The examples tag has no wiki summary.

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**1**answer

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### What are “good” examples of string manifolds?

Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...

**27**

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**4**answers

3k views

### What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...

**5**

votes

**3**answers

209 views

### two essentially different concretizaions

It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete ...

**2**

votes

**1**answer

177 views

### Non-uniruled variety with level one Hodge structure.

I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.

**2**

votes

**2**answers

666 views

### Non-split groups

I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom

**3**

votes

**2**answers

995 views

### Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...

**37**

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**26**answers

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### 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 ...

**46**

votes

**46**answers

16k 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 ...

**46**

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**28**answers

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### 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, ...

**27**

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**3**answers

2k views

### Wanted: example of a non-algebraic singularity

Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...

**0**

votes

**0**answers

213 views

### Example of function with a certain behavior.

Let $f: R \rightarrow R$. Consider the following properties:
$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$
$$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$
$(2)$ - There is a ...

**4**

votes

**1**answer

348 views

### Examples of “inner products” of parallel morphisms in a dagger category

There is a very interesting abstract notion of the trace of an endomorphism $f : c \to c$ of an object $c$ in a braided monoidal category (although the symmetric case is easier): see, for example, ...

**3**

votes

**2**answers

430 views

### higher order structure by higher order derivatives

Anyone recall a structure determined by a 3rd order partial derivative?
not the general nth order of recent Baranovsky

**5**

votes

**2**answers

680 views

### The rank of a not necessarily finitely generated module.

This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...

**4**

votes

**1**answer

1k views

### An example of a rank one projective R-Module that is not invertible

Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...

**6**

votes

**0**answers

440 views

### Principal $G$-bundles as fully extended TQFTs, and $n$-representations

This is a follow up to this MO question: Fully dualizable objects in classical field theories
Assuming the notation there (which in turn come from Topological Quantum Field Theories from Compact Lie ...

**49**

votes

**6**answers

3k views

### Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications

If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...

**8**

votes

**2**answers

389 views

### Cohen-Macaulay domain with non-Cohen-Macaulay normalization?

Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...

**10**

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**2**answers

899 views

### Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
The modular curves $X(n)$. They are ...

**2**

votes

**3**answers

755 views

### Algebraic structures of greater cardinality than the continuum?

Are there interesting algebraic structures whose cardinality is greater than the continuum? Obviously, you could just build a product group of $\beth_2$ many groups of whole numbers to get to such a ...

**4**

votes

**5**answers

824 views

### Appearances of 'exotic' compact Lie Groups

The structure theorem for compact Lie Groups states that all compact Lie groups are finite central quotients of a product of copies of $U(1)$ and simple compact Lie groups. And yet, as easy as ...

**4**

votes

**1**answer

636 views

### On the Existence of Certain Fourier Series

Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S_{n}(f)$ satisfies $\|S_{n}(f)\|_{L^{1}(T)} \rightarrow \|f\|_{L^{1}(T)}$ but $S_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?

**6**

votes

**3**answers

340 views

### Non-trivial integral forms of algebras

Suppose $\mathcal{A}$ is a $\mathbf{C}$-algebra then an integral form would be a subring $\mathcal{B} \subset \mathcal{A}$ such that the canonical map $\mathcal{B} \otimes_{\mathbf{Z}} \mathbf{C} ...

**8**

votes

**6**answers

804 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 ...

**6**

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**10**answers

885 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 ...

**4**

votes

**2**answers

4k views

### Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...

**8**

votes

**2**answers

592 views

### Statements forced by one condition of a poset, but not the whole thing

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...

**6**

votes

**10**answers

956 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 ...

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votes

**7**answers

2k views

### Applications of the notion of of Gromov-Hausdorff distance

I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...

**60**

votes

**32**answers

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### What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...

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votes

**2**answers

8k views

### Simple bijection between reals and sets of natural numbers

Using the Cantor–Bernstein–Schröder theorem, it is easy to prove that there exists a bijection between the set of reals and the power set of the natural numbers. However, it turns out to be difficult ...

**10**

votes

**4**answers

938 views

### Exotic principal ideal domains

Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that ...

**2**

votes

**1**answer

243 views

### Is there a category with a subobject classifier but which is not finitely complete?

This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned ...

**3**

votes

**4**answers

1k views

### Interesting examples of flasque sheaves?

Does anyone know any interesting examples of flasque sheaves? Ideally, I would like to see one that both arises naturally and is geometric in some sense. On the other hand, I know so few examples ...

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**3**answers

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### Example of a projective module which is not a direct sum of f.g. submodules?

This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...

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votes

**6**answers

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### Counterexamples in algebraic topology?

In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...

**17**

votes

**6**answers

1k views

### Nonfree projective module over a regular UFD?

What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...

**13**

votes

**5**answers

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 ...

**5**

votes

**9**answers

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### Examples of two different descriptions of a set that are not obviously equivalent?

I am teaching a course in enumerative combinatorics this semester and one of my students asked for deeper clarification regarding the difference between a "combinatorial" and a "bijective" proof. ...

**2**

votes

**2**answers

444 views

### Is there a non-trivial example for a 1-homogeneous function satisfying a specific inequality of second order?

Let $\mathbb{R}^n$ be the $n$-dimensional real vector space with Cartesian coordinates $x=(x^1,\ldots, x^n)\in \mathbb{R}^n$. I'm searching for a non-trivial example of a function $A:\mathbb{R}^n ...

**4**

votes

**3**answers

561 views

### Holomorphic function with a.e. vanishing radial boundary limits

Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...

**5**

votes

**2**answers

630 views

### Simple example of a sequence without computable modulus of convergence

Can anyone give a simple example of a sequence that converges, but there's no computable function that gives $N$ as a function of $\epsilon$, i.e., the modulus of convergence is not computable?
In ...

**21**

votes

**3**answers

2k views

### Clearing misconceptions: Defining “is a model of ZFC” in ZFC

There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with ...

**10**

votes

**1**answer

597 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 ...

**4**

votes

**3**answers

1k views

### Nonessential use of large cardinals

In Awfully sophisticated proof for simple facts, we are asked for examples of complex proofs of simple results. To quote from the questioner's post, we are asked for proofs that are akin to "nuking ...

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**3**answers

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### Non finitely-generated subalgebra of a finitely-generated algebra

Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...

**6**

votes

**2**answers

705 views

### Simple examples of equivariant homology and bordism

I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...

**32**

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**2**answers

2k views

### Difficult examples for Frankl's union-closed conjecture

Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...

**27**

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**13**answers

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### 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 ...

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votes

**3**answers

7k views

### Advanced Math Jokes [closed]

I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and ...