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1500 questions
44
votes
3 answers
Does an existence of large cardinals have implications in number theory or combinatorics?
Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any interesting theorems in these areas that can be proved…
Oksana Gimmel
- 1,679
44
votes
2 answers
What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which exists in a world where standard conjectures and…
Joël
- 25,755
44
votes
35 answers
Fixed point theorems
It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and of course everyone should know Picard's Theorem…
Rodrigo A. Pérez
- 3,071
44
votes
8 answers
Should one attack hard problems?
When I applied for a PhD student position I had an interview with two professors. Somehow we touched the problem if $P$ is $NP$ and, once we got there, for some reason both professors made it clear that in their opinion there is absolutely no point…
user10891
44
votes
7 answers
Character table does not determine group Vs Tannaka duality
From the example $D_4$, $Q$, we see that the character table of a group doesn't determine the group up to isomorphism. On the other hand, Tannaka duality says that a group $G$ is determined by its representation ring $R(G)$.
What is the additional…
user19475
44
votes
1 answer
Possible formal smoothness mistake in EGA
EGA IV 17.1.6(i) states that formal smoothness is a source-local property. In other words, a map $X\to Y$ of schemes is formally smooth if and only if there is an open cover $U_i$ ($i\in I$) of $X$ such that each restriction $U_i\to Y$ is formally…
JBorger
- 9,278
44
votes
1 answer
Thurston's senior thesis at New College
I've been collecting some of the many unpublished manuscripts of Bill Thurston over the years. His recent passing inspired me to ask the following. I've seen a number of references (for instance, in his wikipedia biography here) to a senior thesis…
Andy Putman
- 43,430
44
votes
2 answers
Does the curvature determine the metric?
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are
not isometric to one another, but that there is a…
Bernhard Boehmler
- 1,695
43
votes
7 answers
Complete discrete valuation rings with residue field ℤ/p
There are two great first examples of complete discrete valuation ring with residue field $\mathbb{F}_p = \mathbb{Z}/p$: The $p$-adic integers $\mathbb{Z}_p$, and the ring of formal power series $(\mathbb{Z}/p)[[x]]$. Any complete DVR over…
Greg Kuperberg
- 56,146
43
votes
5 answers
How large is TREE(3)?
Friedman, in _Lectures notes on enormous integers shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation denotes iteration). But actually, using the…
Feldmann Denis
- 3,570
- 1
- 18
- 36
43
votes
6 answers
Why do I need densities in order to integrate on a non-orientable manifold?
Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form exists and the concept of a density is introduced,…
ISH
- 843
43
votes
7 answers
Why should I care about Heegaard-Floer theory?
I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these kinds of things have been known since the '60's and…
Mitya
- 1
43
votes
2 answers
Meaning/origin of Seiberg-Witten equations/invariants
Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take an orthogonal frame bundle $P$ of $TX$, a…
Chris Gerig
- 17,130
43
votes
3 answers
History of the four-colour problem
It is stated in many places that the first published reference to the four-colour problem (aka the four-color problem) was an anonymous article in The Athenæum of April 14, 1860, attributed to de Morgan.
I was poking around in earlier issues of The…
Brendan McKay
- 37,203
43
votes
1 answer
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such inclusion structures should be simpler if they…
Thomas Riepe
- 10,731