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9
votes
2answers
456 views

What is a sieve and why are sieves useful?

I have been trying to understand what is exactly a sieve and why sieves are useful. I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they ...
11
votes
4answers
447 views

Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an ...
18
votes
2answers
765 views

First occurrence of “by the usual compactness argument”?

In his blog, Jeff Shallit asks, what was the first occurrence of the exact phrase, "by the usual compactness arguments," in the mathematical literature? He reports that the earliest appearance he has ...
9
votes
0answers
338 views

Where to find NSF Reports (For mathematical expository purpose)? [closed]

This is not asking about how to apply for funding. I just find that some NSF funding report (near the end of the funding cycle) is quite interesting read. In particular, it's almost like a research ...
159
votes
7answers
78k views

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
9
votes
0answers
412 views

Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The question is in the title. My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people ...
7
votes
6answers
1k views

How do you present a non-existence theorem?

(This question might be too vague, feel free to edit or vote for closing.) In math there are usually lots of non-existence theorems. When someone presents such a theorem, one natural response is "why ...
3
votes
1answer
1k views

Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
16
votes
14answers
2k views

Teaching a pedagogy course

At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
22
votes
2answers
917 views

Finding citations for 'well-known' results

I am almost ready to submit my most recent paper, and I find myself in a problem that has already occurred multiple times in my short publishing career. In this paper, I wish to state a result which ...
4
votes
4answers
2k views

Has anyone found an error in an early version of Neukirch?

I remember a friend in graduate school throwing an early edition of Jurgen Neukirch's Algebraic Number Theory book against a wall (so hard that it split the binding) after he had worked for a number ...
17
votes
6answers
853 views

How should one present calculations?

It is often necessary to present calculations, or at least their outlines, in a proof. However, when I need to do it, if the calculation takes more than one line (or perhaps two at the most), I feel ...
17
votes
9answers
3k views

General Equilibrium for Mathematicians

I've been reading up a lot on the recent financial crisis, and central to the story is the existence of general equilibrium models in economics, say, as proven by Arrow and Debreu (and MacKenzie?). ...
9
votes
2answers
642 views

Expository treatment of Schubert Cells Paper

I was wondering about the paper by Bernstein, Gel'fand, and Gel'fand on Schubert Cells. This paper is fairly old(and often cited) so I figured someone must have represented this material. In ...
143
votes
29answers
13k views

Which journals publish expository work?

I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps). Are there any journals which publish expository work, ...
14
votes
4answers
901 views

Introductory text for the non-arithmetic moduli of elliptic curves

I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ ...
2
votes
5answers
815 views

Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)

This is a bit of an ill-defined question, and I feel I should have been able to resolve it by combining Google with a few library trips, but I'm having difficulty narrowing down the search results to ...
1
vote
2answers
318 views

Use of the word “data” not in the statistical sense [closed]

Occasionally I see the use of the word "data" in definitions. For instance, one definition of an exact sequence starts off by saying, "An exact sequence of abelian groups (or modules or vector spaces) ...
46
votes
1answer
4k views

Is it best to run or walk in the rain? [closed]

According to the Norwegian meterological institute, the answer is that it is best to run. According to Mythbusters (quoted in the comments to that article), the answer is that it is best to walk. My ...
1
vote
4answers
2k views

Is it alright for STD error bars to be below zero?

I have some statistical data from which I want to graph the means and use the standard deviations as error bars. However this produces a graph with some of the error bars passing below zero. A ...
8
votes
3answers
1k views

Short Introduction to Planar Algebras

Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.