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5
votes
1answer
82 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
136 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 ...
37
votes
5answers
1k 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
324 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
70 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
155 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
102 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 ...
7
votes
1answer
70 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
111 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
102 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
28 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
232 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
141 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
84 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
76 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 ...
5
votes
6answers
365 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
166 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 ...
3
votes
0answers
112 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
86 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
216 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
253 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
83 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
122 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 ...
47
votes
12answers
3k 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
271 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
144 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
83 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 ...
43
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
123 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 ...
4
votes
0answers
272 views

Why should the taxpayer pay the mathematician? [closed]

I'm sorry if the title sounds overly confrontational. There will be some confusion and a bit of anger throughout the post, but I hope it will be clear what I'm asking by the end of it. I've been ...
3
votes
2answers
154 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
173 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
81 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, ...
10
votes
6answers
12k 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 ...
2
votes
1answer
581 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
292 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 ...
47
votes
5answers
4k 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
742 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, ...
9
votes
3answers
948 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: ...
4
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
180 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
283 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 ...
17
votes
2answers
653 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
208 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
104 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 ...
58
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 ...