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1500 questions
80
votes
6 answers
How to write a good MathSciNet review?
When reviewing for MathSciNet, I routinely find myself just paraphrasing and abbreviating the introduction provided by the author, and occasionally adding a few words about the quality of the research or the cleverness of the argument (which the…
Jakub Konieczny
- 1,582
80
votes
23 answers
Algebraic geometry examples
What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that are unusual or nonstandard, as I already know…
Richard Borcherds
- 20,442
80
votes
1 answer
Converse to Euclid's fifth postulate
There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, or more precisely Playfair's version of it,
states…
Mohammad Ghomi
- 7,047
80
votes
2 answers
Vladimir Voevodsky's works
Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology.
Voevodsky was awarded the Fields…
user60504
80
votes
13 answers
How does an academic mathematician educate him/herself about job opportunities outside academia?
One of the contradictions of being a math professor is that a big part of your job is to train people to do things which are quite different from what you do yourself professionally; this is especially true for undergrads, but to some measure also…
Ben Webster
- 43,949
80
votes
10 answers
Existence of a zero-sum subset
Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, a_n\}$, with the property that for each $i$ there…
Gjergji Zaimi
- 85,056
80
votes
5 answers
How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
Equivalently, we say in this case that τ is coarser
than σ, that σ is finer…
Joel David Hamkins
- 224,022
80
votes
22 answers
Are there proofs that you feel you did not "understand" for a long time?
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was inspired by some other posts seeking for a hint to…
Amir Asghari
- 2,277
- 3
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79
votes
26 answers
What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, of course -- the more I learn, the more stuff I'd…
Allen Knutson
- 27,645
79
votes
6 answers
Does every polyomino tile R^n for some n?
This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer.
A polyomino is usually defined to be a finite set of unit squares, glued together…
Timothy Chow
- 78,129
79
votes
9 answers
Breakthroughs in mathematics in 2023
At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a nice tradition.
Continuing this, let me ask…
Bogdan Grechuk
- 5,947
- 1
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- 42
79
votes
12 answers
Why are characters so well-behaved?
Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a tool for studying representations of a group; they…
Saul Glasman
- 2,148
79
votes
10 answers
What are the uses of the homotopy groups of spheres?
Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:
Have the homotopy groups of spheres ever been applied to anything, including in algebraic topology…
Andrew Stacey
- 26,373
79
votes
15 answers
Why torsion is important in (co)homology ?
I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of algebraic geometry.
In this community wiki question I…
Qfwfq
- 22,715
79
votes
1 answer
Topological cobordisms between smooth manifolds
Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--Whitney and Pontrjagin numbers.
Novikov has shown that…
Oscar Randal-Williams
- 17,724