The motivation tag has no usage guidance.

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### Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{...

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### Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form?
The subject certainly originated from physics, but is there a deeper reason for why it is still an ...

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### Examples of high level math that can be motivated to laypeople

One of the difficulties of mathematics over other sciences is that our problems are harder to motivate to a general audience. A biologist studying a particular pathway in the body can say that he's ...

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### Motivation for the Preprojective Algebra

Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of $\bar{Q}...

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### Motivation for equivariant homotopy theory?

I'm in the process of learning equivariant homotopy theory, so I was wondering: what is the importance of equivariant homotopy theory, and what has it been applied to so far? I know of HHR's solution ...

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### applications of Berkovich spaces

What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$

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### How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...

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### How to understand a rooting of a dessin d'enfant?

As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...

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### Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...

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### Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

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### Why tropical geometry?

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...

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### What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, ...

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### Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...

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### How to motivate the skein relations?

I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...

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### Are quivers useful outside of Representation Theory?

Dear All!
There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for ...

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### Fundamental motivation for several complex variables [closed]

I have 3 general abstract reasons to care about complex analysis in a single variable:
The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is ...

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### Why doesn't this group have a name?

$$\{A\in GL_n(\mathbb{C}) : |det(A)|=1\}$$ This seems to me to be a perfectly natural group to study; it is easy to define and contains $U(n), SL_n$, and all the torsion. Is there any good reason ...

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### Why are Galois Representations so important in Number theory ?

Dear everyone,
Motivation :
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...

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### Snazzy applications of Several Complex Variables techniques

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems which depend on a lot of ...

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### Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...

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### Why are subfactors interesting?

I get asked this question a lot, and am not very happy with any of the answers.
Vaguely I think of subfactor theory as a generalization of representation theory of groups. That is, if you have a ...

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### Is modern computability theory “really” about algorithms?

Apologies if my question seems overly naive, but I haven't seen/heard/read any good answers.
What is modern computability theory "really" about? The study of feasible(even remotely feasible) ...

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### Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of ...

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### Why are spectral sequences so ubiquitous?

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...

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### Motivation for uniform surjectivity of mod l representations associated to elliptic curves

Background
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $G_{\mathbb{Q}}$ be the absolute Galois group $Aut(\overline{\mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup $E\...

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### motivation for compactness [duplicate]

Possible Duplicate:
How to understand the concept of compact space
Hello,
I am learning some analysis on my own and
what is the motivation to consider compactness?
eg. I do not understand ...

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### How to understand the concept of compact space [closed]

the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I ...

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### Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems ...

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### Understanding iterated integrals

I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory.
It seems the ...

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### Motivation for the covariant model structure on SSet/S

I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure?
I'm sorry for not providing ...

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### Applications of Noncommutative Geometry

This is related to Anweshi's question about theories of noncommutative geometry.
Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative rings I have much ...