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.

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24 votes
0 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 ...
semisimpleton's user avatar
16 votes
0 answers
584 views

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. ...
Omar Antolín-Camarena's user avatar
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 ...
10 votes
0 answers
204 views

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 ...
user avatar
10 votes
0 answers
332 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
sphere's user avatar
  • 413
9 votes
0 answers
295 views

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 votes
0 answers
181 views

$p$-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 ...
Yiftach Barnea's user avatar
9 votes
0 answers
772 views

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
2k views

"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 votes
0 answers
422 views

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 votes
0 answers
498 views

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 votes
0 answers
1k views

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 votes
0 answers
377 views

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 ...
Arrow's user avatar
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7 votes
0 answers
580 views

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 ...
user avatar
6 votes
0 answers
227 views

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 ...
Mark Schultz-Wu's user avatar
6 votes
0 answers
252 views

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 $f:\ M\to k$ ...
Duchamp Gérard H. E.'s user avatar
6 votes
0 answers
544 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
310 views

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 ...
6 votes
0 answers
200 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,...
6 votes
0 answers
209 views

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 votes
0 answers
196 views

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 ...
Will Sawin's user avatar
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5 votes
0 answers
515 views

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
598 views

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 votes
0 answers
154 views

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,.....
M. Winter's user avatar
  • 12.5k
5 votes
0 answers
359 views

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?
5 votes
0 answers
77 views

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
224 views

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 ...
Mikhail Tikhomirov's user avatar
5 votes
0 answers
187 views

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 ...
Jarek Kędra's user avatar
  • 1,772
5 votes
0 answers
197 views

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 ...
Per Alexandersson's user avatar
5 votes
0 answers
509 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 ...
Anton Hilado's user avatar
  • 3,269
5 votes
0 answers
227 views

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
216 views

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
87 views

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 ...
jg1896's user avatar
  • 2,683
4 votes
0 answers
187 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 $\mathbf{P}^1$...
Pulcinella's user avatar
  • 5,506
4 votes
1 answer
542 views

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 votes
0 answers
413 views

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 ...
C.F.G's user avatar
  • 4,165
4 votes
0 answers
272 views

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 votes
0 answers
259 views

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
195 views

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 ( \...
Max Muller's user avatar
  • 4,485
3 votes
0 answers
455 views

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 votes
0 answers
78 views

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 votes
0 answers
137 views

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 votes
0 answers
544 views

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
428 views

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 ...
sadman-ncc's user avatar
2 votes
0 answers
141 views

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 votes
0 answers
205 views

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 ...
user avatar
2 votes
0 answers
393 views

Textbooks on solidifying graduate knowledge

I am finishing my undergraduate program soon and start getting ready for graduate school. What I have realized is that although I have passed many subjects and with good grades I feel that ...
2 votes
0 answers
118 views

Conjectures that can be tested with large numbers of Hecke eigenvalues of GSp(4) automorphic forms

As part of my thesis work I have proved Ibukiyama's conjecture implies something about $\mathrm{SO}(5)$ forms associated to certain lattices lifting to $\mathrm{GSp}(4)$ (This was originally a ...
2 votes
0 answers
183 views

Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it. The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
2 votes
0 answers
176 views

Can the Moler and Morrison Algorithm be Improved?

In a nutshell, the Moler and Morrison algorithm is a fast method for calculating euclidean distances in a numerically stable way by using reflections instead of the pythagorean theorem. In order ...
Manfred Weis's user avatar
  • 12.6k