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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
76
votes
How do you decide whether a question in abstract algebra is worth studying?
I'm going to interpret your question in the language of Gowers's "two cultures" essay as follows:
How does one get good at theory-building?
The process of developing a good theory can seem decep …
46
votes
Logic in mathematics and philosophy
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might …
- 82.7k
43
votes
Theorems that are 'obvious' but hard to prove
There is a whole class of examples of the following general form: There is an obvious candidate for the solution to an optimization problem, and the obvious candidate is in fact best, but it's very ha …
39
votes
Your favorite surprising connections in mathematics
The work of Nabutovsky and Weinberger applying computability theory (a.k.a. recursion theory) to differential geometry. For example one of their results is that if you consider the space of Riemannia …
33
votes
Proposals for polymath projects
Update: February 23, 2017. Launched on polymathblog.
Rota's Basis Conjecture. Let $B_1, \ldots, B_n$ be $n$ bases of an $n$-dimensional vector
space $V$ (not necessarily distinct or disjoint). …
30
votes
How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
If you are not already aware of it, I'd recommend reading Representation theory: Its rise and its role in number theory by Langlands himself. He motivates the Langlands program in terms of one of the …
- 82.7k
25
votes
Motivation behind Analytic Number Theory
I will go out on a limb and say that in my opinion, it is the norm, rather than the exception, for a branch of mathematics to be a collection of results that we can prove using the techniques we know, …
- 82.7k
24
votes
The use of computers leading to major mathematical advances II
For experimental mathematics as that term is usually understood, I would commend to your attention the paper by Roger Behrend, Ilse Fischer and Matjaž Konvalinka, "Diagonally and antidiagonally symmet …
23
votes
Accepted
What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?
There's a certain confusion underlying your question, which Andreas Blass's answer is trying to point out. Let me see if I can explain it in different words.
You say, “the negation of Con(ZFC) proves …
- 82.7k
22
votes
Breakthroughs in mathematics in 2021
Having just listened to some of Jacob Tsimerman's Minerva lectures, I became aware of the recent arXiv preprint, Canonical Heights on Shimura Varieties and the André–Oort Conjecture, by Jonathan Pila, …
18
votes
Theorems that impeded progress
The proof that a particular computational problem is NP-complete can cause people to stop trying to make theoretical progress on it, instead focusing all their attention on heuristics that have only e …
15
votes
What are some fundamental "sources" for the appearance of pi in mathematics?
As a counterpoint to gowers's devil's advocacy, I'd mention that some formulas for $\pi$ have been discovered experimentally, and in some cases we still don't know how to prove them. For example, in …
13
votes
What is an important mathematical question?
I want to point out that you raised two questions, and in my opinion they are very different questions.
So I really want to know how to decide whether a question is worth studying?
How do I deci …
13
votes
Theorems that impeded progress
Like RBega2 I hesitate to say that this is definitely an example, but the paper "Natural Proofs" by Razborov and Rudich, which showed that certain kinds of proof techniques would be insufficient to pr …
13
votes
Every mathematician has only a few tricks
Scott Aaronson has taken a stab at articulating his own methodology for upper-bounding the probability of something bad. He was inspired by a blog post by Scott Alexander bemoaning how rarely experts …