MathOverflow Community Digest

Top new questions this week:

Can the nth projective space be covered by n charts?

That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$? I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...

reference-request at.algebraic-topology gn.general-topology gt.geometric-topology differential-topology  
asked by Saúl Rodríguez Martín 33 votes
answered by Steven Landsburg 34 votes

Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?

Consider the hierarchy of relative geometric constructibility by straightedge and compass. Namely, given a geometric figure $B$, a set of points in the plane, we define that geometric figure $A$ is ...

ac.commutative-algebra fields geometric-constructions  
asked by Joel David Hamkins 28 votes
answered by Pace Nielsen 15 votes

Is there a noncommutative Gaussian?

In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not ...

pr.probability free-probability  
asked by Meow 14 votes
answered by Terry Tao 21 votes

The character table of the symmetric group modulo m

Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$. Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...

rt.representation-theory linear-algebra matrices symmetric-groups characters  
asked by Mare 11 votes
answered by Mark Wildon 19 votes

Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$

This is inspired by this Math.SE question, for $a=1$. Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series $$\sum_{n=1}^\infty \frac{(2+\...

sequences-and-series irrational-numbers  
asked by Clement C. 11 votes

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...

fa.functional-analysis cv.complex-variables harmonic-analysis taylor-series  
asked by André Henriques 11 votes
answered by Christian Remling 12 votes

Is it ever unnecessary to mathematically formalize a concept?

From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics. In all of the cases ...

pr.probability lo.logic gm.general-mathematics machine-learning  
asked by Alginpeter 10 votes
answered by Iosif Pinelis 14 votes

Greatest hits from previous weeks:

Are there other nice math books close to the style of Tristan Needham?

I've been very positively impressed by Tristan Needham's book "Visual Complex Analysis", a very original and atypical mathematics book which is more oriented to helping intuition and insight than to ...

books math-communication big-list  
asked by Marco Disce 163 votes
answered by L J 45 votes

Best algebraic geometry textbook? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...

ag.algebraic-geometry books big-list textbook-recommendation  
asked by sanokun 223 votes
answered by Javier Álvarez 217 votes

Nonequivalent definitions in Mathematics

I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research. By that I mean discovering that in different ...

big-list definitions  
asked by Angeliki Koutsoukou Argyraki 131 votes
answered by Gerry Myerson 159 votes

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

books ca.classical-analysis-and-odes big-list textbook-recommendation real-analysis  
asked by Ryan 35 votes
answered by lhf 31 votes

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...

nt.number-theory elliptic-curves diophantine-equations rational-points  
asked by alex alexeq 151 votes
answered by Michael Stoll 117 votes

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

ag.algebraic-geometry nt.number-theory intuition motivation exposition  
asked by James D. Taylor 288 votes
answered by Marty 183 votes

Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g....

applications mathematics-education popularization big-list  
asked by Scott Morrison 193 votes
answered by Terry Tao 57 votes

Can you answer these questions?

For any initial ideal $I$ of the ideal of maximal minors, is it true that $I^n = I^{(n)}$?

Let $X$ denote a generic $n \times m$ (with $n \leq m$) matrix and $R = k[X]$, where $k$ is any field. Let $J := I_n (X)$. It is well-known that $J^t = J^{(t)}$ for all $t$ (where $-^{(t)}$ denotes ...

ac.commutative-algebra determinantal-ideals  
asked by Rellek 1 vote

Topology of densest graphs whose optimal $3D$-matching can be calculated efficiently

let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$. Question: has $G:=\...

combinatorial-optimization extremal-graph-theory matching-theory  
asked by Manfred Weis 1 vote

Normal bundle of a Fano threefold as Brill-Noether loci

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...

ag.algebraic-geometry moduli-spaces derived-categories fano-varieties  
asked by user41650 1 vote
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