All Questions
Tagged with big-list reference-request
111 questions
3
votes
0
answers
270
views
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 ...
2
votes
1
answer
526
views
What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
- 850
19
votes
6
answers
2k
views
Book recommendation introduction to model theory
Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus....
- 9,111
11
votes
4
answers
950
views
Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
- 60.5k
14
votes
5
answers
5k
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Who says understanding physics helps mathematicians? (A reference request) [Take the word "who" literally.]
If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that ...
-8
votes
2
answers
859
views
Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
4
votes
0
answers
95
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 ...
- 3,318
14
votes
2
answers
1k
views
Famous papers published in annotated form?
I very much enjoyed reading through The Annotated Turing which goes through Turing's "On Computable Numbers, with an Application to the Entscheidungsproblem" in careful detail. I saw this ...
- 323
1
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0
answers
198
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A zoo of derivations
Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...
49
votes
2
answers
5k
views
Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
6
votes
0
answers
259
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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 $f:\ M\to k$ ...
- 3,738
1
vote
1
answer
228
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Named sets of permutations
I am looking into interesting subsets of permutations,
and there are several classes of permutations which are named.
For example, there are
Derangements,
Alternating,
Grassmann permutations (at most ...
33
votes
6
answers
7k
views
Most important results in 2022
Undoubtedly one of the news that attracted the most attention this year was the result of Yitang Zhang on the Landau–Siegel zeros (see Consequences resulting from Yitang Zhang's latest claimed ...
1
vote
0
answers
153
views
Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $
I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with ...
- 4,829
17
votes
3
answers
2k
views
Theoretical results on neural networks
With this question I'd like to have a recollection of theoretical rigorous results on neural networks.
I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
16
votes
3
answers
2k
views
What are the main open problems in the theory of amenability of groups?
I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today.
A survey or a list of questions would be welcome.
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 ...
11
votes
9
answers
5k
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. ...
5
votes
0
answers
788
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, ...
23
votes
1
answer
3k
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) ...
14
votes
29
answers
7k
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?...
22
votes
17
answers
5k
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 ...
195
votes
18
answers
17k
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 ...
132
votes
22
answers
11k
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 ...
12
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 ...
- 5,822
0
votes
1
answer
1k
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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 ...
11
votes
1
answer
900
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 ...
13
votes
2
answers
1k
views
Contrasting theorems in classical logic and constructivism
Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples?
What are some examples of most contrasting ...
77
votes
30
answers
6k
views
Atlas-like websites on specific areas of mathematics
In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples:
GroupNames: https://...
19
votes
1
answer
882
views
Importance of textbooks in health of a sub-discipline
I am interested in published articles, and also more informal writing (blog posts, talk slides etc.) which discuss the importance of textbooks (where this word encompasses research monographs etc.) in ...
84
votes
11
answers
12k
views
What are examples of (collections of) papers which "close" a field?
There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...
14
votes
3
answers
3k
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Errata for Bott and Tu's book "Differential Forms in Algebraic Topology"
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel.
Is there a good list of errata for Bott and Tu available? ...
- 315
13
votes
2
answers
1k
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A comprehensive list of random walk inequalities?
I am interested in finding a comprehensive list of all noticeable random walk inequalities.
ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$
I can only seem to find books/papers that list ...
- 231
1
vote
1
answer
312
views
Topological Invariants for Group
Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
user57432
0
votes
0
answers
116
views
Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
81
votes
18
answers
25k
views
What programming language should a professional mathematician know? [closed]
More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable?
If I am right,...
21
votes
6
answers
5k
views
Lebesgue measure theory applications
I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example.
Theorem: Let $X$ be a differentiable submanifold of $\...
- 757
40
votes
11
answers
12k
views
Contemporary philosophy of mathematics
Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in ...
2
votes
0
answers
445
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 ...
196
votes
12
answers
31k
views
Do you know important theorems that remain unknown?
Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an ...
19
votes
2
answers
9k
views
The Ultimate L in a Nutshell: On Descriptive Articles
Everybody who catches a fleeting glimpse of Woodin's central papers on Ultimate $L$ (i.e. Suitable Extender Models I & II), admits that they aren't so tempting for lazy readers who don't like to ...
24
votes
2
answers
3k
views
Roadmap to Hill-Hopkins-Ravenel
How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the paper ...
- 3,309
82
votes
17
answers
11k
views
Examples of algorithms requiring deep mathematics to prove correctness
I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.
5
votes
2
answers
499
views
Critical points in $ZF$ without Choice
Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...
- 6,132
15
votes
5
answers
4k
views
Applications of space filling curves
I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.
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
79
votes
15
answers
9k
views
Sophisticated treatments of topics in school mathematics
Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
9
votes
5
answers
1k
views
List of generic properties of Riemannian metrics
I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
10
votes
3
answers
1k
views
Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
75
votes
22
answers
19k
views
Essays and thoughts on mathematics
Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...