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Questions tagged [big-picture]

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

2
votes
1answer
31 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 ...
1
vote
1answer
85 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 ...
2
votes
0answers
57 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 ...
6
votes
2answers
84 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 ...
1
vote
1answer
79 views

Two questions regarding the convention concerning concrete categories

Definition of Concrete Categories Let $\mathbf{X}$ be a category. A concrete category over $\mathbf{X}$ is a pair $(\mathbf{A}, U)$, where $\mathbf{A}$ is a category and $U : \mathbf{A} \to \...
2
votes
1answer
71 views

Importance of Axiom of Choice (or its weak form) in Real Analysis

By Wikipedia article on Axiom of dependent choice, it is necessary to have it for development of real analysis. The ${\mathsf {DC}}$ says: Axiom (${\mathsf{DC}}$). For any nonempty set $X$ and ...
4
votes
2answers
120 views

In algebraic geometry, what kind of theory can only be described by topos but not a site?

A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books ...
13
votes
5answers
2k views

Intuition Wanted: Why Define Integrals Component-Wise

In our analysis course, we just defined the following: Let $g := (g_1, \ldots, g_n): [a, b] \to \mathbb{R}^n$, where $g_1, \ldots, g_n: [a,b] \to \mathbb{R}$ are integrable. Then we call the ...
34
votes
8answers
3k views

Why are algebraic structures preserved under intersection but not union?

In general, the intersection of subgroups/subrings/subfields/sub(vector)spaces will still be subgroups/subrings/subfields/sub(vector)spaces. However, the union will (generally) not be. Is there a "...
1
vote
0answers
97 views

How does the locally compact group play a role in many branches of math? [closed]

It seems that Fourier analysis/harmonic analysis plays an important role in math, at least in number theory and statistics. It also seems to me that talking about it is talking about locally compact ...
-1
votes
1answer
92 views

What mathematics cannot be reduced to pigeonhole?

Pigeonhole is a fundamental principle without which state of mathematics will be much different. However what examples of good mathematics has not yet been proved and cannot be proved with pigeonhole ...
4
votes
4answers
135 views

A graphical representation of Mathematics as a whole?

Does a graphical representation of mathematics, its fields and subfields, exist ? Meaning, for instance a graph where vertexes are fields of mathematics (e.g convex geometry, Lie algebra, Kahler ...
1
vote
3answers
79 views

What can be learned for number theory from geometrical constructions (and vice versa)?

Even though this question of mine was not so well received at MO I'd like to pick two examples and make a question out of them here. Consider these two pairs of geometrical constructions which yield ...
3
votes
3answers
101 views

How do we get past how **every** outcome is very unlikely?

Edit: This question is about rejecting the null hypothesis. Last month my evil twin and I were at a game show. The rules are as follows: There is a sealed booth with two magic boxes. Box A has a ...
13
votes
1answer
202 views

On the clarification of Manin's remark about Gödel’s incompleteness theorems

In his paper Georg Cantor and his heritage Yuri I. Manin writes (see page 7, 3rd paragraph), Baffling discoveries such as Gödel’s incompleteness of arithmetics lose some of their mystery once one ...
2
votes
0answers
66 views

The shape and group structure of an elliptic curve over $\overline{\mathbf{F}_p}$ and intermediary extensions

Let $p$ be prime, let $q$ be a power of $p$ and let $E/\mathbf{F}_q$ be an elliptic curve defined over the finite field $\mathbf{F}_q$. Let $\overline{\mathbf{F}_q}$ be the algebraic closure of $\...
4
votes
0answers
77 views

Motivation behind Primary Decomposition

I am reading Atiyah's Commutative Algebra chapter on Primary Decomposition. I understand the proofs but I have no intuition as to How did one come up with definition of a primary ideal $q$ as $...
3
votes
1answer
71 views

Why care distribution functions more than random variables?

This may be wrong, but I have often heard some saying " we mainly care about CDFs". Similarly, in textbooks, one sees $X \sim N(0,1)$, without any reference to sample space. But why - and how do we ...
5
votes
1answer
132 views

Mandelbrot and Julia fractals for $z_{n+2} = z_{n+1}^2 + z_n^2 + c$

The Mandelbrot and Julia type fractals are very Well known. But such fractals follow from $$z_n = f(z_{n-1},c)$$ In other words a recursion that only depends on the previous value and a constant. (...
1
vote
0answers
50 views

Does the defintion of chain equivalence have anything to do with liebniz product rule of differentiation?

I have this vague/ big picture question which might be too vague to answer (I don't mind it being deleted in this case). Do chain equivalences have anything to do with Leibniz product rule of ...
2
votes
1answer
148 views

The context & motivation for the Tits alternative in combinatorial group theory

The Details: Definition 1: A class $\mathcal{G}$ of groups satisfies the Tits alternative if for any $G$ in $\mathcal{G}$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup ...
1
vote
1answer
58 views

There is a natrual connection on the tangent bundle?

I come up with a (maybe stupid) question: let $M$ be a smooth manifold, then the exterior differential $d$ is a natural connection on $\Omega^k(M)$, hence by dualizing we get a natural connection on $...
0
votes
1answer
39 views

On multiplicative and additive properties of cyclotomic polynomials

Is there explicit relation between $\Phi_{a+b}(x)$, $\Phi_{ab}(x)$, $\Phi_{a}(x)$ and $\Phi_{b}(x)$ at general coprime or non-coprime $a,b\in\Bbb Z$? If $a,b$ are distinct primes then we have $x^{ab}-...
3
votes
0answers
100 views

What makes a math problem important?

Given that pure mathematics is, by definition, not concerned with applications, how does one decide that one problem is more valuable than another? Is it just a matter of certain topics becoming ...
0
votes
0answers
43 views

Analogies and differences between polynomial evaluation and modulo operations

Given a polynomial $f(x)\in\Bbb Z[x]$, prime $p$ and integer $a$ are there analogies and differences between evaluating $f(x)$ at $x=a$ and computing $a\bmod p$? Both look like some kind of function ...
0
votes
0answers
32 views

Formulas without efficient algorithms?

Suppose the fastest known method to multiply two decimal numbers is an exponential time algorithm. Would it still be beneficial to mathematics having formulas like the following for positive integers? ...
1
vote
0answers
68 views

Why, intuitively, are propositional resolution proofs so long?

I'm trying to gain an intuitive understanding of why propositional resolution proofs tend to be so long. As every essential prime implicant can be produced via resolution, intuitively I would have ...
11
votes
1answer
382 views

Mathematical disciplines with high thresholds

Are there mathematical disciplines that are extremely inaccessible, with very high thresholds even for those who have the necessary prerequisites? Number theory and graph theory, for example, are ...
3
votes
2answers
220 views

Books on the philosophy of geometry

I am looking for recent books ( say published after 2000) on the philosophy of geometry, most books on the philosophy of mathematics seem to ignore or bypass geometry at all or am I just looking with ...
12
votes
2answers
194 views

$\pi$ when not in base 10

Very novice amateur mathematician here. My daughter (8 yo) is a math junkie and is trying to wrap her head around irrational numbers. We were talking about $\pi$, and I rambled on about how folks have ...
3
votes
3answers
595 views

Why introduce the $p$-adic numbers?

My current intuition about the p-adic numbers comes from the following three facts: You can describe $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with the $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ ...
4
votes
0answers
118 views

Mathematics Textbooks that exemplify “Understanding” Mathematics.

I hope this question isn't considered too strange. Gowers argues in "The Two Cultures of Mathematics" That mathematicians can be broadly categorized as those interested in understanding mathematics ...
5
votes
1answer
124 views

Fields with an overlap between logic and algebra?

I was curious if there is any field of study that uses both logic and algebra. To clarify, logic and analysis overlap in areas like descriptive set theory, and there are a lot of logic ideas in areas ...
5
votes
1answer
117 views

Structure preserving maps, and their use for understanding math.

A warning, I've labeled this a soft question, because I'm not quite sure I know what I'm asking. In general, the study of some mathematical structure X (a function, a shape, whatever) becomes more ...
5
votes
2answers
756 views

Gaining a Deep Understanding of Theorems

As one might infer from my user name, I am currently pursuing an autodidactic path in pure mathematics while awaiting the real math classes next year at university. I recently bought Dummit and Foote'...
1
vote
0answers
45 views

A simple clarification on joint probability inequality.

Suppose we have $N$ balls and we have an experiment that is set up in the following way. Your goal is to pick a particular ball that is fixed. At each trial you pick $N^{1-c_1}$ balls and replace. We ...
4
votes
2answers
813 views

Why continuous function can be considered as “topological homomorphism”?

In Munkres's book on topology, the notion of homeomorphism is stated to be analogous to the notion of isomorphism in context of modern algebra. I was wondering what will be the analogous concept of ...
6
votes
0answers
290 views

What happens in your imagination when you do Mathematics? [closed]

I know that this is more a soft question and that the answer depends on the answerer but I think that for every discipline it is not only essential to discuss the discipline itself but also its ...
0
votes
1answer
49 views

Is there a relation between Elliptic Curves and Frobenius Numbers?

A CS professor yesterday asked me this query. I think there is no direct relation if any. The frobenius number is the largest number that cannot be represented by $au+bv$ where $gcd(a,b)=1$ holds and ...
3
votes
0answers
104 views

What does the probabilistic approach to PDE give us that we can't obtain otherwise?

My apologies for the naive question. As someone from a PDE background, I am just wondering what core things can be said using probabilistic methods about certain PDE problems that couldn't be said ...
8
votes
1answer
663 views

Why should I learn modern category theory if my interest mainly is structured sets?

A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately ...
2
votes
0answers
158 views

Are all theorems usable? [closed]

The (revised) question to answer: Can anyone give an example of a serious proof using this funny (revised) theorem? For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ ...
4
votes
1answer
47 views

Problems reducible to polynomial root finding

In the past, I have noticed several problems for which the solution goes something like this: Reduce the problem to a polynomial equation Find the roots of the polynomial Interpret appropriately in ...
0
votes
1answer
302 views

Theoretically, can mathematical equations be used to graph any picture imaginable? [duplicate]

I know that quadratic formulas make parabolas, linear makes straight lines and sin, cos and tan make cool curves but after messing around with WolframAlpha and online equation graphing tools I can ...
0
votes
1answer
536 views

Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
0
votes
1answer
129 views

Significance of Rank of Matrix

Why we determine the Rank of Matrix ? Instead of this just asking for my info : What is the easiest way to find Rank of Matrix ?
4
votes
1answer
482 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
7
votes
2answers
399 views

Algebraic topology & Riemannian geometry project idea?

I'm taking a first course on Riemannian geometry this semester. For a final project, I would like to do something that involves algebraic topology. However, the only results I know in algebraic ...
8
votes
1answer
211 views

Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
4
votes
0answers
329 views

Synthetic differential geometry and algebraic geometry

I am reading here and there about basic synthetic differential geometry. One of the central ideas seems to be that it should be developed in a suitable topos, hence, in particular, a cartesian closed ...