All Questions
22,770 questions
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 ...
252
votes
37
answers
178k
views
Best algebraic geometry textbook? (other than Hartshorne)
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...
234
votes
16
answers
57k
views
What elementary problems can you solve with schemes?
I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...
222
votes
8
answers
35k
views
How to memorise (understand) Nakayama's lemma and its corollaries?
Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
- 14.3k
182
votes
33
answers
32k
views
What should be learned in a first serious schemes course?
I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (...
157
votes
11
answers
20k
views
Why are flat morphisms "flat?"
Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...
- 12.6k
156
votes
4
answers
12k
views
Analytic tools in algebraic geometry
This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...
- 55.9k
150
votes
2
answers
22k
views
What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...
- 13.6k
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
140
votes
0
answers
10k
views
Grothendieck -sad news [closed]
Sorry for that this is not a real question. But I thought people would like to know.
Alexandre Grothendieck died today: http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-...
135
votes
6
answers
23k
views
what mistakes did the Italian algebraic geometers actually make?
It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations (...
- 41.4k
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
128
votes
15
answers
51k
views
A learning roadmap for algebraic geometry
Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...
116
votes
2
answers
31k
views
Why is the Hodge Conjecture so important?
The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...
- 1,263
108
votes
7
answers
21k
views
What is the field with one element?
I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-...
- 5,062
107
votes
8
answers
15k
views
What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...
- 118k
103
votes
3
answers
6k
views
Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...
- 24.2k
102
votes
6
answers
11k
views
Is there an analogue of curvature in algebraic geometry?
I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
- 29.2k
101
votes
2
answers
11k
views
Riemann hypothesis via absolute geometry
Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over ...
- 5,232
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
96
votes
4
answers
5k
views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
- 4,316
93
votes
0
answers
17k
views
Hironaka's proof of resolution of singularities in positive characteristics
Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...
- 8,071
89
votes
5
answers
16k
views
Why higher category theory?
This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
87
votes
15
answers
37k
views
The importance of EGA and SGA for "students of today"
That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.
...
87
votes
12
answers
12k
views
Why do we make such big deal about the 'unsolvability' of the quintic?
The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...
- 1,389
86
votes
13
answers
24k
views
How has modern algebraic geometry affected other areas of math?
I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...
86
votes
4
answers
15k
views
Etale cohomology -- Why study it?
I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...
- 2,799
84
votes
1
answer
5k
views
Is there a complex surface into which every Riemann surface embeds?
This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \...
- 19.3k
83
votes
5
answers
13k
views
how does one understand GRR? (Grothendieck Riemann Roch)
I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer ...
- 12.4k
81
votes
4
answers
8k
views
Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?
Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...
- 7,329
80
votes
23
answers
19k
views
Algebraic geometry examples
What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that ...
80
votes
15
answers
15k
views
Why torsion is important in (co)homology ?
I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
80
votes
2
answers
7k
views
Vladimir Voevodsky's works
Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
79
votes
9
answers
21k
views
Results that are widely accepted but no proof has appeared
The background of this question is the talk given by Kevin Buzzard.
I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.
...
78
votes
5
answers
14k
views
Is there a "geometric" intuition underlying the notion of normal varieties?
I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me ...
- 14.3k
78
votes
6
answers
6k
views
Rigidity of the category of schemes
Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets,...
- 63.1k
76
votes
5
answers
9k
views
Is there an intuitive reason for Zariski's main theorem?
Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version:
Zariski's main ...
- 6,348
76
votes
6
answers
9k
views
Errata to "Principles of Algebraic Geometry" by Griffiths and Harris
Griffiths' and Harris' book Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My ...
76
votes
2
answers
6k
views
Is it known that the ring of periods is not a field?
I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...
- 32.5k
75
votes
17
answers
10k
views
Facts from algebraic geometry that are useful to non-algebraic geometers
A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...
75
votes
4
answers
16k
views
What's the "Yoga of Motives"?
There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-...
- 15.1k
75
votes
4
answers
6k
views
When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?
If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or ...
- 47.5k
75
votes
4
answers
5k
views
What are reasons to believe that e is not a period?
In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....
- 2,493
75
votes
5
answers
3k
views
When the automorphism group of an object determines the object
Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
74
votes
29
answers
8k
views
Proofs where higher dimension or cardinality actually enabled much simpler proof?
I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
74
votes
16
answers
8k
views
Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...
74
votes
3
answers
10k
views
Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?
This question is partly motivated by Never appeared forthcoming papers.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance ...
- 2,413
74
votes
1
answer
6k
views
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
- 63.1k
73
votes
6
answers
6k
views
A bestiary of topologies on Sch
The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? ...
- 35.4k
73
votes
2
answers
8k
views
The inverse Galois problem and the Monster
I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...
- 4,994