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4
votes
5answers
797 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
1answer
631 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
3answers
298 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
5answers
672 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
votes
10answers
826 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
2answers
3k 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
2answers
506 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 ...
4
votes
4answers
639 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 ...
10
votes
7answers
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): ...
43
votes
29answers
9k views

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 ...
5
votes
2answers
6k 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
4answers
887 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
1answer
230 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
4answers
991 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 ...
8
votes
3answers
1k views

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, ...
71
votes
6answers
4k views

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 ...
16
votes
6answers
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 ...
12
votes
5answers
2k 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
9answers
1k views

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
2answers
403 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
3answers
529 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
2answers
550 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 ...
19
votes
3answers
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
1answer
575 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
3answers
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 ...
12
votes
3answers
1k views

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
2answers
678 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 ...
30
votes
2answers
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 ...
16
votes
11answers
5k 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 ...
5
votes
3answers
5k 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 ...
12
votes
3answers
1k views

Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that $$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$ is and exact sequence in $C$. Does it follow that ...
4
votes
4answers
579 views

An example of a non-paracompact tvs (over the reals, say)

What is an example of a non-paracompact topological vector space? I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
10
votes
3answers
348 views

Algebraic Curves and Phase Diagrams of Physical Systems

Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized ...
16
votes
5answers
4k views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...
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 ...
21
votes
2answers
2k views

Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
16
votes
3answers
3k views

Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$? The question closely related to ...
23
votes
11answers
4k views

Motivating the de Rham theorem

In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing ...
4
votes
1answer
686 views

Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
1
vote
1answer
1k views

Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
19
votes
19answers
3k views

Examples of categorification

What is your favorite example of categorification?
29
votes
17answers
5k 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 ...
30
votes
7answers
5k views

Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set). Any countable set ...
4
votes
1answer
322 views

Categories with products that preserve quotients

It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve ...
9
votes
3answers
943 views

Concavity of $\det^{1/n}$ over $HPD_n$.

One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...
117
votes
67answers
36k views

Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
4
votes
0answers
280 views

Example of a Grothendieck pretopology satisfying a weak saturation condition

Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...
13
votes
3answers
795 views

Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
5
votes
3answers
429 views

Nonmetrizable uniformities with metrizable topologies

I'm looking for such pathological examples of uniform spaces which are not metrizable, but whose underlying topology is metrizable. Willard in his General Topology text constructs such a uniformity ...
1
vote
0answers
550 views

Example of smooth, proper but non-projective curve over an affine, connected base?

Would someone please give an example of a smooth, proper but non-projective curve $C/S$, where $S$ is affine and connected? I believe that whatever your example, $C/S$ must have genus $1$, admit no ...