## Top new questions this week:

### Nelson's proof of Liouville's theorem

The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in ...

ho.history-overview harmonic-functions

### Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind)

Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like? I realize that this question probably falls outside of the scope ...

math-philosophy

I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in ...

career

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...

dg.differential-geometry gr.group-theory rt.representation-theory lie-groups euclidean-lattices

### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

nt.number-theory analytic-number-theory computational-complexity computation

### Table of (integral) cohomology groups of K(Z,n)

Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients? It seems like a natural counterpart to the table of the homotopy groups of spheres, ...

homotopy-theory cohomology

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

real-analysis fourier-analysis fourier-transform

## Greatest hits from previous weeks:

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

big-list mathematics-education

### What are some important but still unsolved problems in mathematical logic?

In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...

lo.logic soft-question big-list open-problem open-problems-list

### Computing the ordinal of a rational language well-partially-ordered by the subword relation

Let $\Sigma$ be a finite set or "alphabet", $\Sigma^*$ the free monoid on $\Sigma$ or set of "words". If $w,w'\in \Sigma^*$, write $w\leq w'$ when $w$ is a "subword" of $w'$, i.e., can be obtained by ...

co.combinatorics order-theory formal-languages ordinal-numbers

### The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in ...

lo.logic set-theory large-cardinals axiom-of-choice
Please see edit below. So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models ...