the Integral Comparison Test for the convergence of a series of non-negative terms. Determine for which positive real numbers the series converges. In each of the following cases determine whether the series is convergent or divergent: i , ii , iii . I don’t know exactly what was referred to in the course as the integral comparison test, but since all the sequences being summed are monotone decreasing I’ll go for a neat statement that assumes that and coincides with what Wikipedia refers to as the integral test . Let be a decreasing ... [Link]

This is the second in a series of posts that started here. In the first post I explained what I’m up to. Now let me just continue with some more questions. I’m now on to the harder Section II questions. Here’s the first one I want to look at. Even though it makes the posts shortish, I think I’m going to stick to one long question per post. 9F. Prove the Axiom of Archimedes. Let be a real number in and let be positive integers. Show that the limit exists, and that its value depends on whether is rational or ... [Link]

When I was a mathematics undergraduate, I became aware of a huge cultural difference between mathematicians and engineers. That sounds like the beginning of a joke you’ve heard twenty times already, but it isn’t. The difference was that when mathematicians were set questions, they were expected to work out how to solve them, and if they couldn’t do so then it was too bad — the best they could do about it was ask their supervisors. But engineers had model answers for everything, available with the latest technology, which in those days was microfiche. In case you have no idea ... [Link]

Today April 27, 2012 it is precisely 213 years 7 months, and 29 days to the completion of the declaration of the rights of man, which makes it a perfect occasion to celebrate this remarkable human creation. Here is a beautiful lecture by Jonathan Israel about the history of basic human rights: The History of Basic Human Rights: The Declaration of the Rights of Man, 1789 [Link]

I just got back from an excellent trip to the U.K., which started with my first visit ever to Wales, continued with my second PSSL, and concluded with my first ever visit to Scotland thanks Tom! . Homotopy type theorists may be interested to have a look at my slides from Swansea and also the slides from a seminar we had at UCSD last quarter , but I won’t say any more about that now. Instead I want to discuss briefly several of the PSSL talks. One talk that I particularly enjoyed was John Bourke’s. He’s found a way to ... [Link]

I know that I posted a photo just a few days ago, but I’ve decided that I like this one better. [Link]

I’m glad to be able to report that “A new proof of the density Hales-Jewett theorem” has recently appeared in Annals of Mathematics. Unfortunately it’s behind a paywall, but you can find an almost final version on the arXiv. I might add that my enthusiasm for this way of working is undimmed. The reason there has been no Polymathematical activity on this blog for quite a while is that I’ve been busy with more conventional projects, but in the not too distant future I’d like to do some more open research. Also, Gil Kalai and I have a plan to ... [Link]

I have never known what it’s like to speak only one language. I speak four altogether: Polish my native language and Russian, English, and French, learned in that order. I’ve lived half of my life now in English, technically my third language but in truth second and competing for first. This sort of thing is common in Europe and among academics, and especially among American or Canadian academics of European origin such as me. It’s less common elsewhere, and I’ve heard all kinds of myths and misconceptions. I’m writing this mostly for those who haven’t had the experience. I got ... [Link]

Interested in biological diversity? Want to know more about how diversity can be quantified? Maybe diversity comes up in your work. Maybe you’ve heard rumours that there’s serious mathematics involved, and you want to know more. Or maybe you’re just curious. If so, come to a meeting in Barcelona! It’s running 2-6 July, and there are grants to cover attendance expenses. If you want one, please apply as soon as possible. We also have free slots for contributed talks. We’ve assembled what is already a head-spinningly varied group of people, from livestock breeding experts to ostensibly pure mathematicians to evolutionary ... [Link]

Dinitz’ conjecture The following theorem was conjectured by Jeff Dinitz in 1979 and proved by Fred Galvin in 1994: Theorem: Consider an n by n square table such that in each cell i,j you have a set with n or more elements. Then it is possible to choose elements from such that the chosen elements in every row and in every colummn are distinct. Special case: if all are the same set, say {1,2,…,n} then this is possible. Such a choice is called a Latin square for example . 1 2 3 4 2 3 4 1 3 4 1 ... [Link]

Last summer I wrote a post about EPSRC’s plans to direct their funding towards certain areas and not others, and in particular on its effect on mathematicians, the most dramatic of which was to restrict their fellowships, which had previously been available throughout mathematics, to statistics and applied probability. The strongest argument I could see in favour of EPSRC’s position was that they were reviewing the various subareas of mathematics before deciding which should be grown, which maintained and which shrunk, and that so far only statistics and applied probability had been reviewed with a decision that it should be ... [Link]