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

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3
votes
0answers
22 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
29 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 ...
11
votes
0answers
143 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
68 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 ...
1
vote
1answer
52 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 ...
6
votes
1answer
56 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 ...
3
votes
1answer
78 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 ...
4
votes
3answers
127 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 ...
2
votes
0answers
79 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
33 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 ...
5
votes
0answers
64 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 ...
35
votes
2answers
1k 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 ...
6
votes
1answer
184 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: ...
2
votes
2answers
45 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
19 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
27 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? ...
2
votes
1answer
78 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
144 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
69 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 $B$ ...
2
votes
1answer
43 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 ...
6
votes
5answers
508 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 ...
3
votes
1answer
168 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 ...
12
votes
3answers
950 views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
2
votes
1answer
33 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 ...
2
votes
0answers
43 views

Protomodular categories

The axioms for abelian categories are nice and clear. The axioms for protomodular categories - and therefore semi-abelian categories - are beyond me entirely. I'm looking for a breakdown of the ...
1
vote
1answer
97 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
4
votes
1answer
192 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
3
votes
1answer
199 views

Cramer and Riemann Conjecture Implication

Cramer's conjecture gives $$p_{n+1}-p_n= O(\log^2 p_n)$$ while Riemann Hypothesis yields just $$p_{n+1}-p_n= O(\sqrt p_n\log^2 p_n).$$ Does Cramer conjecture on prime gaps imply Riemann Hypothesis ...
-1
votes
1answer
386 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 - ...
17
votes
4answers
4k 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 ...
2
votes
1answer
51 views

Does the concept of “dynamic average” makes any sense?

While making an excel table about how many times an event happens per day I thought that it could be interesting to see what is the growth rate of those events. If in $2$ days the event happens two ...
1
vote
0answers
122 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
7
votes
1answer
357 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
44
votes
14answers
4k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
5
votes
1answer
131 views

Why are infinite sums so much harder to calculate than the associated infinite integral?

It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent ...
2
votes
1answer
106 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
9
votes
1answer
340 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 ...
3
votes
2answers
208 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
1
vote
4answers
208 views

Physical meaning of limit

Does the concept of "limit" have a well-defined physical meaning (like, for example, the derivative)?
1
vote
0answers
112 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 $x$ ...
1
vote
2answers
80 views

Enlightening ideas and methods that change one's appoach to problems, theorems or mathematics as a whole

I would like to collect a "big-list" of ideas and methods from different areas (although I'm particularly interested in elementary number theory, algebra, calculus, linear algebra, geometry, physics, ...
5
votes
1answer
177 views

Problems in elementary number theory and methods from physics

I was wondering if there are intuitive "physical" arguments to solve problems from number theory (elementary number theory in particular, but also advanced topics). To make an example, a proof of ...
0
votes
2answers
137 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
5
votes
0answers
346 views

Classification of Geometry [closed]

I'm asking for big picture in geometry here. I've studied the first three chapters of John Lee's smooth manifolds but still I cannot see the path ahead. My questions are mainly about classification of ...
0
votes
2answers
250 views

What is global differential geometry?

What is the difference between local and global differential geometry? I cannot find their (exact) definitions. There are some other terms in geometry like "rigid" (e.g. that structure is more rigid ...
4
votes
4answers
214 views

Why is differential geometry called differential geometry?

Why is differential geometry called differential geometry? Why it is not called differential and integral geometry? Isn't integration and finding areas as important as differentiation? Is it the case ...
4
votes
1answer
105 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 ...
0
votes
1answer
65 views

Quote objects in concrete categories

I refer to 'Analogy of ideals with Normal subgroups in groups' which was a very enlightening question for me. When I was young I was too avid on abstract algebra and I did too many courses at the same ...
2
votes
0answers
128 views

Must diagrams be commutative?

Given a category C and a function $ \Theta : Mor(\textbf{C}) \times Mor(\textbf{C}) \longrightarrow Mor(\textbf{Rel}) $ and suppose that the relation, with $(r,s)\in\Theta(u,\bar{u})$ and so forth, ...
23
votes
6answers
3k views

Mathematics - The big picture

My knowledge in math sums up to several university level courses and a lot of self-study. Sometimes it feels like Mathematics is a huge subject with a lot of different areas which some of them ...