Questions tagged [big-picture]
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.
381 questions
21
votes
4
answers
2k
views
Why and how do (classical) reverse mathematics and intuitionistic reverse mathematics relate?
Broadly speaking, the idea of “reverse mathematics” is to find equivalents to various standard mathematical statements over a weak base theory, in order to gauge the strength of theories (sets of ...
- 38.7k
23
votes
13
answers
5k
views
Great theorems with elementary statements: 2026-onward
My 2021 book
Landscape of 21st Century Mathematics, Selected Advances, 2001–2020
collects great theorems with elementary statements published in 2001-2020. I now finishing the second edition of this ...
41
votes
12
answers
8k
views
Examples for the use of AI and especially LLMs in notable mathematical developments
The purpose of this question is to collect examples where large language models (LLMs) like ChatGPT have led to notable mathematical developments.
The emphasis in this question is on LLMs, but ...
5
votes
1
answer
359
views
Why do we care about residually solvable/nilpotent groups?
I'm interested in a certain property $X$. In the introduction to basically every paper on $X$ there's a paragraph that goes something like:
$X$ is related to residually-solvable groups in this way, ...
- 491
13
votes
1
answer
424
views
Convergence spaces, pseudotopologial spaces, etc.: what's the big picture? when do we encounter them and what are they good for?
I've often wanted to learn more about convergence spaces, but I've found myself lost in a maze of definitions (sometimes with conflicting names across sources) with no intuition about what each one is ...
- 38.7k
1
vote
1
answer
523
views
Why not begin studying sets with Hierarchy Theory?
By $\sf HT^\psi$ I mean the Hierarchy Theory of $\psi$ height. This is a set theory written in mono-sorted first order logic with equality and membership, with the following axioms:
Specification: $\...
- 13.1k
0
votes
0
answers
189
views
Geometric framework coupling probability distributions and dual observables: a presymplectic structure and bracket dynamics
In many physical systems, we distinguish between the state of the system — often described by a probability distribution — and the observables. I explore a geometric framework where probability ...
5
votes
1
answer
344
views
A natural geometric duality between probability distributions and random variables
Motivation
The goal of this work is to develop a unified geometric framework for finite probability distributions and finite random variables, utilizing differential geometry and information geometry. ...
0
votes
0
answers
95
views
Can Singleton augmented Mereology provide a truth argument for set theory?
Let "$ \phi \text { is one-to-one between } \pi, \psi $", stands for meeting both of: $$ \forall x \pi(x) \exists!y \psi(y): \phi(x,y) \\ \forall y \psi(y) \exists!x \pi(x): \phi(x,y) $...
- 13.1k
19
votes
4
answers
2k
views
Elementary consequences of famous technical theorems and/or conjectures
Recently, Will Sawin gave a perfect answer to my question about elementary consequences of Langlands program. Then Timothy Chow asked a similar question about non-abelian class field theory, and ...
- 8,625
2
votes
2
answers
542
views
Questions about some parallel between polynomial and differential equation
Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations ?
Do the relations between ...
- 3,306
2
votes
0
answers
137
views
Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
- 1
8
votes
1
answer
596
views
Reference request: Expository paper on the use of functional analysis in differential and integral equations
Some textbooks on functional analysis do not hint that a major raison d'être of the subject is its use in the study of differential and integral equations. The reader could go all the way through ...
- 12.8k
12
votes
3
answers
3k
views
What governs our "perception?" about the platonic realm of sets?
Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the ...
- 13.1k
-8
votes
2
answers
1k
views
Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...