MathOverflow Weekly Newsletter - Tuesday, April 28, 2015

Top new questions this week: Is the set $ AA+A $ always at least as large as $ A+A $? Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ... nt.number-theory co.combinatorics additive-combinatorics   asked by Oliver Roche-Newton 58 votes answered by Terry Tao 18 votes Volume of the unitary group I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following: $$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} ... reference-request mp.mathematical-physics moduli-spaces   asked by Max 25 votes answered by user25309 24 votes Can topological cyclic homology compute Picard groups? Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism $$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$ where $Pic(\mathcal{O}_K)$ is the ... at.algebraic-topology algebraic-k-theory   asked by Craig Westerland 18 votes answered by Oscar Randal-Williams 10 votes An unfair marriage lemma I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem: Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ... reference-request co.combinatorics graph-theory matching-theory   asked by Sergei Ivanov 18 votes answered by bof 14 votes A combinatorial question about orthonormal bases Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ... co.combinatorics   asked by Tom Goodwillie 12 votes answered by James Cranch 24 votes Mysterious identity between numbers of odd/even meander systems Definitions: An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting closed semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ... co.combinatorics meanders   asked by მამუკა ჯიბლაძე 11 votes answered by Gabriel C. Drummond-Cole 6 votes For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$? It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ... nt.number-theory diophantine-approximation transcend.-number-theory   asked by Jordan 11 votes answered by Will Sawin 13 votes Greatest hits from previous weeks: Geometric Interpretation of Trace This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ... linear-algebra matrices traces   asked by B. Bischof 135 votes answered by Rado 65 votes Dividing a square into 5 equal squares Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper. mg.metric-geometry discrete-geometry tiling polygons   asked by sanz 9 votes answered by Q.Q.J. 33 votes Can you answer these? How to prove that a projective module is not free? Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ... ra.rings-and-algebras noncommutative-algebra noncommutative-rings   asked by Joakim Arnlind 6 votes Are Bökstedt's THH manuscripts available? In many papers dealing with topological Hochschild homology, the original unpublished manuscripts by Bökstedt are cited. To name one example, in McClure and Staffeldt's On the topological Hochschild ... at.algebraic-topology   asked by lentic catachresis 5 votes The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to ... co.combinatorics projective-geometry finite-geometry   asked by Bob Smith 5 votes

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