For questions about gaining and achieving motivation.

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5
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1answer
69 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
5
votes
1answer
553 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 ...
0
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0answers
29 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
1
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0answers
43 views

Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
2
votes
2answers
44 views

Do coalgebras arise outside the study of bi/Hopf-algebras?

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not ...
22
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4answers
905 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
1
vote
2answers
75 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
2
votes
3answers
338 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
18
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2answers
828 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
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1answer
37 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
1
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0answers
44 views

How do I visualize algebraic products?

I think I have no or a little problem with analyzing given algebraic products. That is, I know properties of direct, semi-direct and free products. For example, the free product $G\ast H$ is a group ...
16
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6answers
334 views

What problems, ideas or questions first got you interested in algebraic geometry?

Obviously, a lot of people are very interested in algebraic geometry. I suppose this means it is a fascinating area. However the few times I have tried to read introductory books and/or articles in ...
2
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0answers
64 views

A good way to explain $\varepsilon$-$\delta$ for chemistry / biology students?

I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry ...
2
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0answers
59 views

Intuition for the Positive Real Number $\epsilon$ in Topology

Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by ...
2
votes
0answers
49 views

Set theory, motivating factors [duplicate]

Set theory seems to pop up in many different fields of mathematics. As someone with a CS degree, I've only encountered very basic set theory; dealing with non-specific sets, and their intersections, ...
4
votes
2answers
66 views

Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?

Suppose $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\},$ that $X$ is a Banach space over $\mathbb{K}$, and that $f : X \leftarrow X$ is a bounded linear transform. Then the spectrum of $f$ is defined as the ...
28
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10answers
1k views

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
7
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1answer
68 views

Motivation behind parameters

This article shows a technique of evaluating a definite integral by introducing a suitable parameter. This however doesn't throw light on motivation for introducing that particular parameter. For ...
19
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5answers
3k 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 ...
1
vote
1answer
47 views

Why do we consider only real or complex Banach spaces?

In wikipedia, normed vector space is defined as a vector space over a subfield of $\mathbb{C}$ equipped with a norm. However, Banach space is defined as a complete normed space over $\mathbb{R}$ or ...
2
votes
1answer
81 views

What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?

The definition of the Lebesgue measure on $\mathbb R^n$ is fundamentally tied up with the following assumption: The measure of the cartesian product of $n$ intervals should be the product of the ...
8
votes
5answers
2k views

Why is the tensor product constructed in this way?

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 ...
28
votes
12answers
490 views

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
1
vote
1answer
29 views

Growth conditions for partial differential equations

Hi I am interested in what the exact purpose is of growth conditions associated with solving partial differential equations. For example the following pde: $$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla ...
1
vote
1answer
99 views

Linear systems of differential equations

I would like to see an example of a real physical situation where one can find a set of variables evolving according to a system of linear differential equations. I wasn't able to find any such ...
4
votes
3answers
599 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
1
vote
1answer
42 views

Why are special numbers important? (Such as fermat prime, mersenne prime)

Whenever I studied topics in mathematics, I found those topics are important in purely mathematical sense and I could see some motivations. However, I cannot see neither motivation nor importance of ...
3
votes
1answer
96 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
2
votes
1answer
177 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
6
votes
1answer
57 views

Motivation for and differences between properties of measure, outer measure

This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the big picture relationship between properties of measure and outer measure. I ...
7
votes
3answers
339 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
4
votes
0answers
59 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
0
votes
1answer
43 views

Uses of Mersenne primes in math

There is an international search for Mersenne primes. The project is huge. But what are the uses of Mersenne Primes in math? Do they have any other properties other than being of the form $2^n-1$?
21
votes
3answers
997 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 ...
0
votes
2answers
62 views

Where do I start learning and how do I stay motivated?

I just finished intermediate algebra in college. I liked it and breezed through it. I feel like at this level of math I can only expect dull and unenthusiastic teachers (which has been the case ...
1
vote
0answers
90 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
17
votes
4answers
2k views

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ ...
2
votes
0answers
55 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
1
vote
2answers
124 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
7
votes
4answers
912 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
1
vote
5answers
120 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
5
votes
3answers
143 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
1
vote
1answer
72 views

Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
2
votes
2answers
72 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
11
votes
3answers
629 views

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
5
votes
5answers
451 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
2
votes
1answer
164 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
3
votes
3answers
232 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 $P(A,B) = P(A)P(B)$. ...
16
votes
5answers
1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
55
votes
12answers
31k 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 ...