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

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16
votes
4answers
6k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
1
vote
0answers
169 views

Why would I learn modern category theory if my interest mainly is structured sets, what would I have to gain? [closed]

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 ...
5
votes
2answers
136 views

Loop spaces have the homotopy type of a topological groups

Every based loop space has the homotopy type of a topological group. I would like to understand this fact, and this is what this question is about : why is it true, and how does one prove it? I ...
2
votes
0answers
141 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
35 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 ...
50
votes
14answers
3k views

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
9
votes
1answer
82 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 ...
0
votes
1answer
61 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
30 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
42 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 ?
27
votes
1answer
379 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
26
votes
9answers
42k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
7
votes
6answers
752 views

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
7
votes
2answers
739 views

What is “Bourbaki's style in mathematics”?

I know Nicolas Bourbaki "is the pseudonym of a group of (mainly) French mathematicians who publish an authoritative account of contemporary mathematics." But what characterizes "Bourbaki's style in ...
86
votes
7answers
9k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
3
votes
1answer
52 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
87 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 ...
76
votes
9answers
4k views

Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
2
votes
0answers
87 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 ...
7
votes
8answers
1k views

Fundamental Theorem of Trigonometry [closed]

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
3
votes
1answer
37 views

Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
5
votes
2answers
352 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set in analogy with pointed sets. Given two spotted sets, then a morphism $\...
7
votes
1answer
2k views

Eilenberg–Zilber as abstract nonsense - why is it important?

The Eilenberg–Zilber theorem in singular homology, relating the monoidal structure of the category of chain complexes with the chain complex of the cartesian product of the underlying spaces, is used ...
2
votes
1answer
40 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in $\mathbb{R}^{2n}...
3
votes
0answers
35 views

Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
2
votes
0answers
36 views

A question on Axiom XI of Veblen's paper on the axioms of geometry

Recently I have started reading Oswald Veblen's A System of Axioms for Geometry. There it is written that (see page 346), Axiom XI. If there exists an infinitude of points, there exists a certain ...
12
votes
0answers
188 views

How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
1
vote
2answers
73 views

Extracting an infinite subsequence

Suppose that $\{a_i\}_{i\in\Bbb N}$ is a sequence of real numbers such that for any $i\in\Bbb N$, there exists $j\in\Bbb N$ with $j>i$ and $a_j>a_i$. How to prove that $\{a_i\}$ contains an ...
7
votes
1answer
356 views

Recommendations for an “illuminating” (explained in the post) group theory/abstract algebra resource?

I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: http:...
2
votes
1answer
64 views

Naive categorical question about prime numbers, primes, and irreducibles

My question is about the "right" way to think of prime numbers/elements. Looking at primes in $\mathbb Z$, there are two ways of characterizing them: $p$ is prime iff its only divisors are $\pm 1,\...
6
votes
1answer
67 views

Are all instances of torsion special cases of the same concept?

The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that $2\...
5
votes
3answers
147 views

Is it possible for extremely ingenious but elementary proofs for famous problems to exist?

As Erdős put it, "Mathematics is not ready for such problems." when faced with the great conjecture of Collatz. So is it impossible altogether for simple but ingenious proofs for famous problems ...
4
votes
1answer
101 views

Geometric justification for the prime spectrum and “generic points”

I realize there have been plently of discussions about this, but most of them are over my head and I never understand the geometric intuition behind them. I'm trying to make a big list of "...
7
votes
0answers
87 views

Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...
1
vote
0answers
84 views

Homology/cohomology for the uninitiated [closed]

I have heard of (co)homology occurring in many mathematical contexts and I vaguely suspect that it non-trivially relates different subjects. Also that it somehow relates to category/topos theory, ...
4
votes
1answer
56 views

Geometric intuition for left/right exactness

Sheaf cohomology measures the obstruction of the global section functor from being exact. Since it's left exact, it is exact iff it preserves epis. In particular, $H^1$ measure the failure to be ...
37
votes
2answers
2k views

How and why does Grothendieck's work provide tools to attack problems in number theory?

This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing. I have always been fascinated with Grothendieck and the way he did mathematics. I've ...
9
votes
2answers
892 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
2
votes
2answers
54 views

Understanding group homomorphisms and isomorphisms conceptually

This is a question similar to: Understanding the Laplace operator conceptually Homomorphisms and isomorphisms are easy to define: 1) given two groups $(G, *)$ and $(H, \cdot)$, a homomorphism is a ...
0
votes
1answer
21 views

Directional Derivatives With Respect to Negative Vectors

I understand this is probably a silly question but I'm with it struggling nonetheless. Consider the directional derivative of $f(x)=x^2$ at $x=1$ with respect to $u=-1$. I can see that this is equal ...
1
vote
0answers
34 views

GRH versus RH - Big picture

What is the relation between generalized Riemann Hypothesis and Riemann Hypothesis? Does proving one have implication the other? Are there results which implies failure of one to failure of other? ...
26
votes
4answers
809 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
2
votes
1answer
117 views

A question on the generalization of Cartesian Product

In Halmos’s Book, it is written that, The notation of families is the one normally used in generalizing the concept of Cartesian product. The Cartesian product of two sets $X$ and $Y$ was defined ...
2
votes
1answer
180 views

A map $h:S^1\to X$ Induces a Trivial Homomorphism of Fundamental Groups Iff it is Nullhomotopic.

I recently started reading Algebraic Topology from Part II of Munkres' book Topology(Second Edition). A part of Lemma 55.3 in the book proves the following: Let $h:S^1\to X$ be a continuous ...
4
votes
1answer
87 views

Boolean algebras, Stone theorem and being isomorphic to a field of sets

I'm a little bit confused about duality between boolean algebras and topological spaces or sets. I know the following theorem (which is due to Stone, as far as I know): Every boolean algebra $B$ ...
1
vote
0answers
115 views

An adequate difference between $\forall x\in A:P(x)$ and $(\forall x)(x\in A\rightarrow P(x))$?

Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ ...
2
votes
1answer
44 views

Interpretation of alternative group structures on a given group

Let $(G,e,\circ) $ be a group with $e$ the identity element and $a \in G$ and $\circ$ the group operation. Then we can form a new group $(G_a,a,\circ_a)$ with the same underlying set as $G$ and $x \...
10
votes
1answer
426 views

What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
-1
votes
1answer
516 views

I need help organising these books by topic [closed]

Okay, this should be a quick and easy question for those of you who've studied calculus. I have a list of books that I want to order by topic, the books are as follows: Michael Spivak - ...
3
votes
1answer
212 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to "...