Tensor functors between categories of quasi-coherent sheaves – arXiv:1202.5147

“algebraic geometry is 2-affine”

Chris Schommer-Pries

This paper by Martin Brandenburg and Alexandru Chirvasitu looks interesting. There is a connection to a categorified algebraic geometry. 

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Legitimizing Napkin Drawing: The Curious Dispersion of Laffer Curves (1978-2008)

YANN GIRAUD

Image The Neo-Laffer curve, drawn by Martin Gardner in Scientific American (1981)

The Laffer curve is a graphical representation of how government revenues vary with the level of taxation. Allegedly, it was first drawn on a cocktail napkin by one of US President Ronald Reagan’s advisors in the 1970s.  Since then, it has been routinely reproduced in economics textbooks. This article provides an historical account that shows a sharp contrast between the formal triviality of the curve and the complexity of its circulation through various communities of economists, policy advisors, propagandists, and journalists. In this paper, I show that the dispersion of the Laffer curve presents two peculiarities: first, unlike many other diagrams used in economics, popular instantiations of the Laffer curve preceded its “academization” in professional economics; second, in spite of numerous transformations in the process of circulation, the curve’s canonical presentation as a symmetrical, bullet-like diagram was reinforced over…

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The Stacks Project gets ever awesomer with new viz

mathbabe

Crossposted on Not Even Wrong.

Here’s a completely biased interview I did with my husband A. Johan de Jong, who has been working with Pieter Belmans on a very cool online math project using d3js. I even made up some of his answers (with his approval).

Q: What is the Stacks Project?

A: It’s an open source textbook and reference for my field, which is algebraic geometry. It builds foundations starting from elementary college algebra and going up to algebraic stacks. It’s a self-contained exposition of all the material there, which makes it different from a research textbook or the experience you’d have reading a bunch of papers.

We were quite neurotic setting it up – everything has a proof, other results are referenced explicitly, and it’s strictly linear, which is to say there’s a strict ordering of the text so that all references are always…

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Laying train tracks

Geometry and the imagination

This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether the track closes up to make a loop. The pieces of track are all roughly the following shape:

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Three problems and a disclaimer

chorasimilarity

In this post I want to summarize the list of problems I am currently thinking about. This is not a list of regular mathematical problems, see the disclaimer on style written at the end of the post.

Here is the list:

1. what is “computing with space“? There is something happening in the brain (of a human or of a fly) which is akin to a computation, but is not a logical computation: vision. I call this “computing with space”. In the head there are a bunch of neurons chirping one to another, that’s all. There is no euclidean geometry, there are no a priori coordinates (or other extensive properties), there are no problems to solve for them neurons, there is  no homunculus and no outer space, only a dynamical network of gates (neurons and their connections). I think that a part of an answer is…

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Furstenberg’s Correspondence Theorem

I Can't Believe It's Not Random!

In 1977 Furstenberg gave a new proof of Szemerédi’s theorem using ergodic theory. The first step in that proof was to turn the combinatorial statement into a statement in ergodic theory. Thus Furstenberg created what is now known as Furstenberg’s correspondence principle. While this was not (by far) the most difficult part of the proof of Szemerédi’s theorem, it was this principle that allowed many generalizations of Szemerédi’s theorem to be proved via ergodic theoretical arguments. Most of those generalizations had to wait a long time before seeing a combinatorial proof, and for some, no combinatorial proof was ever found (yet).

In this post I will state and prove the correspondence principle and then I will use it to give another proof of Sárközy’s theorem, discussed in my previous post.

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Kleisli Categories

  Bartosz Milewski's Programming Cafe

In the previous installment of Categories for Programmers, Categories Great and Small, I gave a few examples of simple categories. In this installment we’ll work through a more advanced example. If you’re new to the series, here’s the Table of Contents.

Composition of Logs

You’ve seen how to model types and pure functions as a category. I also mentioned that there is a way to model side effects, or non-pure functions, in category theory. Let’s have a look at one such example: functions that log or trace their execution. Something that, in an imperative language, would likely be implemented by mutating some global state, as in:

string logger;

bool negate(bool b) {
     logger += "Not so! ";
     return !b;
}

You know that this is not a pure function, because its memoized version would fail to produce a log. This function has side effects.

In modern programming…

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