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28
votes
15answers
2k views

What are examples of good toy models in mathematics?

This post is community wiki. A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...
41
votes
46answers
14k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
8
votes
6answers
749 views

Spaces of filters

This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better ...
0
votes
0answers
27 views

Infinite dimension optimization example

I am searching for an example problem for infinite dimension optimization for continuous time control. I have the algorithm tested on real systems but I would like a problem where the Lyapunov ...
0
votes
1answer
80 views

Example of a polycyclic group which is not of polynomial growth? [closed]

The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?
2
votes
1answer
153 views

Divergence invariant lifting of a vector field via a submersion

What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true: For every vector field $X$ on $S^{2}$ there is a non vanishing vector field ...
16
votes
4answers
5k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
7
votes
0answers
184 views

Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
12
votes
3answers
288 views

Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$

The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some ...
0
votes
0answers
51 views

Example of A semigroup with a unique left invariant measure [on hold]

I'm looking for an example of a locally compact semigroup that has a unique left invariant measure. I know that every locally compact group will admit a unique left invariant measere namely the Haar ...
4
votes
3answers
788 views

An example where GCD depends on the domain

First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$. I want to find an example of an GCD-domain $R$, a subdomain $S ...
108
votes
75answers
34k views

Best online mathematics videos?

I know of two good mathematics videos available online, namely: Sphere inside out (part I and part II) Moebius transformation revealed Do you know of any other good math videos? Share.
24
votes
4answers
2k views

Recent, elementary results in algebraic geometry

Next semester I will be teaching an introductory algebraic geometry class for a smallish group of undergrads. In the last couple weeks, I hope that each student will give a one-hour presentation. ...
6
votes
10answers
875 views

Examples of $G_\delta$ sets

Recall that a subset A of a metric space X is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are three ...
32
votes
19answers
6k views

Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
34
votes
24answers
6k views

Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
178
votes
63answers
90k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
14
votes
2answers
887 views

A simplicial complex which is not collapsible, but whose barycentric subdivision is

Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is? Every collapsible complex is necessarily contractible, and subdivision preserves the ...
75
votes
6answers
7k views

Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...
27
votes
13answers
6k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
3
votes
5answers
273 views

Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point

Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points? Using ...
81
votes
42answers
17k views

What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics? There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
27
votes
14answers
4k views

Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
11
votes
2answers
389 views

A sequence of subsets of an infinite group

Is there an infinite group $G$ such that there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ? link
4
votes
0answers
179 views

Examples of unproven but likely true existential sentence (in the sense of incompleteness)

Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all ...
2
votes
0answers
108 views

Is there a smooth rationally connected, proper variety $M$ over $\mathbb{C}$ such that c_1(L)([A]) = 1 which is not projective space?

Is there a smooth, rationally connected, projective variety $M$, which is not a projective space and is equipped with an ample line bundle $L$, such that the homology class of the curves $[A]$ which ...
47
votes
33answers
4k views

Dimension Leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...
0
votes
0answers
103 views

Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition

there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the ...
14
votes
5answers
3k views

Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...
3
votes
1answer
131 views

Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
1
vote
1answer
152 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
6
votes
2answers
193 views

How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be? The reason I am interested in this is that my ...
95
votes
59answers
16k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
10
votes
3answers
587 views

Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
6
votes
16answers
2k views

Vector spaces without natural bases

Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural". To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
61
votes
23answers
22k views

What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
10
votes
1answer
586 views

Nondifferentiability set of an arbitrary real function

A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...
5
votes
2answers
183 views

smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group

I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group. Related questions: Schemes with trivial brauer group and Brauer ...
2
votes
2answers
217 views

Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation: $\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if ...
45
votes
27answers
4k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
2
votes
0answers
238 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...
6
votes
10answers
847 views

Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind: -Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
5
votes
0answers
211 views

Homeomorphisms of product spaces: an example

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...
7
votes
6answers
411 views

Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.) Let $A$ be a set ...
45
votes
28answers
4k views

Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
4
votes
1answer
191 views

Unravelling some hypotheses on a variety

In Le group de Brauer II, Grothendieck states Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] ...
28
votes
5answers
2k views

`Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$.

[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold, say some space ...
4
votes
3answers
143 views

A model with $\kappa$ many automorphism and a rigid element.

The following should be known, but I could not find an example. Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in ...
0
votes
1answer
118 views

Simple but topologizable [closed]

Do you have an example of an infinite simple group with at least 3 distinct group topologies on it‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
11
votes
7answers
2k views

Simple show cases for the Yoneda lemma

I've been given a very simple motivating and instructive show case for the Yoneda lemma: Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...