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16
votes
2answers
584 views

Motivation for/history of Jacobi's triple product identity

I'm taking a short number theory course this summer. The first topic we covered was Jacobi's triple product identity. I still have no sense of why this is important, how it arises, how it might have ...
2
votes
2answers
98 views

What is the significance of permutable subgroups? (and $X$-permutable subgroups?)

Let $G$ be a group and $H$, $K$, $X$ be subgroups of $G$. We say $H$, $K$ are permutable if $HK=KH$. or we say $H, K$ are X-permutable if $‎\exists x, x\in X$ such that $H^{x}K=KH^{x}.$ Why are ...
6
votes
1answer
90 views

Recovering the structure of an object from its morphism:Yoneda Lemma

I've heard that Yoneda lemma informally states that one can recover the internal structure of an object by looking at the morphism coming out from that object. But this is not clear to me from the ...
6
votes
2answers
299 views

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = ...
2
votes
1answer
175 views

Why do we need continuous random variables since they can be approximated by discrete ones?

I do not understand the motivation of developing the theory of continuous random variables. Given simple discrete random variables, the continuous ones can be well approximated.
5
votes
1answer
169 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
11
votes
7answers
2k views

Why is it important to study combinatorics?

I was having a discussion with my friend Sayan Mukherjee about why we need to study combinatorics which admittedly, is not our favourite subject because we see very less motivation for it(I am not ...
3
votes
6answers
116 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
4
votes
2answers
239 views

Motivation for the study of the Chern connection

Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following ...
1
vote
1answer
163 views

Intuition behind symmetric and antisymmetric tensors

I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
1
vote
1answer
49 views

It is not possible to introduce multiplication in $v_n$(For $n>2$) so as to satisfy all field properties

In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells It can be shown that except $n=1, 2$, it is not possible to introduce multiplication ...
4
votes
3answers
186 views

What is the point of extremal epimorphisms in category theory? Why not just use strong epis instead?

I've been trying to get my head around the various types of epimorphisms you get in category theory, but I can't see why anyone uses "extremal" epis as opposed to the slightly less general notion of ...
5
votes
5answers
1k views

Motivation for the Tensor Product [duplicate]

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
26
votes
6answers
2k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
1
vote
3answers
131 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
6
votes
2answers
141 views

Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has ...
7
votes
4answers
1k views

Order of nontrivial elements is 2 implies Abelian group

If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation ...
2
votes
4answers
89 views

Motivating complex structure on $\mathbb{R}^2$

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations ...
3
votes
2answers
114 views

Use of $\mathbb N$ & $\omega$ as index sets

Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? ...
47
votes
12answers
17k views

Why study linear algebra?

Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study ...
10
votes
2answers
340 views

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
3
votes
3answers
133 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
1
vote
2answers
456 views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
4
votes
1answer
272 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
3
votes
1answer
314 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
0
votes
2answers
151 views

motivation of limit points

Lets use Wikipedia's definition of a limit point and let $\lim(A)$ denote the set of limit points of $A$. $a\in \lim (A) \leftrightarrow a\in\operatorname{cl}(A\setminus\{a\})$, $\lim (A)\cup A = ...
3
votes
0answers
173 views

Algebraic Geometry question

Why do we study projective normality of a projective variety ? Does it have anything to do with non-singularity ? Any other purpose to study this ?
3
votes
0answers
412 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
7
votes
1answer
503 views

History behind Exact Sequences.

I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the ...
9
votes
2answers
591 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
16
votes
2answers
641 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 ...
9
votes
1answer
348 views

How to motivate the axioms for the inner product

Typically, one doesn't just write down lists of axioms and then sees if there are enough interesting examples that satisfy them; they evolve over time, usually from a couple of very ...
3
votes
2answers
561 views

Why do we need (the abstract concept of) random variables (in discrete probability models)?

What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that ...
12
votes
2answers
510 views

Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
0
votes
2answers
88 views

Interpretation of condition on positive random variable

Let $W$ be a random variable such that $\mathbb{P}(W > 0) = 1$ and $\mathbb{E}(W) = 1$. Is there an interpretation or motivation for the condition $$\mathbb{E}(W \log (W) ) < c$$ where $c \in ...
17
votes
4answers
2k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
15
votes
4answers
826 views

Fun math for young, bored kids?

For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ...
4
votes
2answers
132 views

Why a norm and not some other function that defines a metric?

If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on ...
12
votes
6answers
985 views

Motivation of the Gaussian Integral

I read on Wikipedia that Laplace was the first to evaluate $$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx$$ Does anybody know what he was doing that lead him to that integral? Even better, can ...
22
votes
4answers
1k views

Motivation behind topology

What is the motivation behind topology? For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we ...
4
votes
3answers
283 views

Motivation for the Mapping Class Group

Question: What is the motivation for studying the mapping class group? In particular, what types of questions does it attempt to answer and what kind of invariant is it? Motivation for this ...
8
votes
2answers
160 views

Demonstrating the value of abstracting away from elements/subsets to maps

Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction: an actual element, e.g. $s\in S$ an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$ an ...
38
votes
7answers
3k views

What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?

Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) ...
29
votes
7answers
3k views

Uses of quadratic reciprocity theorem

I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic ...
8
votes
5answers
1k views

Motivation behind the definition of complete metric space

What is motivation behind the definition of a complete metric space? Intuitively,a complete metric is complete if they are no points missing from it. How does the definition of completeness (in ...