Most Popular

1500 questions
46
votes
4 answers

What are "good" examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly: What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (or not spin) (and why)? For comparison, I'd…
Otis Chodosh
  • 7,077
46
votes
14 answers

How to select a journal?

What are good criteria for selecting a journal to submit a paper to? One criterion per answer, please. It is easy to group journals by subject and prestige, but is there a thought-process that you use to determine which journal is good for your…
Ben Weiss
  • 1,588
46
votes
0 answers

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of the usual ring-theoretic scheme theory to…
46
votes
5 answers

Publishing journals articles without transferring copyright.

I'm a grad student getting close to submitting my first journal article (which will be single-authored). My understanding is that it's standard practice for authors to transfer the copyright of their paper to the journal in which it is published. I…
John
  • 453
46
votes
7 answers

Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness

I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector fields and one-forms on Euclidean space.…
46
votes
3 answers

Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many observables in quantum mechanics - such as position,…
46
votes
2 answers

Formal group laws and L-series

Let E be an elliptic curve, let L(s)=anns denote its L-function, and set f(x)=anxnn. Then Honda has observed that F(X,Y)=f1(f(X)+f(Y)) defines a formal group law. The formal group law of an…
46
votes
6 answers

Is there a quaternionic algebraic geometry ?

Let H be the skew-field of quaternions. I'm aware of the Theorem 1. A function f:HH which is H-differentiable on the left (i.e. the usual limit h1(f(x+h)f(x)), for h0, exists for every…
Qfwfq
  • 22,715
46
votes
0 answers

Cochains on Eilenberg-MacLane Spaces

Let p be a prime number, let k be a commutative ring in which p=0, and let X=K(Z/pZ,n) be an Eilenberg-MacLane space. Let F be the free E-algebra over k generated by a class η in (homological)…
Jacob Lurie
  • 17,538
  • 4
  • 74
  • 77
46
votes
8 answers

Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As testing for the presence of any given minor can…
Gordon Royle
  • 12,288
46
votes
6 answers

Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

In his 1967 paper A convenient category of topological spaces, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a good replacement of the category Top topological spaces, in order to do homotopy theory. The most…
46
votes
5 answers

‘Naturally occurring’ K(π,n) spaces, for n2

[edited!] Given a group π and an integer n>1, what are examples of Eilenberg–MacLane spaces K(π,n) that can be constructed as "known" manifolds? (Or if not a manifold, say some space people had a pre-existing desire to study before…
Romeo
  • 2,714
46
votes
3 answers

Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of mathematical theoretical analysis, but there are quite…
46
votes
3 answers

How to read an article and make it actually useful?

I've been wondering for a while: how should mathematicians read an article in order to "take most" from it? For example, when I did my Master's thesis I based it on an article (I'm into analysis) and of course I analyzed every and each part of it,…
46
votes
7 answers

Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics. I'm going to focus the post (and modulate my…