Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.

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272 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
14
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435 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
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346 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
9
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194 views

What is the most cited mathematical paper?

Just out of curiosity: What is the paper with the largest number of citations in all of mathematics? I think it is Shannon's A Mathematical Theory of ...
8
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0answers
134 views

Known exact values of the $\operatorname{Li}_3$ function

We know some exact values of the trilogarithm $\operatorname{Li}_3$ function. Known real analytic values for $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
7
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97 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
6
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48 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
6
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0answers
98 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
6
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561 views

Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
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43 views

Books with “project”-like questions

I'm looking for a big list of resources for advanced undergraduate - beginning graduate (and even beyond, really) with a particular feature. Namely, I really like solving "project"-like problems that ...
5
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0answers
112 views

What are some great graduate textbooks with solutions in the back to the problems?

I can think of Aubin's A Course in Differential Geometry, as well as Knapp's books. Any other great ones you know of? Especially in the GSM series from AMS (blue and yellow covers).
5
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0answers
124 views

The mathematical heritage of Lewis Carroll

Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix ...
5
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0answers
121 views

Known algebraic geometry's results in Characteristic $p$

What are the most well known results in classical (à la Weil) algebraic geometry in characteristic $p$, which are thought to be true (but not yet proved) in characteristic 0? Thanks
5
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332 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
4
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57 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
4
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0answers
2k views

Lists of open problems in set theory

Are there any publicly available lists of open problems in set theory besides the following ones? (And if so, what are they?) http://www.math.wisc.edu/~miller/res/problem.pdf ...
3
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83 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
3
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0answers
101 views

Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
3
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0answers
99 views

Undergraduate Schools for the Mathematically Inclined

I'm a rising senior and working on generating a list of colleges to apply to, but it seems to me that (with few notable exceptions) my two main criteria are mutually exclusive. Are there any schools ...
3
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73 views

Math for kids with Cuisenaire rods

I work with kids and i am searching some cool stuff to do with Cuisenaire rods. Thinking about an application i thought that i can show to my students what will be the sum of first $N\in\mathbb{N}$ ...
3
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53 views

Basis-dependent results on matrices

It is common fashion to try to formulate results about matrices in a basis-free way, using linear algebra. What are some good examples of situations where this is impossible? I illustrate what I have ...
2
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0answers
27 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
2
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0answers
25 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
2
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0answers
51 views

GRE Mathematics Practice Exams

I will be taking the subject test in the near future. Can you recommend me some sources (online or print) from which I can find realistic practice exams? I would like to get my hands on as many ...
2
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0answers
60 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
2
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0answers
62 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references? Edit: Something in the spirit of http://arxiv.org/abs/1403.3006, but not necessarily for functions of ...
2
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84 views

Deep questions in number theory not accessible by combinatorial results

Number theory and arithmetic geometry were invented to solve many questions about properties of numbers. What are the some of the foundational results or estimates that are accessible to powerful ...
2
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0answers
99 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
2
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0answers
46 views

Methods to prove axiom independence

What methods have been used to prove the independence of axioms? For instance, in many abstract algebra books the axiom of choice is stated to be independent of all the other axioms of set theory, but ...
2
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0answers
88 views

Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ ax^2+bxy+cy^2+dx+ey+f=0 $$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
2
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0answers
50 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
2
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0answers
49 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
2
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0answers
504 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
2
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0answers
53 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
2
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0answers
84 views

Books similar to “Primes of the form $x^2+ny^2$”

Are there any other books which are similarly to the book "Primes of the form $x^2+ny^2$"? Basically, I want a book which starts with a very important classical problem ( in this case which primes can ...
2
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0answers
154 views

Properties about Matrices that can be proved by only using Block Multiplication of Matrices

I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
2
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0answers
225 views

A rigorous book (or preferrably set of notes) on classic multivariable calculus-analysis?

This is different to (Theoretical) Multivariable Calculus Textbooks as I want a classical treatment of line and surface integrals without the notion of a differential form. Prerequisites: Paths, ...
2
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0answers
244 views

Motivating questions for some topics in undergraduate calculus

Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other ...
2
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0answers
270 views

Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
2
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0answers
106 views

List of large classes of functors providing morphisms

I recently posted this question on mathoverflow and it was closed as being too localized. I am hoping to more precisely say what I mean here. I recently learned, through my Topology coursework, that ...
1
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0answers
57 views

What are some really cool problems that involve “least squares and Eigenvalue problems”?

I am required to find a research topic in this domain, so I'm really interested in finding out what kind of problems are covered in this domain, and how others are using these techniques to solve ...
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0answers
30 views

Other Diophantine problems that use a Pell equation

What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like, $$20^2+21^2 = 29^2$$ These are ...
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0answers
166 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
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0answers
47 views

Methods for evaluating polynomials quickly

I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion. For example, one of my favorite methods is the "nested form of a ...
1
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0answers
48 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
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0answers
60 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
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0answers
110 views

What is a Toy Model for the mathematician's practice? Definition and examples

Wikipedia says Toy model (physics): "In physics, a toy model is a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also ...
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0answers
114 views

Equivalence relation over groups $a\asymp_sb :\rightarrow\exists n\in\Bbb Z:as^n=b$: terminology and decision problem

Let's define this relation over the elements of an infinite group $(G,\cdot,e)$ $$a\asymp_sb :\rightarrow\exists n\in\Bbb Z(as^n=b)$$ where $a^n$ is defined as follow 1)$a^0=e$ 2)$a^{n+1}=aa^n$ ...
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0answers
74 views

Important integral inequalities list.

What are the most important and usefull integral inequalities? I know Chebyshev and Schwarz. Google search provides very few results, mathworld doesn't provide a list.
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0answers
27 views

Practical determinations of trigonometric identities

I am looking for articles, or any reference, that detail practical determinations of trigonometric identities, with particular emphasis on trigonometric functions raised to the power of 3 or higher. ...