MathOverflow Weekly Newsletter
MathOverflow Weekly Newsletter

Top new questions this week:

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 13 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 17 votes
answered by Oscar Randal-Williams 10 votes

A Polynomial With Positive Prime Density

Let $P(x)$ be a non-constant polynomial with real coefficients. Can natural density of $$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$ be positive?

nt.number-theory prime-numbers  
asked by Ghiasi M 17 votes
answered by Terry Tao 21 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 16 votes
answered by user25309 14 votes

Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of $$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...

nt.number-theory co.combinatorics roots-of-unity  
asked by Jens Reinhold 15 votes
answered by Richard Stanley 26 votes

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...

dg.differential-geometry cv.complex-variables complex-geometry several-complex-variables  
asked by MKO 15 votes
answered by Francesco Polizzi 15 votes

Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not ...

ag.algebraic-geometry complex-geometry  
asked by mqx 13 votes
answered by Francesco Polizzi 16 votes

Greatest hits from previous weeks:

Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be ...

ho.history-overview big-list  
asked by Joseph O'Rourke 46 votes
answered by Gerry Myerson 38 votes

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...

asked by Ricardo Menares 87 votes
answered by Mark Bell 9 votes

Can you answer these?

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...

na.numerical-analysis integration approximation-theory spherical-geometry  
asked by Manos 4 votes

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

reference-request fa.functional-analysis ap.analysis-of-pdes sp.spectral-theory operator-theory  
asked by anonymous 3 votes

behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $ ...

dg.differential-geometry ca.analysis-and-odes ap.analysis-of-pdes  
asked by Math604 3 votes
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