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

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3
votes
1answer
67 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 ...
39
votes
13answers
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 ...
4
votes
0answers
72 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 ...
0
votes
0answers
40 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.
8
votes
0answers
132 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 ...
2
votes
2answers
47 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
19 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
5 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
3answers
41 views

Physical meaning of limit

Does the concept of "limit" have a well-defined physical meaning (like, for example, the derivative)?
0
votes
0answers
95 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
66 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
156 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
91 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 ...
4
votes
0answers
110 views

Classification of Geometry

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
73 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
167 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
18 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 ...
3
votes
1answer
31 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
59 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
105 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, ...
22
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
90 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
148 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 ...
38
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 ...
18
votes
3answers
359 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
75 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
167 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. ...
2
votes
2answers
120 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
79 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
122 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 ...
3
votes
0answers
117 views

Unifying concepts in mathematics [closed]

Background Unfortunately my background's in engineering, so we've only been taught bits and pieces of math needed to be fluent in the science, but as I've started studying abstract algebra and real ...
1
vote
0answers
31 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 ...
7
votes
1answer
321 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 ...
4
votes
1answer
161 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
86 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
86 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
550 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 ...
0
votes
3answers
174 views

Why are group theory and ring theory a part of abstract algebra?

I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What ...
4
votes
0answers
124 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
3
votes
0answers
87 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
10
votes
2answers
252 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
4
votes
2answers
327 views

What is elliptic bootstrapping?

While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with ...
0
votes
1answer
84 views

Understanding a diagram on Convolution

Could someone please explain what is happening at the "f*g" row and below? The image is located here as linked from the Wikipedia page. I want to teach myself about Fourier Transforms / Series, and ...
6
votes
1answer
137 views

Are there treelike representations of axioms, theorems, lemmas and corollaries?

I was watching Bill Shillito Lectures on Higher Mathematics, in the second episode he says the basic stuff about axioms and theorems - that axioms are unproved statements in which we build theorems ...
48
votes
12answers
4k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
2
votes
1answer
56 views

Graphic interpretation of path fibration.

Let $S^2$ the unit sphere. We can consider the associated path fibration $$ \Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a ...
14
votes
1answer
302 views

What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: ...
7
votes
3answers
150 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 ...