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1500 questions
56
votes
12 answers
Why is it useful to study vector bundles?
I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is there any big reason why the study of…
user709
56
votes
10 answers
de Rham cohomology and flat vector bundles
I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed forms in $\Omega^k(M)$ modulo the set of exact…
Spinorbundle
- 1,909
56
votes
4 answers
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is asymptotically almost surely connected. The way I know…
Matthew Kahle
- 7,755
56
votes
9 answers
Examples in mirror symmetry that can be understood.
It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider mirror symmetry as science, what are some examples…
aglearner
- 13,995
56
votes
4 answers
Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various nice operations. But whereas complete intersections…
Martin Brandenburg
- 61,443
56
votes
16 answers
Examples of using physical intuition to solve math problems
For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects it's subsequent bounce
would be considered…
Luke Grecki
- 71
56
votes
10 answers
A clear map of mathematical approaches to Artificial Intelligence
I have recently become interested in Machine Learning and AI as a student of theoretical physics and mathematics, and have gone through some of the recommended resources dealing with statistical learning theory and deep learning in neural…
AI Bert
- 485
56
votes
13 answers
Cardinalities larger than the continuum in areas besides set theory
It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably infinite. Are there any well known results within say,…
Daniel Miller
- 5,609
56
votes
21 answers
Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has many useful applications. I wonder if there are…
zhaoliang
- 11
56
votes
1 answer
A mysterious connection between primes and $\pi$
The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
Conjecture (December 15, 2019). Let $s(n)$ be the sum of all primes $p\le n$ with $p\equiv1\pmod4$,…
Zhi-Wei Sun
- 14,451
56
votes
14 answers
Does any method of summing divergent series work on the harmonic series?
It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I…
Qiaochu Yuan
- 114,941
56
votes
2 answers
What is prismatic cohomology?
Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in this forum:
What is prismatic cohomology and what…
Dr. Evil
- 2,651
56
votes
2 answers
A condition that implies commutativity
Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for the cases $n=2, 3,4$ is the subject matter of…
José Hdz. Stgo.
- 8,732
56
votes
7 answers
Capitalization of theorem names
I hope this question is suitable; this problem always bugs me. It is an issue of mathematical orthography.
It is good praxis, recommended in various essays on mathematical writing, to capitalize theorem names when recalling them: for instance one…
Andrea Ferretti
- 14,454
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- 111
56
votes
3 answers
Work of plenary speakers at ICM 2018
The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from 2014 and 2010. Can you, please, for the benefit of others give a short…
C. Eratosthene
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