## 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

### 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

### 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

### 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

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

### 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

## 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

### 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

### 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

### 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

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