MathOverflow Weekly Newsletter - Tuesday, March 31, 2015

Top new questions this week: Interplay between Loop Quantum Gravity and Mathematics It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ... ag.algebraic-geometry at.algebraic-topology dg.differential-geometry mp.mathematical-physics   asked by Bilateral 20 votes answered by Carlo Beenakker 5 votes Identities for power series like $\sum_n z^{n^3}$ Probably, one of the first power series that every mathematician encounter is the geometric series $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ Also, a particular ... reference-request power-series theta-function   asked by Fry 19 votes Mathematicians wearing hats on arbitrary total orders I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$: Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat ... co.combinatorics lo.logic   asked by Adam P. Goucher 17 votes answered by Joel David Hamkins 25 votes Enriques surfaces over $\mathbb Z$ Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the ... ag.algebraic-geometry nt.number-theory algebraic-surfaces k3-surfaces   asked by Will Sawin 16 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 15 votes answered by Terry Tao 26 votes Thales' semicircle theorem in higher dimensions Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle. Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle ... reference-request mg.metric-geometry   asked by Joseph O'Rourke 14 votes answered by Will Sawin 7 votes (Non)existence of mirrors with more than two foci Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ... dg.differential-geometry mg.metric-geometry problem-solving   asked by Thibaut Demaerel 13 votes answered by Douglas Zare 10 votes Greatest hits from previous weeks: Not especially famous, long-open problems which anyone can understand Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list ... big-list open-problem open-problems-list   asked by David Feldman 168 votes answered by Yemon Choi 106 votes 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 Can you answer these? Mass Transportation Through Wonderful Roller There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B. Wonderfulness of roller comes from this property ... co.combinatorics differential-equations   asked by Ghiasi M 6 votes On a permutation module for GL(n,q) Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ... gr.group-theory rt.representation-theory   asked by Benjamin Steinberg 6 votes Can integers be distorted to make primes more regular? Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as $$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ... nt.number-theory prime-numbers   asked by Yaakov Baruch 5 votes

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