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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337 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 ...
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725 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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514 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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136 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
7
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0answers
178 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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0answers
119 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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0answers
96 views

Using multiple integrals for tough single integrals

I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ So, I am wondering if ...
6
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0answers
123 views

Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
6
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0answers
59 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
110 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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0answers
769 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 ...
5
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77 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
5
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0answers
48 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
119 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
140 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
129 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
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41 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
4
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0answers
3k 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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0answers
146 views

Comprehensive Linear Algebra Text

Occasionally I come across a fact from linear algebra that I have not seen before. These facts are often obscured in search engines by either introductory texts or unrelated papers, and it is ...
3
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0answers
122 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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186 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 ...
3
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145 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
147 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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0answers
79 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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56 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
63 views

Algebra 2 Test-Taking Shortcuts

Intro: I will be taking an Algebra 2 multiple choice final very soon. The material covered will be, a review from Algebra 1, composition of functions, variation, polynomial division, logarithms, ...
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19 views

Analogies between finite groups and Lie groups

I believe there are some striking analogous facts between finite groups and Lie groups. One analogue almost too basic to mention is the appropriate notion of subobjects. In elementary group theory ...
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28 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
146 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
70 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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100 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
186 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
votes
0answers
138 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
125 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
57 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 ...
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0answers
62 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
875 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
54 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
91 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
164 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
267 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
255 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
319 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
111 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 ...
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0answers
26 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
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0answers
40 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear ...
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0answers
37 views

Evaluating definite integrals using complex contour integrals

In most cases one only has to consider the complex function where we put $z$ instead $x$. For example, calculating the integral: $$\int_{0}^{\infty}\frac{dx}{x^{4}+1}$$ Here i just integrate the ...
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0answers
15 views

Examples of generalized geometric series.

I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices ...
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0answers
65 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
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70 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...