# Questions tagged [big-picture]

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 "...
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### 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 ...
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### 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 ...
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### 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 ...
107 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 ...
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### 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 ...
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### 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 ...
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### 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. (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\pi$ 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 ...
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### 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 $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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'...
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 ...