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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53 votes
6 answers
5k views

Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
72 votes
13 answers
10k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
35 votes
17 answers
2k views

Equivalent definitions of Gromov hyperbolicity

Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible. I'm happy for the definitions to require some niceness ...
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 ...
5 votes
5 answers
1k views

Interesting topics for (very) short talks [closed]

Part of the requirements for my Honours is that I record a short 4-7 minute digital talk, which is then distributed to all the other students and staff at my university’s mathematics department. The ...
0 votes
4 answers
2k views

Math journals which publish/reject quickly [closed]

I would like to publish a math paper quickly. The level of journal is not that important (except that it should not send out spam with its own ads). I am looking for a math journal which decides ...
53 votes
10 answers
7k views

Changes forced by the pandemic

The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
6 votes
3 answers
548 views

Anomalous phenomena [closed]

What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern? This is most interesting when ...
11 votes
9 answers
4k views

List of problems for graduate topics?

When I study a new topic, I never feel satisfied until I have spent some time solving a long list of problems. I am looking for either a problem book or a list of problems on graduate math topics. ...
34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
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, ...
35 votes
12 answers
3k views

No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
90 votes
11 answers
13k views

What are possible applications of deep learning to research mathematics?

With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
21 votes
9 answers
2k views

Naturally occurring examples of badly behaved categories

What are some examples of naturally occurring badly behaved (possibly higher) categories? When working with a specific category like ${\bf Set}$ or ${\bf Cat}$, we usually understand/explain them by ...
10 votes
2 answers
1k views

How professional mathematicians deal with discouragement? [closed]

All professional mathematicians feel discouraged occasionally due to some issue. My question is: How do professional mathematicians deal with discouragement? In this link , Andrew Wiles say ...
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,...
46 votes
10 answers
11k views

What kid-friendly math riddles are too often spoiled for mathematicians?

Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples: What is $1+2+\cdots+100$? Is it possible to tile a mutilated chess board with dominoes?...
22 votes
1 answer
2k views

A list of proofs of the Hasse–Minkowski theorem

I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
13 votes
29 answers
6k views

Which great mathematicians had great political commitments? [closed]

Some mathematicians claim that their field has nothing to do with political concerns; others are deeply involved in political life. Are there many great mathematicians with great political commitments?...
21 votes
17 answers
4k views

Which great mathematicians were also historians of mathematics?

As the question title suggests, which great mathematicians were also historians of mathematics? We all know plenty of great mathematicians, but not many historians of mathematics. Not to mention that ...
27 votes
1 answer
935 views

"Non-categorical" examples of $(\infty, \infty)$-categories

This title probably seems strange, so let me explain. Out of the several different ways of modeling $(\infty, n)$-categories, complicial sets and comical sets allow $n = \infty$, providing ...
12 votes
6 answers
715 views

Conditions equivalent to finiteness

We've all probably come across some conditions that naturally imply finiteness, or are equivalent to it. For ZFC examples: A set $X$ can be ordered in such a way that the ordering is well-founded and ...
194 votes
18 answers
15k views

Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
45 votes
4 answers
4k views

How to invoke constants badly

In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways ...
10 votes
2 answers
628 views

Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?

Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that $$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$ That is, they become diffeomorphic after ...
Chris Schommer-Pries's user avatar
131 votes
22 answers
10k views

Books that teach other subjects, written for a mathematician

Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read? Or say I want to learn about Einstein's theory of relativity, but I don't even know much ...
11 votes
4 answers
460 views

Autobiographies and correspondences of mathematicians [duplicate]

Lately I have enjoyed reading several autobiographies and correspondences of mathematicians. I'd like to find more, so I thought I'd ask here which others you have come across and enjoyed. P.S. I have ...
2 votes
2 answers
616 views

Easy to explain conjectures that are still unsolved [duplicate]

Mathematics has many open conjectures which are ridiculously hard to even understand. But this is not always the case. An example is: Collatz conjecture. I would like to see some more examples. So ...
41 votes
6 answers
2k views

Are there "natural" sequences with "exotic" growth rates? What metatheorems are there guaranteeing "elementary" growth rates?

A thing that consistently surprises me is that many "natural" sequences $f(n)$, even apparently very complicated ones, have growth rates which can be described by elementary functions $g(n)$ ...
23 votes
14 answers
4k views

Math talk for all ages

I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
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 ...
36 votes
5 answers
4k views

Advice for researchers outside academia

Perhaps some personal background is relevant to this question. A couple of years ago, I graduated with a master's degree in Applied Mathematics from a good Dutch university. Even though I obtained ...
18 votes
4 answers
603 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 ...
Tim Phalange's user avatar
36 votes
4 answers
2k views

Online, evolving, collaborative foundational text projects

There are two online, evolving, collaborative "foundational text" projects for research mathematicians that I am aware of: (1) The Stacks Project for algebraic geometry (2) Kerodon for ...
58 votes
82 answers
18k views

Prominent non-mathematical work of mathematicians

First of all, sorry if this post is not appropriate for this forum. I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things. Recently 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
28 votes
1 answer
2k views

Recent uses of applied mathematics in pure mathematics

In this answer Yves de Cornulier mentioned a talk about the possible uses of persistent homology in geometric topology and group theory. Persistent homology is a tool from the area of topological data ...
6 votes
1 answer
471 views

The connections between Kolmogorov complexity and mathematical logic

We know that Kolmogorov Cmplexity (KC) has connections to mathematical logic since it can be used to prove the Gödel incompleteness results (Chaitin's Theorem and Kritchman-Raz). Are there any other ...
Jori's user avatar
  • 179
146 votes
18 answers
13k views

Suggestions for special lectures at next ICM

(I am posting this in my capacity as chair of the ICM programme committee.) ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's ...
43 votes
10 answers
5k views

What are some examples of proving that a thing exists by proving that the set of such things has positive measure?

Suppose we want to prove that among some collection of things, at least one of them has some desirable property. Sometimes the easiest strategy is to equip the collection of all things with a measure, ...
91 votes
14 answers
8k views

Time-saving (technology) tricks for writing papers

I have over the years learned some tricks which saves a lot of time, and I wish I had known them earlier. Some tricks are LaTeX-specific, but other tricks are more general. Let me start with a few ...
5 votes
1 answer
842 views

Books to develop a deep understanding of Algorithmic Information Theory?

I'm mathematical physicist working with hydrodynamics modelling. Recently, I had to turn to modelling of flows with particles and some questions I have I think are related to Algorithmic Information ...
163 votes
38 answers
27k views

Short exact sequences every mathematician should know

I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An ...
11 votes
11 answers
1k views

Lattices on classical combinatorial families

I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs. I am mosty interested in lattices ...
Martin Rubey's user avatar
  • 5,533
42 votes
11 answers
6k views

Important (but not too well known) inequalities

After seeing the question Important formulas in combinatorics, I thought it might be of interest to have a similar list of inequalities, although not restricted to combinatorics. As with that list, ...
0 votes
1 answer
320 views

Examples of additive categories [closed]

I already this question here but I didn't get any satisfactory answer, so I will try in MO now. There are a lot of interesting and creative examples of categories, such as for example, the category ...
6 votes
1 answer
563 views

Explanations simple enough that non-mathematical audiences can understand [closed]

The following (debatable) quote is attributed to Einstein: "You do not really understand something unless you can explain it to your grandmother." I feel very enlightened when there is a ...
77 votes
15 answers
13k views

Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
163 votes
46 answers
31k views

Every mathematician has only a few tricks

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...

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