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1500 questions
71
votes
13 answers
Logic in mathematics and philosophy
What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of the 20th century.
Background and motivation
Logic is…
Gil Kalai
- 24,218
71
votes
6 answers
Kahler differentials and Ordinary Differentials
What's the relationship between Kahler differentials and ordinary differential forms?
Abtan Massini
- 1,756
71
votes
11 answers
How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this subject and keeping my own lecture notes. So I feel…
Pete L. Clark
- 64,763
71
votes
10 answers
Relating category theory to programming language theory
I'm wondering what the relation of category theory to programming language theory is.
I've been reading some books on category theory and topos theory, but if someone happens to know what the connections and could tell me it'd be very useful, as…
Michael Hoffman
- 1,785
71
votes
3 answers
Does iterating the derivative infinitely many times give a smooth function whenever it converges?
I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(x) := \lim_{n \to \infty} d^n f(x)$$ converges.…
Paul Cusson
- 1,735
71
votes
3 answers
What is Koszul duality?
Okay, let's make sure I'm on the same page with those who know homological algebra.
What is Koszul duality in general?
What does it mean that categories are Koszul dual (I guess representations of Koszul dual algebras are the examples?) What are…
Ilya Nikokoshev
- 14,934
71
votes
1 answer
Dualizing the notion of topological space
$\require{AMScd}$
Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements on the open sets of a topological space $X$ are…
71
votes
3 answers
Cohomology and fundamental classes
Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup generated?
Andrea Ferretti
- 14,454
- 13
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- 111
71
votes
4 answers
What are fixed points of the Fourier Transform
The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
pavpanchekha
- 1,461
70
votes
10 answers
Is every finite group a group of "symmetries"?
I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually true:
Does there exist, for every finite group G, a…
Andrew McIntyre
- 742
70
votes
28 answers
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, rings, groups, whatever.)
(2) ${\cal X}_0$ is an…
Steven Landsburg
- 22,477
70
votes
7 answers
Identifying poisoned wines
The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned one by feeding the wines to the rats. The poisoned…
Qiaochu Yuan
- 114,941
70
votes
10 answers
Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of the full fundamental group."
Can anyone explain to…
Harold Williams
- 2,696
70
votes
3 answers
The story about Milnor proving the Fáry-Milnor theorem
This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, Fáry was a professor in France and proved the result…
Greg Kuperberg
- 56,146
70
votes
1 answer
Nontrivial finite group with trivial group homologies?
I stumbled across this question in a seminar-paper a long time ago:
Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace 1\rbrace$?
I believe this to still be an open problem. …
Chris Gerig
- 17,130