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

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2
votes
1answer
33 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
278 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
85 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 ...
11
votes
3answers
751 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 ...
0
votes
0answers
18 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
37 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
90 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
140 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
136 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
249 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
2k 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
41 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
106 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 ...
6
votes
1answer
223 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
3k 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
115 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
74 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
232 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
105 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 ...
0
votes
0answers
25 views

Canonical Structure to figure out proof strategy

Even though the integers form a Euclidean domain, most results about can be derived from the weaker fact that it is a PID (I do realize that establishing it forms a PID uses the fact that it is an ED, ...
0
votes
0answers
15 views

Power spectral density and measure in continuous time

Let $X_t$ be a continuous-time wide-sense-stationary stochastic process. The name of the power spectral density (defined as the Fourier transform of the auto-correlation function) suggests it can be ...
1
vote
4answers
92 views

Physical meaning of limit

Does the concept of "limit" have a well-defined physical meaning (like, for example, the derivative)?
1
vote
0answers
110 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
70 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
2answers
165 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
107 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
183 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
133 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
185 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 ...
0
votes
0answers
23 views

Morphisms of $\mathsf{Meas}$ and Dynamical Systems

The morphisms of a category $\mathsf{Meas}$ whose objects are measure spaces are defined to be equivalence classes of a.e-equal measurable maps that pull back null sets to null sets. Why is pulling ...
4
votes
1answer
75 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
61 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
116 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
2k 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 ...
5
votes
1answer
96 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
8
votes
1answer
215 views

How did Kurt Gödel's Incompleteness Theorem affect the mathematical world?

Hi I am looking not to understand the Incompleteness Theorem, but to find out more about how and what this has effected the mathematics world. I am in high school, in Honors Algebra II, and I am ...
40
votes
5answers
2k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
22
votes
4answers
535 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 ...
3
votes
2answers
76 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
0
votes
4answers
190 views

Is mathematics a science? [duplicate]

Is mathematics a science? I have long considered this to be open to debate, but my professor said that he once heard the quote, ...
1
vote
0answers
17 views

When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
3
votes
2answers
278 views

What is the importance of the spectral theorem?

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...
8
votes
1answer
87 views

Non-polynomial representations of $GL_n$

Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the ...
5
votes
2answers
134 views

Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$?

In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic ...
1
vote
0answers
46 views

Characterisations of the RSK correspondence

I know of the following three definitions of the RSK correspondence: (i) Row insertion (or more generally, plactic insertion) (ii) Viennot's construction (iii) Fomin's growth diagrams However, all ...
8
votes
1answer
474 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 ...
6
votes
2answers
345 views

Is there a field of 'real analytic geometry'?

I am wondering whether there is a field of 'real analytic geometry', and if not, why not? There are branches of geometry corresponding to increasingly large sets of functions: polynomial (algebraic ...
3
votes
0answers
93 views

What do we learn from Mathematics? [closed]

I have been doing mathematics for some time now. I am currently a UG in Mathematics and Computer Science. I like doing mathematics but as I progress, I notice that I don't remember certain theorems at ...
1
vote
1answer
114 views

Connecting Coordinate Geometry and Plane Geometry

What is it that allows us to take theorems proven in Euclidean geometry (i.e. with Euclid's five postulates or Hilbert's Axioms) and then apply them outside of Euclidean geometry. For example in ...
6
votes
7answers
793 views

Fundamental Theorem of Trigonometry

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 ...