Questions tagged [big-list]
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
68 questions with no upvoted or accepted answers
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Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
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16
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0
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596
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What are some examples of weak ω-categories?
As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...
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13
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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 ...
10
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answers
274
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Open problems in complete theories
It is well-known that every complete recursively enumerable first-order theory is decidable. Does that mean that such theories are "trivial", or are there still interesting open problems ...
user507115
10
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0
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214
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Examples of automorphic representations to keep in mind
I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new ...
user140615
10
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788
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Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?
I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...
9
votes
0
answers
299
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List of modern points of view simplifying or clarifying classical topics
There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
9
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answers
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$p$-groups and the arithmetic of $p$
I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
- 5,450
9
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"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
8
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0
answers
558
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What is the nicest bijection $\textbf{R}^p \to \textbf{R}^q$ that you know?
It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist).
The problem with all these examples of bijections is that typically the ...
8
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0
answers
560
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Landau's century-old problems: Anything comparable?
Landau's four problems
are now over a century old (1912), and each still unsolved.
This seems remarkable, even though he was not the originating author all four
(maybe only the 4th?). Still, he ...
8
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0
answers
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Examples of uncountable abelian $p$-groups
Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?
By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
7
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0
answers
483
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Theories of manifolds w/ extra structure and singularities
Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
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7
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0
answers
358
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Which journals publish mathematics book reviews?
Which mathematics journals publish book reviews? So far I have the following:
Notices of the American Mathematical Society
Bulletin of the American Mathematical Society (From looking at its website ...
7
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0
answers
407
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Applications of Monadicity theorems
This is crosspost of this MSE question.
Having carefully read the proof of Beck's monadicity theorems and some related variations, I'm now hungry for cool applications.
For instance, I found these ...
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7
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0
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667
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What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?
It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is,
What would be the consequences if Second Hardy-Littlewood ...
user57432
6
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0
answers
244
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What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
- 2,183
6
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Usefulness of total algebras and exotic generating series
In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
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6
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0
answers
584
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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
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0
answers
211
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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,...
6
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Theorems which are not numerically verified
Perhaps one of the best forms of justification for pure mathematics, in my experience, is the ability to demonstrate the truth of some statements despite the lack of numerical evidence.
A rather ...
5
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568
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What sets are known to have cardinality equal to $\mathbb{N}$ or $\mathbb{R}$ but open as to which?
A long time ago a similar question was asked on math.stackexchange.
There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically.
An ...
5
votes
0
answers
212
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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
5
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0
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788
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Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
5
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0
answers
161
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Where have you encountered "arrangement spaces"?
I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).
Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
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5
votes
0
answers
77
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Numerical and computational approaches to limit cycle theory
I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach.
I would like to ask ...
5
votes
0
answers
230
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On a Robin Forman's remark on combinatorial simplicial complexes
In a very captivating introduction to discrete Morse theory, Robin Forman makes the following remark:
...However, that does not explain why so many simplicial complexes that arise in combinatorics ...
- 5,590
5
votes
0
answers
191
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The Hofer topology
Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}(M,\omega)$ be the group of compactly supported Hamiltonian diffeomorphisms equipped with the Hofer metric. I would like to collect ...
- 1,782
5
votes
0
answers
199
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Examples of combinatorial bijections found by considering functors
Let us assume that I have two sets of combinatorial objects, $A$ and $B$,
and I am looking for a bijection (in particular a map) $\psi:A \to B$ between these sets, usually required to preserve some ...
- 15.8k
5
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0
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534
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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 ...
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5
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0
answers
227
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Quotations about the class number formula, etc.
I'm looking for interesting and/or expressive quotations from mathematicians about the class number formula. I'm interested both in quotations from historical mathematicians and from modern ...
5
votes
1
answer
226
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Examples of noncommutative Bezout domains
I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
4
votes
0
answers
95
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List of equivalent conditions for the invariant subalgebra to be polynomial
Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
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4
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1
answer
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Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...
4
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0
answers
421
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What are your common strategies/remedies when your new theory/idea stuck in most cases?
Sorry if this is not a suitable post for MO.
Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the ...
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4
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answers
274
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What arithmetic would you do in parallel?
This is a post asking for references, and soliciting problems and people interested in accelerated computing. I will add the big-list tag and make it community-wiki. If this interests you strongly, ...
4
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0
answers
270
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Has the external knit product been used to construct a previously unknown group?
In the Wikipedia article
Zappa–Szép product
, the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid ...
4
votes
0
answers
362
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Theorems conditional on false conjectures
What is an example of a theorem that was conditional on a conjecture that later turned out to be false?
4
votes
0
answers
273
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What are some interesting examples of cooperative games that can be naturally generalised to a stochastic version of it?
In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \...
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3
votes
0
answers
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Proofs of the loop-suspension adjunction in infinity-categories
$\DeclareMathOperator{\Map}{Map}$$\DeclareMathOperator{\Fun}{Fun}$$\DeclareMathOperator{\const}{const}$$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\lim}{lim}$In Elements of $\infty$-...
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Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
3
votes
0
answers
638
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What are some of the big open problems in $4$-manifold theory?
I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
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3
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Applications of Ambrose-Singer theorem on holonomy
I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...
3
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0
answers
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A complex differential theorem applying to compact projective manifolds but not all compact Kahler manifolds
The following question may be soft, but I hope it is precise enough. The Hodge conjecture, if proven, would be a theorem in complex differential geometry that holds for all compact projective ...
3
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0
answers
137
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What other axioms for set theory can be written in the form: "If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic"?
The "injective continuum function hypothesis" (ICF) is the following statement.
ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$
...
3
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0
answers
557
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Applications of the class number formula, etc
This is a big list of applications of the class number formula and its generalizations. I'll start:
The solution to Gauss's class number problem for imaginary quadratic fields, and more generally the ...
2
votes
0
answers
158
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What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
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Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background
The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
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2
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What practically computable homotopy and/or (co)homology theories are known for finite (di)graphs, metric spaces, etc?
Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I ...
2
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What problems are easier assuming zeros of a zeta function don’t behave as we expect?
What are some examples of problems which are easier to solve assuming zeros of zeta functions lie off the critical line or do not have expected vertical distribution.
There are some very well known ...
user156885