The motivation tag has no wiki summary.
9
votes
7answers
359 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
51 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?
1
vote
2answers
59 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
38 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
0answers
28 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 ...
3
votes
2answers
56 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 ...
3
votes
5answers
258 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 ...
19
votes
5answers
388 views
Motivation for spectral graph theory.
Why do we care about eigenvalues of graphs?
There must be some reason. There is an entire mathematical discipline about them.
I always assumed that spectral graph theory is an extension of graph ...
1
vote
3answers
97 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
92 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 ...
6
votes
4answers
425 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 ...
3
votes
4answers
83 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
94 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? ...
35
votes
11answers
2k 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
224 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 ...
1
vote
2answers
94 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 ...
3
votes
1answer
160 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 ...
2
votes
0answers
151 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
136 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
166 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 ?
2
votes
0answers
153 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 ...
3
votes
1answer
270 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 ...
8
votes
2answers
441 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 ...
13
votes
1answer
374 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 ...
5
votes
1answer
175 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 ...
2
votes
2answers
284 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
324 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
81 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 ...
14
votes
4answers
914 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 ...
14
votes
4answers
527 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
114 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 ...
11
votes
6answers
591 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 ...
17
votes
4answers
837 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
202 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 ...
7
votes
2answers
143 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 ...
35
votes
7answers
2k 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) ...
27
votes
7answers
2k 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 ...
6
votes
5answers
877 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 ...
