Questions tagged [big-picture]

Questions to get the "big picture" about a subject.

Instead of a single specific problem, you may wish to get an overview what a certain result is good for in the long run, or what in the (non-intrinsic) purpose or motivation of a certain area of research is, etc., that is you want to get the "big picture". The big-picture tag should not be used alone and is not an excuse for asking an overly broad question, though.

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Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them in order of appearance in my education/in…
184
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6 answers

What were some major mathematical breakthroughs in 2016?

As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news…
YukiJ
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Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some…
Greg L
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89
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Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain…
user23211
85
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12 answers

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be applied (directly or indirectly) or has given me Aha!…
PhD
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81
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"Where" exactly are complex numbers used "in the real world"?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious. So what…
PhD
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Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global…
66
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How and why does Grothendieck's work provide tools to attack problems in number theory?

This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing. I have always been fascinated with Grothendieck and the way he did mathematics. I've heard Mochizuki's work on the abc conjecture heavily…
Arrow
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65
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Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and conceptual understanding) on the Laplace operator, and…
62
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What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic…
user78743
49
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3 answers

Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds requires that they be paracompact. I've looked all…
Daniel Moskovich
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Largest "leap-to-generality" in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that would generalize existing ones and provide a…
Alexandre Halm
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What do cones have to do with quadratics? Why is $2$ special?

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations in 2 variables). Recently I came across the…
D.R.
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Why are algebraic structures preserved under intersection but not union?

In general, the intersection of subgroups/subrings/subfields/sub(vector)spaces will still be subgroups/subrings/subfields/sub(vector)spaces. However, the union will (generally) not be. Is there a "deep" reason for this?
MathematicsStudent1122
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Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.) The ideals of a…
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