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174
votes
7answers
85k views

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 ...
73
votes
18answers
6k views

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 ...
59
votes
10answers
10k views

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 ...
38
votes
5answers
5k views

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 ...
33
votes
5answers
4k views

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 ...
28
votes
4answers
3k views

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, ...
28
votes
4answers
4k views

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 ...
26
votes
6answers
3k views

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 ...
21
votes
9answers
2k views

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 ...
21
votes
5answers
1k views

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 ...
16
votes
8answers
3k views

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 ...
13
votes
3answers
2k views

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 ...
13
votes
2answers
809 views

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 ...
12
votes
4answers
2k views

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) ...
12
votes
5answers
1k views

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 ...
11
votes
7answers
2k views

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 ...
10
votes
7answers
2k views

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 ...
10
votes
3answers
842 views

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 ...
9
votes
3answers
837 views

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 ...
7
votes
6answers
6k views

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 ...
5
votes
2answers
385 views

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 ...
4
votes
1answer
2k views

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 ...
4
votes
0answers
394 views

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 ...
3
votes
5answers
1k views
3
votes
1answer
198 views

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 ...
2
votes
3answers
882 views

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 ...
1
vote
1answer
148 views

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 ...