How NOT to reference papers

September 16, 2014

Originally posted on Igor Pak's blog:

In this post, I am going to tell a story of one paper and its authors which misrepresented my paper and refused to acknowledge the fact. It’s also a story about the section editor of Journal of Algebra which published that paper and then ignored my complaints. In my usual wordy manner, I do not get to the point right away, and cover some basics first. If you want to read only the juicy parts, just scroll down…

What’s the deal with the references?

First, let’s talk about something obvious. Why do we do what we do? I mean, why do we study for many years how to do research in mathematics, read dozens or hundreds of papers, think long thoughts until we eventually figure out a good question. We then work hard, trial-and-error, to eventually figure out a solution. Sometimes we do this in a matter of hours and sometimes…

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Some Strange Math Facts

September 15, 2014

Originally posted on Gödel's Lost Letter and P=NP:

250px-STAN_ULAM_HOLDING_THE_FERMIAC

Stanislaw Ulam was a Polish-American mathematician whose work spanned many areas of both continuous and discrete mathematics. He did pioneering research in chaos theory and Monte Carlo algorithms, and also invented the concept of a measurable cardinal in set theory. His essential modification of Edward Teller’s original H-bomb design is used by nearly all the world’s thermonuclear weapons, while he co-originated the Graph Reconstruction conjecture. His name is also associated with the equally notorious 3n+1 conjecture. Thus he was involved in some strange corners of math.

Today Ken and I want to talk about some strange facts observed by Ulam and others that we did not know or fully appreciate.

Perhaps you can use them, perhaps you may enjoy them, but they are all kind of fun. At least we think so. Ulam’s autobiographyAdventures of a Mathematician shows his sense of fun, and he was…

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ICM2014 — Kollár, Conlon, Katz, Krivelevich, Milnor

September 4, 2014

Originally posted on Gowers's Weblog:

As the ICM recedes further into the past, these posts start to feel less and less fresh. I’ve had an enforced break from them as over the course of three days I drove my family from the south of France back to Cambridge. So I think I’ll try to do what I originally intended to do with all these posts, and be quite a lot briefer about each talk.

As I’ve already mentioned, Day 3 started with Jim Arthur’s excellent lecture on the Langlands programme. (In a comment on that post, somebody questioned my use of “Jim” rather than “James”. I’m pretty sure that’s how he likes to be known, but I can’t find any evidence of that on the web.) The next talk was by Demetrios Christodoulou, famous for some extraordinarily difficult results he has proved in general relativity. I’m not going to say anything about the talk, other…

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Hales-Jewett and a generalized van der Warden Theorems

August 27, 2014

Originally posted on I Can't Believe It's Not Random!:

The Hales-Jewett theorem is one of the most fundamental results in Ramsey theory, implying the celebrated van der Waerden theorem on arithmetic progressions, as well an its multidimensional and IP versions. One interesting property of the Hales-Jewett’s theorem is that it is a set theoretical statement, having no structure, and hence making it versatile for applications. Recently I realized that there exists an equivalent formulation of this theorem using some algebraic structure, and indeed it can be seen as an analogue of van der Waerden’s theorem. The main purpose of this post is to establish that equivalence. In an initial section I present the deduction of the multidimensional van der Waerden from the Hales-Jewett theorem, both to set the mood and to set the stage to later establish the analogy between van der Waerden’s theorem and the equivalent formulation of the Hales-Jewett theorem.

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Jim Geelen, Bert Gerards, and Geoff Whittle Solved Rota’s Conjecture on Matroids

August 11, 2014

Originally posted on Combinatorics and more:

gcrota

Gian Carlo Rota

Rota’s conjecture

I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor.  The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors).  Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements.  The notices paper gives an excellent self-contained introduction to the conjecture.

This is a project that started in 1999 and it…

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Szemeredi Regularity and Roth’s Theorem

December 8, 2013

Originally posted on Mental Wilderness:

I’m giving a talk today at the Part III seminars (at the University of Cambridge). Notes are available here: https://dl.dropboxusercontent.com/u/27883775/math%20notes/cnt.pdf

Abstract: The Szemeredi Regularity Lemma is the graph-theoretic analogue of the dichotomy between order and randomness. It states that any large enough graph can be described using a structured component of bounded complexity with small error, the error being the pseudorandom component of the graph. I’ll sketch a proof using the energy increment argument, and then apply it to prove Roth’s Theorem: that any set of positive density in the naturals has a 3-term arithmetic progression.

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Analysing Escher

August 17, 2013

Originally posted on Complex Projective 4-Space:

I have just been involved in the production of an action thriller about the International Mathematical Olympiad. Since certain intervals of time were inactive and unthrilling, I decided to pass the time by writing a few cp4space articles, amongst other things. This particular one was distributed across an evening and several train journeys.

Anyway, when I was at the actual IMO a few years ago, the opportunity presented itself to visit the M. C. Escher museum in The Hague. In the process, I acquired five postcards featuring the art of M. C. Escher, which I intend to put to good use in the immediate future. Until then, however, I shall merely analyse them:

Snakes, 1969

This was Escher’s final print, and has always been particularly aesthetically pleasing. However, until now I haven’t analysed it in great detail. Ignoring the snakes themselves, which are intertwined throughout the structure in a strikingly…

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The Stacks Project gets ever awesomer with new viz

July 30, 2013

Originally posted on 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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Logical Graphs : 1

July 29, 2013

johnpappas:

An extremely interesting post regarding logical graphs.

Originally posted on Inquiry Into Inquiry:

Moving Pictures of Thought

A logical graph is a graph-theoretic structure in one of the styles of graphical syntax that Charles Sanders Peirce developed for logic.

Introduction

A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic.

In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.

In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird’s eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they…

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Graph regularity

July 29, 2013

johnpappas:

Excellent post on regularity lemma and triangle removal lemma.

Originally posted on Yufei Zhao:

In this blog post I will give a brief introduction to Szemerédi’s Regularity Lemma, a powerful tool in graph theory. The post is based on a talk I gave earlier today at a graduate student lunch seminar.

Consider the following problem. Suppose you’re given a very large graph. The graph has so many vertices that you won’t be able to access all of them. But nevertheless you want to find out certain things about the graph. These situations come up in real world applications. Perhaps we would like to know something about a social network, e.g., Facebook, but we don’t have the resource to go through every single node, as there are simply too many of them. For the purpose of this blog post though, we won’t talk about applications and instead stick to the mathematics.

Suppose we are interested answering the following question about the very large graph:

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