Questions tagged [big-picture]
Questions to get the "big picture" about a subject.
216 questions
-4 votes
1 answer
160 views
What is the Big Picture of Linear Algebra? [closed]
I am currently learning about linear algebra. To better understand linear algebra and create a network of mathematical concepts, I would like to know what the Big Picture of Linear Algebra is. Which ...
- 41
0 votes
0 answers
26 views
Graph of Optimization Problem Transformations
In this lecture, Professor Stephen Boyd begins to draw a commutativity diagram to raise the question of whether transforming a program with strong duality into an equivalent problem and computing its ...
- 2,184
5 votes
1 answer
293 views
When are $V_α$ and $V_β$ elementarily equivalent?
As someone unfamiliar with set theory, I was surprised not to find any information online about the following question, which seems natural to me. I might be missing the right search keywords, and ...
- 1,232
0 votes
0 answers
86 views
What are the implications of falsity of weak Cramer's conjecture?
Let $p_n$ be the $n$th prime and $\Delta_n=p_{n+1}-p_n$ Cramer's conjecture gives $$\Delta_n= O(\log^2 p_n)$$ while Riemann hypothesis implies $$\Delta_n= O(\sqrt{p_n}\log^2 p_n).$$ Let weak Cramer's ...
- 6,341
3 votes
0 answers
92 views
Abstract Algebra Resources Focusing on Connections to Other Mathematical Branches
I wanna brush up on my algebra skills by connecting the standard definitions/theorems with other branches of mathematics and I would like to know some book, lecture notes or other references with that ...
- 518
4 votes
1 answer
143 views
Why are rotation numbers not homomorphic?
If $f,g$ are degree-1 monotone maps of the circle, why do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$? I mean, you might say that we have no right to expect an equality. After all, it's not ...
- 27.7k
2 votes
0 answers
58 views
Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$
Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$ Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
- 1,199
4 votes
0 answers
167 views
The number $2$ in cohomology theories
I've started feeling this rather curious mystique coming from an unaddressed - at least in my experience - excessive presence of the number $2$ in a few different areas of maths. My curiosity really ...
- 1,548
1 vote
1 answer
119 views
Why does the foliation $\mathcal{F}$ of this Lorentzian manifold also solve the backwards heat equation?
Consider a linear parabolic partial differential equation: $$t \partial_{tt}\varphi_t(x)=\pm x\partial_x \varphi_t(x)$$ which (essentially) takes the form of the backwards heat equation (minus sign) ...
- 1,199
10 votes
0 answers
408 views
Why is studying centralizers the/a key to classifying finite groups?
In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says One problem, as least with the current methods of ...
- 11.2k
0 votes
0 answers
237 views
Relations between the three different descriptions of 2nd cohomology group in Group Cohomology
I am coming at the 2nd cohomology group in Group Cohomology from the perspective of the Group Extension Problem (or rather the group central extension problem, which perhaps more closely related to ...
- 11.2k
33 votes
3 answers
3k views
Why are $p$-adic numbers ubiquitous in modern number theory?
I'm currently at a stage where I think I'm quite comfortable with the appearance of local non-archimedean fields in the maths I encounter, having seen a fair bit of technology built upon their ...
- 1,548
0 votes
1 answer
328 views
What is the importance of the spectral theorem? (not the importance of diagonalization)
(In this question, $(*)$ means normal when working over $\mathbb C$ and means self-adjoint when working over $\mathbb R$.) This question is related but despite the same title what that question ...
- 2,206
4 votes
0 answers
1k views
Layer Cake Representation Intuition
Let $(X, \mathcal{F}, \mu)$ be a measure space. If $f: X \to [0, +\infty)$ is non-negative and measurable, then $$ \int_X f(x) d\mu(x) =\int_0^\infty \mu(\{ x \in X: f(x)\geq t\})dt $$ It is not very ...
- 1,296
2 votes
1 answer
171 views
Theorems about finite sets the proof of which require the notion of infinite set
I believe that there should exist theorems about finite sets which are not provable without the notion of infinite sets. I am curious if I am right. What are the examples of such theorems if they ? ...
- 1,081