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

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7
votes
3answers
152 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
3
votes
1answer
85 views

Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
44
votes
13answers
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, ...
4
votes
1answer
144 views

covariant derivative vs. exterior derivative

I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla ...
3
votes
2answers
180 views

What tools are used to show a type of convergence is or is not topologizable?

There are many types of convergence. For example, in measure theory and probability theory, there are many types of convergence of measurable mappings (random variables). in measure theory and ...
11
votes
1answer
179 views

What does it mean for a set to have “structure”?

I understand that a set is like a list of things, except that the order doesn't matter and that you can't have any duplicates in a set. For example: $\{3, 1, 4, 2\}$ is the same set as $\{1, 2, 3, ...
2
votes
1answer
84 views

Vantage point of character theory

I am not sure whether I can frame my question properly, or whether at this point my understandings permit me to comprehend the perspectives of the answers to come, but somehow I find it pretty amazing ...
3
votes
2answers
147 views

How bad is this analogy for logical independence?

It is an amazing and well-known fact that the Continuum Hypothesis is logically independent of Zermelo-Frankel set theory with the Axiom of Choice (ZFC), assuming it is consistent. In a similar vein, ...
9
votes
6answers
16k views

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

I've always enjoyed solving problems in the complex world 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 ...
4
votes
1answer
697 views

Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.

This is kind of a big picture question. I just counted up all the symbols used in normal mathematics and, give or take, there are probably around 150 of them, tops. And that's really stretching ...
1
vote
0answers
79 views

Two questions about generalizing multilinear maps.

A multilinear map from the product $V_1\times\ldots\times V_n$ of vector spaces over the same field $K$ to another vector space $W$ over $K$ is a map $\phi$ such that if we fix vectors $$v_1\in ...
4
votes
2answers
345 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
55
votes
5answers
5k 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 ...
14
votes
4answers
822 views

If adjunction arises everywhere, where is it in the fundamental theorems?

MacLane's slogan "adjunction arises everywhere" is widely known, and adjunction has been identified as a key concept (maybe the key concept?) in category theory, eg, in the books by Goldblatt Topoi, ...
10
votes
3answers
1k views

Detailed diagram with mathematical fields of study

Some time ago, I was searching for a detailed diagram with mathematical fields of study the nearest one I could find is in this file, second page. I want something that shows information like: ...
5
votes
2answers
1k views

Main differences between analytic number theory and algebraic number theory

What are some of the big differences between analytic number theory and algebraic number theory? Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much ...
1
vote
3answers
196 views

A quick question about categoricity in model theory

I just want to see if I am using the term "categoricity" correctly in the following context: (1) I was thinking about why someone might reject a simple resolution to Skolem's Paradox. (2) The ...
4
votes
1answer
293 views

Approach to Learning Co/Homology

I have decided to begin studying co/homology and I'm trying to work out the best approach to doing this. As I understand the situation, any system that satisfies the Eilenberg-Steenrod axioms ...
18
votes
3answers
801 views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...
20
votes
0answers
238 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 ...
2
votes
0answers
109 views

Is the notion of a dual space related to the set of polynomial functions on an affine algebraic variety?

Let $M$ be an affine algebraic variety and consider the ring of polynomial functions on $M$, $\mathcal{O}(M):=\{f: M\to k : f\text{ a polynomial}\}$. If $k$ is algebraically closed we can recover our ...
63
votes
8answers
3k 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 ...
6
votes
3answers
376 views

How exactly do logic and mathematics interact and what happens when we change the logic?

[Note: this question turned out to be pretty huge, so if you think it would be better to split it up into smaller questions, please comment. The questions here are quite conceptually intertwined and ...
51
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
27
votes
3answers
2k views

Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
6
votes
2answers
263 views

How do I investigate the metamathematics of Euclid's proof of infinitude of primes?

Is primeness a predicative property? Earlier this year, I jotted down some thoughts in a paper whether Euclid's proof of infinitude of prime numbers is tautological arguing that prime numbers are ...
3
votes
1answer
301 views

Properties of computable numbers

If we enumerate* all the computable numbers, those for which there exist a turing machine that outputs its digits to arbitrary precision. What is known about the asymptotic density of rationals, ...
25
votes
3answers
993 views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
3
votes
1answer
134 views

Bounding projective spaces

For which $n$ does there exist a (topological, smooth, PL, complex) manifold $M^n$ such that $\partial M = \mathbb{R}\mathbb{P}^m$. Obvously, $m = n -1 $ (at least an in the real case). There are a ...
16
votes
4answers
2k views

Advanced algebraic topology topics overview

Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold: it's a beautiful theory; it gives geometric justification to (or ...
5
votes
1answer
456 views

Complex Analysis and Algebra

There are two results in Complex Analysis that have a counterpart in Algebra: -If we consider the ring of holomorphic functions in an open set $\mathcal H(U)$ with the usual sum and product, every ...
11
votes
16answers
1k views

Landmarks of subjects of mathematics

In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it). For example in ...
5
votes
2answers
515 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
11
votes
2answers
2k views

Categories of mathematics

I am interested in understanding how mathematics is divided into many categories, such as what categories are particular cases of what, what categories do not or have little overlap with what. This is ...
26
votes
5answers
2k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
10
votes
4answers
2k views

What's more general than category theory?

First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then ...