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Questions tagged [big-picture]

Questions to get the "big picture" about a subject.

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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 ...
Chris Culter's user avatar
2 votes
0 answers
48 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 ...
zeta space's user avatar
4 votes
0 answers
121 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 ...
Thomas Manopulo's user avatar
1 vote
1 answer
75 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) ...
zeta space's user avatar
9 votes
0 answers
226 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 ...
D.R.'s user avatar
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0 votes
0 answers
160 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 ...
D.R.'s user avatar
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31 votes
3 answers
2k 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 ...
Thomas Manopulo's user avatar
0 votes
1 answer
240 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 ...
Carla_'s user avatar
  • 612
3 votes
0 answers
486 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 ...
Mathematics_Beginner's user avatar
2 votes
1 answer
135 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 ? ...
Evgeny Kuznetsov's user avatar
1 vote
1 answer
237 views

Ruler and compass cannot trisect an angle, proof without field theory

It is well known that ruler and compass cannot trisect an angle. The standard proof of it uses field theory. It is just a contradiction of tower law. However, I wonder if we can prove this only using ...
Ja_1941's user avatar
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4 votes
0 answers
172 views

Traditional and modern approaches to class field theory

I read in Gerald Janusz's book Algebraic Number Fields where he says there are two approaches to class field theory. The traditional approach, as in his book, uses L-series, Dirichlet density and ...
Ja_1941's user avatar
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1 vote
0 answers
143 views

A useful way of characterizing primes?

It's proved that if $P(n)$ is the sum of all different prime factors of $n$, then for all odd primes $p$ there is an inequality $$p\ge P(p^2-1)$$ An inequality holding for odd primes Therefore there ...
Lehs's user avatar
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1 vote
0 answers
147 views

Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?

If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
Anixx's user avatar
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5 votes
0 answers
122 views

The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)

$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
D.R.'s user avatar
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8 votes
0 answers
135 views

Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
D.R.'s user avatar
  • 8,935
4 votes
2 answers
117 views

Build a "rich" first-order logic within a given category

I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal. For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
rfloc's user avatar
  • 1,171
4 votes
0 answers
109 views

Purely algebraic context in which homology arises naturally?

I know that homology is most commonly introduced from a topological context (and/or Stokes theorem related context), but in say a homological algebra course/text, you are just given the definitions (...
D.R.'s user avatar
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1 vote
2 answers
77 views

Crux of linear algebra

What is the connection between solving systems of linear equations and vector spaces? And what do matrices have to do with all of that? I know this is a crux of linear algebra, and therefore, not so ...
Someone who learns's user avatar
1 vote
1 answer
210 views

What questions does Topology answer?

I was reading the great Thurston's article on Mathematical Education and one thing he points out is: People appreciate and catch on to a mathematical theory much better after they have first grappled ...
Douglas's user avatar
  • 414
7 votes
3 answers
2k views

The main idea behind Big O notation

Well, when we use Big O notation we never know even approximate number of steps of a given algorithm, right? For example, if we have $O(n)$ algorithm, then we don't know how fast this algorithm itself ...
mathgeek's user avatar
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3 votes
3 answers
2k views

Why to calculate "Big O" if we can just calculate number of steps?

I never really understood how we use Big O to estimate algorithm efficiency. I know that $f\left(n\right) = O\left(g\left(n\right)\right)$ as $n \to +\infty$ if there exists some constant $C > 0$, ...
mathgeek's user avatar
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14 votes
4 answers
1k views

Proof of Stewart's Theorem using elementary geometry

I was reading Stewart's Theorem which states that Given a triangle with side lengths $a$, $b$, $c$ and a cevian of length $d$ which divides $a$ into two segments $m$ and $n$ as shown in the figure ...
Sayan Dutta's user avatar
  • 9,592
2 votes
0 answers
52 views

Definitions and concepts that are different for various types of spaces - sources for seeing "the bigger picture"?

I have quite unusual question. Almost everytime I am studying math, I lowkey want to "categorize" the stuff and look at it as a bigger picture. Do you know about any resource (diagram, ...
Tereza Tizkova's user avatar
6 votes
1 answer
569 views

Trying to understand the idea behind $T$-conductor

I was reading Linear Algebra by Hoffman Kunze and I found a strange definition Let $W$ be an invariant subspace for $T\in \mathcal L(V)$ and let $\alpha$ be a vector in $V$. The $T$-conductor of $\...
Sayan Dutta's user avatar
  • 9,592
3 votes
1 answer
171 views

Functors making functions natural transformations and vice-versa.

I apologize in advance if this is naive. In this answer Conjugation in a groupoid it is said that given a groupoid $\mathcal G$, and an arbitrary function $\mu:\mathcal G_0\to \bigcup_{x\in \mathcal ...
MphLee's user avatar
  • 2,512
1 vote
2 answers
88 views

What's the typical role of the constant $e^{-\gamma}$?

I often encounter this constant in my research, but I wonder what typical roles does it play in other areas of mathematics? Wikipedia mentions probability theory but nothing exact. Also, I am ...
Anixx's user avatar
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0 votes
0 answers
141 views

Connecting the various interpretations of normal subgroup

What is a normal subgroup? I have heard all of the following: A subgroup $N$ of $G$ is normal if $gNg^{-1} = N$ for all $g \in G$. Or equivalently, a subgroup is normal if it is invariant under ...
twosigma's user avatar
  • 3,272
46 votes
8 answers
5k views

What do cones have to do with quadratics? Why is $2$ special?

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations ...
D.R.'s user avatar
  • 8,935
0 votes
0 answers
86 views

Differential analog to the Lebesgue spaces $\mathcal{L}^p$

I am looking for a differential analogue to the Lebesgue spaces $\mathcal{L}^p$ in the following sense: The integral behaves badly for functions with too much mass i.e. such whose tails decay too slow ...
G. Chiusole's user avatar
  • 5,456
0 votes
0 answers
90 views

What does the loop integral of geodesic curvature intuitively mean in the absence of a surface normal?

The geodesic curvature $k_g$ can be interpreted as the rate of rotation of the tangent vector about the surface normal, as discussed in this article. The article goes on to say that the integral $\int ...
TribalChief's user avatar
0 votes
0 answers
192 views

What are the historical trends of contemporary mathematics?

Question: What would future historians see as the general directions of mathematics in the period of 1970s -- 2020s? What I look for in an answer: a pointer to a review paper on this topic would be ...
MaudPieTheRocktorate's user avatar
1 vote
0 answers
79 views

What is the motivation for studying this subgroup of $SL_n(\Bbb R)?$

Consider a diagonal matrix $A$ with entries $e^{s_1},e^{s_2},\cdot\cdot\cdot$ on the diagonal s.t. $\sum_{i \ge1} s_i=0.$ From what I understand this is a subgroup of $SL_n(\Bbb R).$ This is because ...
zeta space's user avatar
4 votes
1 answer
1k views

Hyperbolic curve and hyperbola?

Def: A hyperbolic curve is an algebraic curve obtained by removing $r$ points from a smooth, proper curve of genus $g,$ where $g$ and $r$ are nonnegative integers such that $2g−2+r > 0.$ How ...
zeta space's user avatar
4 votes
0 answers
135 views

Does anyone know what this diagram could be about?

Does anyone know what this diagram could be about? I found it about a year ago on some blog and I tried to relocate the source of the picture but was unable to. My best guess is that maybe $R(x)$ and ...
zeta space's user avatar
2 votes
1 answer
210 views

What is the big picture behind the quotient of a ring by its maximal ideal being a field?

A ring $(X, +, \times, 0, 1)$ is an algebraic structure such that $(X, +, 0)$ is a commutative group, $(X, \times, 1)$ is a monoid, and $\times$ distributes over $+$. A sub-ring is a subset $Y \...
bzm3r's user avatar
  • 2,672
1 vote
3 answers
91 views

Is a specific choice of ultrafilter necessary in order to get "concrete results" in nonstandard analysis?

Suppose we have sequences of real numbers which are indexed by the natural numbers. We can then define an ultrafilter $\mathcal{U} \subset 2^{\mathbb{N}}$ (where $2^{\mathbb{N}}$ is the powerset of ...
bzm3r's user avatar
  • 2,672
2 votes
1 answer
50 views

On two conceptions of measurability

This is a very broad question about two conceptions of a measurable function. Let $(X, \mathcal X)$ be a measurable space, and let $f$ be a real-valued function of $X$. Throughout the post, I assume ...
aduh's user avatar
  • 8,750
3 votes
1 answer
112 views

On the "duality" between formation of quotient structures and formation of substructures as stated in Joy of Cats

In the book Joy of Cats, the authors write (page 118), One of the nice insights that can be gleaned from category theory is that the formation of quotient structures (such as groups of cosets and ...
user avatar
2 votes
2 answers
438 views

Intuition behind the definition of initial morphism and embedding as stated in Joy of Cats

In Definition 8.6 of Joy of Cats the authors write (I have changed the wording to avoid explaining the conventions that the authors introduce earlier which would lengthen the post unnecessarily), ...
user avatar
2 votes
0 answers
85 views

A map (big picture) of the relationship of all areas in mathematics (including probability, stat, and information theory, etc)

Update 10/15: What I could remember is the picture was under a section named "interesting stuff" (something like this), with a link toward the picture (if I remember it correctly). Also, the professor ...
J.Z.'s user avatar
  • 123
4 votes
1 answer
221 views

Classification of monoidal closed structures

The motivations for this question are somewhat vague probably. I am trying to have a better understanding of monoidal closed structures. In order to do so I would like to collect unusual examples and ...
Ivan Di Liberti's user avatar
1 vote
1 answer
62 views

Alternative Proof to "Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage"

Given: A ten year comparison between the United States and the Soviet Union in terms of crop yields per acre revealed that when only planted acreage is compared, Soviet yields were equal to 68 ...
user avatar
1 vote
1 answer
133 views

General rule of thumb for tensor products.

When would I want to use a tensor product? For example, say I am trying to derive "something" but, to get to the right derivation, I should use the tensor product. The question comes from quantum ...
BonInSossusvlei's user avatar
15 votes
4 answers
1k views

What does a topology do, and what makes a particular topology the 'right' one?

From Wikipedia: The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies. [Aside: ...
R. Burton's user avatar
  • 5,040
9 votes
0 answers
622 views

A definition of differentiable functions for arbitrary topological spaces

Background It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I came ...
user avatar
2 votes
1 answer
217 views

Looking for intriguing applications of martingales

I've been studying martingales before I start graduate school in statistics, and I'd like to ask, what are the most intriguing or surprising applications of martingales that you've come across, either ...
CLL's user avatar
  • 177
1 vote
1 answer
127 views

Is there a notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$?

It is accepted belief $\mbox{countable }\infty$ is not $\mbox{uncountable }\infty$. Is there notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$ (latter ...
Turbo's user avatar
  • 6,245
2 votes
0 answers
87 views

Do we have to redo category theory when learning about enriched categories? [closed]

I'd like to know whether there exists a common language that encompasses both categories and enriched categories, so that results pertaining to either may be proven in a uniform way. I'd prefer it if ...
Cloudscape's user avatar
  • 5,146
11 votes
2 answers
2k views

Why is Catalan's constant $G$ important?

I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related ...
Hobbyist's user avatar
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