Questions tagged [motivation]

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21
votes
1answer
748 views

Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
9
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0answers
388 views

Why to believe the Fargues geometrization conjecture?

In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues. I can't even concisely state the conjecture so I will ...
8
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0answers
230 views

Motivating derived stacks via Euclidean geometry

Here (see Section 3) triangles in Euclidean plane are used to motivate the notion of DM stack (an equilateral triangle has more symmetries than a generic triangle). Can something similar be done to ...
18
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2answers
1k views

Commutative rings : Topoi = Fields :?

The following is probably a bad question, but hopefully, it might have a very good answer. In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
17
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0answers
528 views

What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
58
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4answers
4k views

Why did Voevodsky consider categories “posets in the next dimension”, and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...
4
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0answers
655 views

Relative cohomology in algebraic topology vs algebraic geometry

There is a notion of relative cohomology in both algebraic topology and algebraic geometry and the flexibility granted by this relative viewpoint is quite powerful in both cases. However, the two ...
26
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1answer
4k views

Why did Euler consider the zeta function?

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
0
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2answers
590 views

Why study orbifolds? [closed]

Question is as in the title. Why study orbifolds? I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. ...
12
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1answer
1k views

Entering to the K-theory realm

I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation and interaction with the field of Algebraic Topology. I mainly had concentrated on ...
1
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1answer
347 views

Intuition behind the diagonal lemma while proving Tarski's theorem about truth [duplicate]

Let $F$ be a first order logic theory with a set of axioms that are primitively recursive and are supposed to capture arithmetic (in particular, we can define a Gödel number. Let $\mathbb N$ be the ...
7
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2answers
884 views

Why the circle for Pontryagin duality? [duplicate]

For a locally compact group $G$, we define the Pontryagin dual as $\hat G = Hom(G,\mathbb T)$ where $\mathbb T$ is the circle group and the homomorphisms are continuous group maps. This duality has a ...
35
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6answers
2k views

What motivations for automorphic forms?

Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this ...
3
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2answers
329 views

Bounded Cohomology Motivation

May I ask what is the basic motivation behind studying bounded cohomology? Is there a simple reason why bounded cohomology is more interesting / useful than usual cohomology? Also, is bounded ...
4
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0answers
351 views

Motivation for studying group of homeomorphisms of topological spaces [closed]

Currently I am reading a paper titled "On the Group of Homeomorphisms of an Arc" by N.J Fine and G.E. Schweigert, that was published by Annals of Mathematics in 1955. This paper talks about the group ...
34
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11answers
4k views

Why is the definition of the higher homotopy groups the “right one”?

If someone asked me the question for the fundamental group, I would answer as follows: The connection to classification of covering spaces. The fundamental group of many spaces is an object of ...
7
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2answers
202 views

Motivations for the study of dual connections

I am intrigued by the notion of dual connections: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfy $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$ for a given (pseudo)-...
20
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1answer
891 views

Why symplectic geometry gives Poisson geometry

One way I've learned to understand Poisson geometry is to consider it as symplectic geometry with no open conditions - i.e. no condition of nondegeneracy. This idea can be applied to many other ...
1
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0answers
141 views

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{...
13
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0answers
1k views

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 ...
6
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3answers
1k views

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 ...
15
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3answers
1k views

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 ...
10
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4answers
1k views

applications of Berkovich spaces

What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
1
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2answers
332 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 ...
24
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10answers
3k 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 ...
12
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4answers
2k 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 $\bar{Q}...
17
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7answers
5k 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 ...
9
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3answers
881 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 ...
281
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7answers
130k 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 ...
8
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1answer
3k 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 ...
7
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0answers
646 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 ...
54
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5answers
15k 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 ...
26
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8answers
6k 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 ...
36
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4answers
5k 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, ...
14
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4answers
3k 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) ...
18
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2answers
1k 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 ...
9
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6answers
2k views

applications of Tate-Poitou duality

What are nice applications of Tate-Poitou duality?
98
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18answers
10k 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 ...
3
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3answers
2k 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 why ...
13
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6answers
20k 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 ...
18
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6answers
2k 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 ...
13
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9answers
3k 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 ...
5
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2answers
490 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 ...
33
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6answers
4k 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 ...
17
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3answers
3k 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 ...
53
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5answers
7k 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 ...
11
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3answers
994 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 $E\...
47
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4answers
8k 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 ...
24
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5answers
2k 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 ...
88
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10answers
19k 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 ...