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103

For four main reasons: If the famous conjecture is proven true, the demonstrated results are proven true too. If the reasoning is correct but demonstrated results are proven false, the famous conjecture is proven false too. Others may be able to prove further results based on the demonstrated results which may themselves be proven false, thus again proving ...


80

Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science contains models where equations are approximated by linear equations (using Taylor expansion arguments) and solving for the system helps the theory develop. Beginning to make a list wouldn't even be relevant ...


39

Before I eat a sandwich it's a good idea to make sure it exists, right? Well, it obviously exists if I can see it, so I may not be aware of the need to establish existence, but I had established existence. In other situations this becomes less silly, particularly in cases where the existence of whatever it is you wish to study or to use is not evident. ...


36

Having studied Engineering, I can tell you that Linear Algebra is fundamental and an extremely powerful tool in every single discipline of Engineering. If you are reading this and considering learning linear algebra then I will first issue you with a warning: Linear algebra is mighty stuff. You should be both manically excited and scared by the awesome ...


35

The results would not be invalidated but would be rendered vacuous, i.e. true but no longer informative. A result says If the Riemann hypothesis is true, then blah blah blah mumbo jumbo. If the Riemann hypothesis ultimately is seen to be false, then it is still true that if the Riemann hypothesis is true, then blah blah blah mumbo jumbo. "Are all cell ...


28

One possible reason for assuming a conjecture and generating results is if you don't believe the conjecture and hope to eventually shoot it down. A great historical example of this is the parallel postulate. This states: Given a line and a point not on that line, there is exactly one line passing through this point which is also parallel to the given ...


28

As far as I know, historically smooth manifolds were the first manifolds studied by people like Poincare, Riemann, up to Whitney. There were a few major events that caused people to take things like topological and PL manifolds seriously, but originally people were not motivated to study these kinds of objects. Here are some of the big events/ideas that ...


25

The "product" of vector spaces is more properly thought of as a direct sum, since the dimension increases linearly according to addition, $\dim(V\oplus W)=\dim(V)+\dim(W)$. Tensor products are much bigger in size than sums, since we have $\dim(V\otimes W)=\dim(V)\times\dim(W)$. In fact, in analogy to elementary arithmetic, we have distributivity $(A\oplus ...


25

I will try to give a rather elementary motivation which I believe to be close to how this identity was actually discovered. Let us consider the sine function $f(z)=\sin z$. As a function of complex argument, it has one period: $f(z+2\pi)=f(z)$. Further, it is holomorphic in the whole complex plane with only simple zeroes given by $\pi\mathbb{Z}$. It ...


23

Motivation can be broken up into essentially two parts: (i) Stable homotopy groups of things appear in nature, and (ii) the stable category is technically convenient. I'll try to give some indication of these below, but none of the bullet points do the topic justice... There's lots of love to be had here. Stable Homotopy Groups Appear in Nature Cobordism. ...


23

While you are deprived of Radin here is a short paper that ties together tilings, quasicrystals and diffraction, and Radin's review of another book on these subjects. Below I interpreted your requirements loosely listing books with a unifying theme that develop ideas organically and combine approaches from different areas of mathematics and applications. ...


22

The definition of modularity looks more natural to me if I think of it as follows (rather than as a modified associativity or a weakened distributivity). Given any element $a$ of a lattice $L$, there is a rather obvious way to map any element $x\in L$ to a "nearest" element $\geq a$, namely send $x$ to $a\lor x$. Think of this map as "projecting " elements ...


22

The full appreciation of Jacobi's triple product identity can not be done without some understanding of the elliptic functions. However, it is possible to develop some parts of the theory of elliptic functions without any complex analysis. Anyways back to the triple product identity, it says that $$ \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = \prod_{n = ...


20

This question already has a number of nice answers; I want to emphasize the breadth of this topic. Graphs can be represented by matrices - adjacency matrices and various flavours of Laplacian matrices. This almost immediately raises the question as to what are the connections between the spectra of these matrices and the properties of the graphs. Let's call ...


20

I don't think there is just one motivation. I will mention three. First, the Jacobi identity says precisely that the bracket is a derivation with respect to itself, where a derivation of an algebra is a map $d$ with $d(a\cdot b)=d(a)\cdot b+a \cdot d(b)$. Thus, writing $\mathrm{ad}(a)$ for the map $b \mapsto [a,b]$, the Jacobi identity may be rewritten ...


20

I think there is more to be found out about this subject. There are many different ways to define fractional derivatives and integrals. I do not know if these come from any deep, fundamental facts, but certainly as a generalization of various formulas. Another way to think about the subject is a list of applications or tricks involving a certain integral ...


20

TL;DR version Linear algebra is your ticket to multidimensional space. Study it if you are into economics, computer graphics, physics, chemistry, statistics or anything quantitative (in today's world, that's everything). Meaning of "Linear" and why it is "Easy" Since you are asking the question, perhaps you would benefit from a discussion of what ...


20

It is true, when you only work on measure theory, or algebraic number theory, or classical analysis, you are unlikely to run into anything larger than $\frak c$. But if you start working in arbitrary fields, and arbitrary modules, or arbitrary rings. Not just finitely generated, or countably generated. Then you need to have a better understanding of how ...


19

You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number. If the real numbers didn't have a number whose square is 2, that would be a rather serious defect; it would mean that the real numbers cannot be used as the setting for the kinds of mathematics where one wants to take a square root of $2$. Basically, things cut both ...


19

Taking inverses reverses the order of multiplication, so if every element is its own inverse multiplication must be commutative.


19

I can't speak much to what traditional Spectral Graph Theory is about, but my personal research has included the study of what I call "Spectral Realizations" of graphs. A spectral realization is a special geometric realization (vertices are not-necessarily-distinct points, edges are not-necessarily-non-degenerate line segments, in some $\mathbb{R}^n$) ...


18

This is a good question. My own take on motivating quadratic reciprocity is recorded here (these are lecture notes from an undergraduate course on introductory number theory). If you look there, you will find that most of what I have said is an elaboration of the two points you bring up. I think a crisp way of explaining what QR does for you is in the ...


18

Topology can mean different things in mathematics, depending on the context. If a mathematician describes themselves as a topologist, this likely means that they study various kinds of shapes (technically, manifolds, or related kinds of spaces), with an eye perhaps to classifying them (a typical example being the classification of closed orientable ...


18

To a topologist, topological groups are interesting in their own right. The group structure actually gives us interesting topological structure, too! One interesting fact is that the fundamental group (an important topological invariant) of a topological group is Abelian, a fact that spectacularly fails to be true in general - any group can be the ...


16

You asked about a natural problem that leads to this integral. Here's a summary of the argument I give in my undergraduate probability theory class. (It's due to Dan Teague; he has the article here.) Imagine throwing a dart at the origin in the plane. You're aiming at the origin, but there is some variability in your throws. The following assumptions ...


16

To provide a bit of a counterpoint to the other answers, here is an example of a similar situation, which might provide a better picture, since the conjecture in question has turned out to be false (so we are better able to get to a conclusion). The conjecture in question is the Lusztig conjecture, which provides a formula for the character of a simple ...


15

One very nice elementary application is Gosper's batting average problem: if a baseball player's (3-digit rounded) batting average is $.334$, what's the smallest number of at-bats that player could have? (Batting average is computed as (number of hits)/(at-bats).) The solution proceeds by noting that a rounded average of $.334$ corresponds to an actual ...


15

Quadratic reciprocity allows you to make precise certain intuitions about the primes. More precisely, it tells you that for every finite set $p_1, p_2, ... p_n$ of primes and every function $f : \{ 1, 2, ... n \} \to \{ -1, 1 \}$ there exists an arithmetic progression such that any prime $q$ in that progression satisfies $\left( \frac{p_i}{q} \right) = ...


15

The reason that $<$concept$>$ is important is that there are enough examples of $<$concept$>$ to justify making the definition. So I will answer this by giving some examples. The first example that we come across is the real numbers, $(\mathbb{R}, +)$ (by extension all the $\mathbb{R}^n$). We see that the map $sub:\mathbb{R}^2\to \mathbb{R}$ ...


14

I'd like to mention Spirographs. The formulas are actually rather simple, but I'm afraid that my Latex-foo is not sufficient to reproduce them here adequately. So I'll just refer to the Wikipedia page, and some example images (also from Wikipedia):



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