## Top new questions this week:

### Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds? A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...

gn.general-topology point-set-topology

### On a drawing in Dixmier's Enveloping Algebras

This image comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes'). Dixmier writes that The curves shown on p. XIV have their origin in the study of U(sl(3)). They are ...

rt.representation-theory lie-groups lie-algebras

### What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...

oa.operator-algebras big-list gm.general-mathematics

### Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...

dg.differential-geometry rt.representation-theory differential-operators

### Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?

In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to ...

nt.number-theory lo.logic set-theory

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$ been enumerated? E.g., for $n{=}2$, there are $f(2)=3$ such matrices: $$\left( \begin{array}{cc} 1 & 0 \\ 0 ... co.combinatorics matrices finite-groups  asked by Joseph O'Rourke 13 votes  answered by Gerhard Paseman 2 votes ### Is \sum_{k=1}^{n} \sin(k^2) bounded by a constant M? I know \sum_{k=1}^{n} \sin(k) is bounded by a constant. How about \sum_{k=1}^{n} \sin(k^2)? real-analysis sequences-and-series  asked by npbool 12 votes  answered by Terry Tao 22 votes ## Greatest hits from previous weeks: ### why study Lie algebras? I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ... dg.differential-geometry lie-groups lie-algebras differential-equations  asked by Olivier Bégassat 57 votes  answered by Deane Yang 74 votes ### Examples of common false beliefs in mathematics The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ... big-list mathematics-education  asked by gowers 452 votes  answered by Tilman 372 votes ## Can you answer these? ### Interesting triple integral Some time ago I stumbled on an alleged identity$$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...

ca.analysis-and-odes integration

### Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...

dg.differential-geometry riemannian-geometry clifford-algebras