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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.
7
votes
Contemporary mathematical themes
The dichotomy between structure and randomness is one such theme. Tao's paper focuses on additive number theory, where the idea is that almost all sets are either highly structured (e.g., contain ari …
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, …
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 …
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
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 …
4
votes
Intuitive and/or philosophical explanation for set theory paradoxes
The history of axiomatic set theory did not proceed in the way that you suggest here. Zermelo, for example, was motivated to form his axiomatic system primarily in order to give a careful proof of hi …
- 82.7k
5
votes
Why is it a good idea to study a ring by studying its modules?
This is closely related to Pete Clark's answer, but stated in a slightly different way that I personally find helpful. I think it's not too hard to convince people that when studying an abstract obje …
- 82.7k
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 …
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). …
12
votes
The advantage of asymmetric objects
In light of your stated motivation, the following may not be what you had in mind, but objects with no symmetries are often easier to handle when it comes to computation and/or enumeration. For examp …
- 82.7k
4
votes
What is the high-concept explanation on why real numbers are useful in number theory?
Some historical context may be useful here.
In the late 19th and early 20th centuries, many mathematicians informally categorized mathematics into three tiers: arithmetic, analysis, and set theory. …
10
votes
Do empirical studies have a place in contemporary mathematics research?
coudy's answer, pointing you to the Experimental Mathematics, is the right answer, and there are already other MO questions about experimental mathematics that are relevant, but I can't resist giving …
6
votes
Why does mathematics seem to have a polarity bias?
I don't have any deep insights to offer, but I'd like to suggest that there are two separate questions being mixed together here.
The first question is why there often appears to be an asymmetry betwe …
- 82.7k
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 …
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 …