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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35
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12answers
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 ...
65
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9answers
6k 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 ...
20
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9answers
1k 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 ...
8
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2answers
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
0answers
128 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
0answers
118 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,...
43
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10answers
10k 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?...
19
votes
1answer
602 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) ...
12
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29answers
4k 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?...
20
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17answers
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 ...
20
votes
1answer
397 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
6answers
580 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 ...
155
votes
15answers
8k 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 ...
42
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4answers
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 ...
9
votes
1answer
373 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 ...
115
votes
22answers
7k 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
4answers
364 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 ...
1
vote
2answers
285 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 ...
29
votes
4answers
833 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 $g(n)$ ...
20
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14answers
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
0answers
87 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 ...
33
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5answers
3k 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 ...
0
votes
0answers
174 views

Consequences of Gauss class number problem

What are the consequences of Gauss Class number problem other than being able to answer the question of writing a prime in the form $x^2 + ny^2$ for every $n$? A related question has been asked here ...
18
votes
4answers
566 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 ...
33
votes
4answers
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 ...
51
votes
73answers
14k 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
0answers
180 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 ...
27
votes
1answer
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 ...
5
votes
1answer
272 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 ...
131
votes
17answers
10k 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
9answers
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, ...
4
votes
1answer
168 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 ...
111
votes
31answers
12k 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 ...
12
votes
11answers
793 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 ...
40
votes
11answers
4k 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
1answer
187 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
1answer
534 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 ...
66
votes
14answers
10k 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 ...
147
votes
46answers
25k 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 ...
54
votes
15answers
4k views

Request for examples: verifying vs understanding proofs

My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
5
votes
5answers
1k 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 ...
69
votes
30answers
6k 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 ...
21
votes
7answers
1k views

Examples of improved notation that impacted research?

The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work. I am aware that there is a related post ...
14
votes
5answers
1k 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 ...
6
votes
0answers
240 views

Interesting things you learned while grading/marking? [closed]

What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
11
votes
3answers
632 views

Series and sequences in physical systems & closed form expressions

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
0
votes
1answer
914 views

Do mathematicians ignore mathematical works from non-mathematicians? [closed]

Is it true that mathematicians ignore and do not like to take a look at or comment on any mathematical work or manuscript from a person outside the field of mathematics (meaning is not a professional ...
42
votes
10answers
4k views

List of long open, elementary problems which are computational in nature

I would like to ask a question of a similar vein to this question. Question: I'm asking for a list of long open problems which are computational in nature which a beginning graduate student can ...
5
votes
1answer
257 views

Abstract mathematical concepts/tools appeared in machine learning research

I am interested in knowing about abstract mathematical concepts, tools or methods that have come up in theoretical machine learning. By "abstract" I mean something that is not immediately related to ...
66
votes
7answers
17k 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....

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