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1500 questions
72
votes
13 answers
The use of computers leading to major mathematical advances II
I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question that I asked 11 years ago.
There are several…
Gil Kalai
- 24,218
72
votes
9 answers
Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)
Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
Yoo
- 1,073
72
votes
6 answers
A better way to explain forcing?
Let me begin by formulating a concrete (if not 100% precise) question, and then I'll explain what my real agenda is.
Two key facts about forcing are (1) the definability of forcing; i.e., the existence of a notion $\Vdash^\star$ (to use Kunen's…
Timothy Chow
- 78,129
72
votes
8 answers
Category theory and set theory: just a different language, or different foundation of mathematics?
This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as precise as possible, I am outlining the background…
Claus
- 6,777
72
votes
4 answers
Non-Borel sets without axiom of choice
This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of choice, and examples of non Lebesgue measurable…
Anweshi
- 7,272
72
votes
6 answers
Surprisingly short or elegant proofs using Lie theory
Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:
A finite extension $K$ of $\mathbb{C}$ forms a finite-dimensional vector space over…
Robin Goodfellow
- 769
72
votes
2 answers
Can you hear the shape of a drum by choosing where to drum it?
I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian manifold) it is in general impossible to use the…
Emilio Pisanty
- 2,308
72
votes
2 answers
The amplituhedron minus the physics
Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume familiarity with quantum field theory,
but perhaps there…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
72
votes
4 answers
Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. (For this famous outer automorphism, see for…
Benoit Jubin
- 1,049
71
votes
10 answers
Nice proof of the Jordan curve theorem?
As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive reason why a very simple proof is not possible?
user2498
- 1,823
71
votes
2 answers
Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, I often get bamboozled by all the different types…
Andrew Stacey
- 26,373
71
votes
16 answers
Is there a nice application of category theory to functional/complex/harmonic analysis?
[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most category theory books and it normally has little…
Dan
- 11
71
votes
9 answers
What is a continuous path?
I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting nervous thinking: it's just a continuous function…
Valerio Capraro
- 5,894
71
votes
3 answers
What exactly is the relation between string theory and conformal field theory?
Maybe it would be helpful for me to summarize the little bit I
think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and
an operator
$$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\otimes m}$$
to a Riemann surface $X$ with $n$…
Minhyong Kim
- 13,471
71
votes
3 answers
Where do all these projection formulas come from?
I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples.
Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\subset Y$, we have $$f(f^{-1}(Y')\cap X')=Y'\cap…
Georges Elencwajg
- 46,833