All Questions
Tagged with big-list ag.algebraic-geometry
60 questions
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
9
votes
1
answer
506
views
Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
7
votes
1
answer
2k
views
Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
1
vote
0
answers
233
views
Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero
When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
- 157
5
votes
0
answers
212
views
Relations between Whittaker functions/W algebras and Stokes data/resurgence
Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
- 5,701
33
votes
3
answers
2k
views
The probabilistic method outside of discrete mathematics
The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive ...
20
votes
1
answer
2k
views
Expected applications of condensed mathematics
As a student of algebraic geometry (in an advanced stage, but still far from an expert on anything), I am quite excited about learning some condensed mathematics. I have been told that the theory has ...
2
votes
1
answer
190
views
Limit of line bundles on smooth curves degenerating to double line
Consider a family of smooth plane conics $f_\lambda(x,y,z)=0$ as a family $T_\lambda = (C,L,v_1,v_2,v_3)_\lambda$ of genus zero curves with a degree 2 line bundle $L$ and an ordered basis $v_i$ for ...
- 21
6
votes
0
answers
584
views
What are the topics in noncommutative algebraic geometry?
Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
6
votes
0
answers
211
views
What is known about "dimension two" vertex algebras?
In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...
18
votes
4
answers
621
views
What are immediate applications of the classification of connected reductive groups?
After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.
That's a non-trivial theory! I'm hoping that now that I am done ...
- 557
5
votes
5
answers
2k
views
Terminology introduced in recent years with more than one meaning
Suppose a term(inology) is recently (in last 20 years) introduced in research mathematics.
It might happen that some one who wish to use it, in the same area of research, for different purposes or ...
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 ...
15
votes
5
answers
2k
views
Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
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.
...
5
votes
1
answer
291
views
The Idea of Kroneckerian geometry
Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$.
Let's ...
2
votes
2
answers
214
views
Measuring failure of a setup to preserve some structure giving interesting notions
I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation:
Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
16
votes
2
answers
539
views
Surprising appearances of Painlevé transcendents
What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations?
I have seen examples from enumeration of classes ...
21
votes
1
answer
1k
views
When simple cohomological computations predict ingenious algebro-geometric constructions?
Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...
9
votes
0
answers
388
views
True on stalks, false on affine opens [closed]
In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g.
...
user142126
14
votes
8
answers
2k
views
Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
48
votes
12
answers
10k
views
How to explain to an engineer what algebraic geometry is?
This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
8
votes
2
answers
3k
views
What does reduction of structure group of principal bundle say?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle.
The notion of reduction of structure group is standard but I will ...
- 6,023
7
votes
1
answer
664
views
What are some open problems in moduli spaces and moduli stacks?
I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ?
Thanks in advance for your help.
- 325
7
votes
1
answer
470
views
Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
- 76
9
votes
4
answers
560
views
When is it easier to work projectively?
There are many instances in which theory over $\mathbb{C}$ is cleaner than theory over $\mathbb{R}$. For example, continuously differentiable functions over $\mathbb{R}$ are not necessarily twice ...
8
votes
1
answer
987
views
Steps in Geometric Complexity Theory
GCT purports to provide a program to show that $NP \not \subset P/poly$.
At the high level what are the steps involved in the program and what stage is each step in?
What difficulties currently are ...
- 13.9k
21
votes
3
answers
7k
views
What are the current breakthroughs of Geometric Complexity Theory?
I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods.
That program seems ...
28
votes
9
answers
5k
views
Applications of algebra to analysis
EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
2
votes
0
answers
527
views
Neat applications of Galois descent?
I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space ...
- 10.5k
6
votes
1
answer
899
views
Interesting implications on the theory of motives if the Hodge conjecture holds
For example,
Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as ...
- 1,720
15
votes
4
answers
6k
views
What is the interface between functional analysis and algebraic geometry?
This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers ...
- 2,246
35
votes
2
answers
3k
views
Equivalent descriptions of Hodge conjecture?
I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...
5
votes
0
answers
534
views
Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme
I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes?
At this point I am still ...
- 3,309
9
votes
2
answers
791
views
Rational points techniques on curves not using their Jacobian
Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
- 2,224
19
votes
3
answers
2k
views
Using schemes to prove things about rings
I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment.
The question is as follows:
What are examples of strict applications ...
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 ...
23
votes
6
answers
2k
views
Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
23
votes
4
answers
4k
views
What information is contained in the Kazhdan-Lusztig polynomials?
The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
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, ...
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
0
answers
2k
views
Applications of cohomology and base change?
What is the theorem on coherent cohomology and base change good for?
One version of the theorem is:
Suppse $f \colon X \to Y$ is a proper morphism of noetherian schemes and $F$ is a $Y$-flat coherent ...
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-...
9
votes
6
answers
5k
views
Examples of naturally occurring Quadratic forms or quadrics.
I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
15
votes
4
answers
1k
views
What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 5,846
41
votes
13
answers
16k
views
What is a good introductory text for moduli theory?
Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...
25
votes
2
answers
2k
views
Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
14
votes
6
answers
5k
views
More upper/lower semi-continuous functions in (algebraic) geometry?
The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.
Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
14
votes
3
answers
2k
views
Singularities of space curves: Open question lists?
For plane curve singularities most questions have been answered,
in large part due to the Newton-Puiseux expansion.
I've heard that there are a number of open problems regarding
space curve ...
61
votes
11
answers
21k
views
What are some open problems in algebraic geometry?
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...