Questions tagged [motivation]
The motivation tag has no usage guidance.
59 questions
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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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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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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 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 ...
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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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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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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 >...
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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
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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 ...
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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 ...
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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 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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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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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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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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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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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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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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 ...
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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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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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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 ...
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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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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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applications of Berkovich spaces
What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
user19475
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applications of Tate-Poitou duality
What are nice applications of Tate-Poitou duality?
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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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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 ...
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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 ...
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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 ...
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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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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
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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)-...
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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 ...
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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 ...
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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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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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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 ...
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