Questions tagged [motivation]
The motivation tag has no usage guidance.
59 questions
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
16
votes
3
answers
1k
views
Ultraproducts of Banach spaces versus model theoretic ultraproduct
Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower ...
18
votes
4
answers
2k
views
What's the use of group cohomology for class field theory?
I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...
- 521
9
votes
3
answers
2k
views
Simple motivation to study arithmetic geometry
Is there a simple-to-understand diophantine equation (in the sense that it's easy to explain to a child) that has a positive integer solution, but to prove that such a solution exists and to find it ...
- 649
7
votes
1
answer
542
views
Are large cardinals about more than just consistency?
The other day, I was reading the preface of Kanamori's The Higher Infinite and noticed that he says large cardinals provide a useful 'measuring stick' for consistency. That raised the question of ...
- 203
57
votes
10
answers
4k
views
Books/websites which have motivating stories of mathematicians overcoming hardships in life
Edit 1: I have received a lot of great answers. I am not accepting any answer because I think there might be in future that some user want to contribute any new answer, as in my opinion some users ...
2
votes
1
answer
291
views
Where does this clever choice of differential come from? (calculating $\mathrm{H}^1_{\mathrm{dR}}(E/k))$
In these notes of Kedlaya, he calculates the de Rham cohomology of an affine part $X$ of an elliptic curve $E$ over a field $K$, given by $y^2 = P(x) = x^3 + ax + b$.
He uses these relations:
$0 = y^...
- 1,142
5
votes
1
answer
686
views
What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?
(I asked it first in MathStackExchange but I haven't get an answer yet)
Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms.
For unramified ...
- 452
11
votes
2
answers
1k
views
What is the significance of Friedlander-Iwaniec and related theorems?
On p.177 of Number Theory Revealed: A Masterclass by Andrew Granville, the author states that "One can ask for prime values of polynomials in two or more variables." (though he later ...
- 2,075
27
votes
1
answer
1k
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 ...
- 11.8k
28
votes
3
answers
5k
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 ...
user140765
8
votes
0
answers
279
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 ...
user138661
20
votes
2
answers
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 ...
- 9,111
17
votes
0
answers
770
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 ...
- 179
68
votes
4
answers
5k
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 ...
- 775
6
votes
0
answers
1k
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 ...
- 7,746
26
votes
1
answer
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 ...
- 293
0
votes
2
answers
822
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. ...
- 6,023
14
votes
1
answer
3k
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 ...
- 309
1
vote
1
answer
617
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 ...
- 7,746
11
votes
2
answers
3k
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 ...
- 7,746
40
votes
6
answers
4k
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 ...
7
votes
3
answers
915
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 ...
- 1,229
3
votes
0
answers
440
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 ...
- 428
42
votes
12
answers
7k
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
votes
2
answers
323
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)-...
- 442
21
votes
1
answer
1k
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 ...
- 4,991
1
vote
0
answers
201
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{...
- 21
16
votes
0
answers
2k
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 ...
- 189
6
votes
3
answers
2k
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 ...
16
votes
3
answers
2k
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 ...
12
votes
4
answers
2k
views
applications of Berkovich spaces
What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
user19475
1
vote
2
answers
374
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 ...
- 3,393
24
votes
10
answers
4k
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 ...
- 66.8k
22
votes
5
answers
4k
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}...
- 345
18
votes
7
answers
6k
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
votes
3
answers
902
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 ...
- 1,261
296
votes
8
answers
143k
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 ...
14
votes
1
answer
5k
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
votes
0
answers
752
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 ...
65
votes
5
answers
18k
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 ...
34
votes
8
answers
8k
views
Are quivers useful outside of Representation Theory?
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 already >...
- 1,437
44
votes
4
answers
7k
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, ...
- 33.8k
13
votes
4
answers
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) ...
- 141
25
votes
2
answers
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 ...
12
votes
6
answers
3k
views
applications of Tate-Poitou duality
What are nice applications of Tate-Poitou duality?
123
votes
18
answers
14k
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
votes
3
answers
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 ...
- 317
17
votes
6
answers
24k
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 ...
- 181
23
votes
6
answers
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 ...