MathOverflow Community Digest

Top new questions this week:

What is the minimal density of a set A such that A+A = N?

Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$? What I know: ...

additive-combinatorics sumsets  
user avatar asked by Zur Luria Score of 22
user avatar answered by Emil Jeřábek Score of 38

Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ... geometric-group-theory free-groups  
user avatar asked by ThorbenK Score of 14
user avatar answered by Carl-Fredrik Nyberg Brodda Score of 21

Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...

nt.number-theory ca.classical-analysis-and-odes analytic-number-theory l-functions dirichlet-series  
user avatar asked by Nomas2 Score of 14
user avatar answered by GH from MO Score of 16

Request for explicit character tables of conjectured, non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...

reference-request finite-groups ho.history-overview characters  
user avatar asked by Sebastien Palcoux Score of 12
user avatar answered by Dave Benson Score of 18

How exactly are realizability and the Curry-Howard correspondence related?

Consider, on the one hand: the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...

lo.logic soft-question computability-theory type-theory realizability  
user avatar asked by Gro-Tsen Score of 12
user avatar answered by Andrej Bauer Score of 8

Is the Tukey order well-founded

Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows: For two ultrafilters $U,W$ on $\omega$, we say ...

set-theory order-theory ultrafilters  
user avatar asked by Tom Benhamou Score of 11
user avatar answered by Gabe Goldberg Score of 10

Can a nowhere locally Hölder function be differentiable almost everywhere?

Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...

user avatar asked by Nate River Score of 10
user avatar answered by user479223 Score of 12

Greatest hits from previous weeks:

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

books ca.classical-analysis-and-odes big-list textbook-recommendation real-analysis  
user avatar asked by Ryan Score of 38
user avatar answered by lhf Score of 32

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

soft-question ho.history-overview big-list mathematics-education online-resources  
user avatar asked by alex Score of 420
user avatar answered by Charles Siegel Score of 69

"Philosophical" meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...

ct.category-theory intuition yoneda-lemma  
user avatar asked by Sam Derbyshire Score of 148
user avatar answered by Urs Schreiber Score of 102

Why does the monster group exist?

Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John: If you were to come back a hundred years after your death, what problem ... finite-groups monster  
user avatar asked by Leibniz's Alien Score of 37

Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...

big-list soft-question recreational-mathematics game-theory  
user avatar asked by Douglas Zare Score of 115
user avatar answered by aorq Score of 96

Theorem versus Proposition

As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(...

soft-question mathematical-writing  
user avatar asked by MRA Score of 64
user avatar answered by user1835 Score of 94

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 ...

reference-request big-list examples intuition alternative-proof  
user avatar asked by Mariano Suárez-Álvarez Score of 397
user avatar answered by Mariano Suárez-Álvarez Score of 552

Can you answer these questions?

Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...

measure-theory hilbert-spaces lebesgue-measure continuity positivity  
user avatar asked by Daniel Goc Score of 1

Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?

The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...

mapping-class-groups equivariant-cohomology equivariant-homotopy  
user avatar asked by Noah Wisdom Score of 2

Roth's Theorem Variations?

One can motivate Roth's theorem as follows. On $[0,1]$ consider the function $f$ that takes $x$ to the cardinality of the set { $p/q : |x - p/q | < 1/q^{2 + \epsilon}$ } . Now one can see that $\...

user avatar asked by David Feldman Score of 1
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