54
votes
54answers
9k views

Books you would like to see translated into English.

I have recently been told of a proposal to produce an English translation of Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and this prompts me to ask a more general question: ...
73
votes
62answers
23k views

Which mathematicians have influenced you the most? [closed]

There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can ...
24
votes
2answers
4k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
10
votes
0answers
220 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
4
votes
0answers
283 views

How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments. Let $X$ be an algebraic variety over ...
2
votes
1answer
193 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...
4
votes
1answer
119 views

Asymptotics of Fresnel integrals

It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ...
79
votes
10answers
11k views

Work of plenary speakers at ICM 2014

The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM ...
48
votes
14answers
5k views

How to write popular mathematics well?

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...
47
votes
28answers
4k views

Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
80
votes
17answers
17k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...
7
votes
0answers
400 views

Evidence that Graph Isomorphism problem is not $NP$-complete

Graph isomorphism problem is one of the longest standing problems that resisted classification into $P$ or $NP$-complete problems. We have evidences that it can not be $NP$-complete. Firstly, Graph ...
52
votes
8answers
5k views

Have you solved problems in your sleep? [closed]

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
32
votes
6answers
2k views

Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...
34
votes
2answers
4k views

How did “normal” come to mean “perpendicular”?

How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to ...
5
votes
3answers
1k views

Why are we interested in the Fundamental Groupoid of a Space?

The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$. ...
19
votes
63answers
12k views

What's your favorite equation, formula, identity or inequality? [closed]

Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?
38
votes
30answers
45k views

What programming languages do mathematicians use? [closed]

I understand this might be a slightly subjective question, but I am honestly curious what programming languages are used by the mathematics community. I would imagine that there is a group of ...
24
votes
4answers
1k views

Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal. Question 1. Does every set of reals contain a measure-zero subset of the same ...
0
votes
1answer
292 views

Number of Dyck paths with k returns and b peaks

The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by $$\frac{k}{2n-k}{2n-k \choose n}.$$ The number of Dyck paths from the ...
0
votes
1answer
199 views

The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads. Let's rewrite the product in a way ...
87
votes
16answers
11k views

What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this ...
81
votes
10answers
9k views

Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
10
votes
1answer
748 views

Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...
57
votes
22answers
15k views

What's the “best” proof of quadratic reciprocity?

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
81
votes
19answers
12k views

Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that ...
6
votes
2answers
381 views

Relationship between Hochschild cohomology and Drinfeld centers

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$. I was reading nlab's entry on Hochschild cohomology ...
43
votes
0answers
3k views

What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...
12
votes
2answers
370 views

How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...
120
votes
6answers
41k views

Where to buy premium white chalk in the U.S., like they have at RIMS?

While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek. At RIMS (in Kyoto) in 2005, they had the best white ...
1
vote
1answer
178 views

Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define ...
5
votes
1answer
233 views

New series for $1/\pi$ based on Ramanujan's ideas

In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in ...
4
votes
2answers
345 views

Topological retraction vs categorical retraction

Let $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a topological retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have ...
59
votes
7answers
9k views

Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...
2
votes
0answers
145 views
+50

Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...
6
votes
4answers
317 views

Breaking up the free Lie algebra into Gl irreps

The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra , the free Lie algebra generated by any choice of ...
8
votes
2answers
583 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
77
votes
9answers
6k views

Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
66
votes
14answers
19k views

Top specialized journals

In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA. What journals ...
17
votes
1answer
1k views

Quantitative lower bounds related to Zhang's theorem on bounded gaps

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ ...
25
votes
1answer
705 views

Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$ with $i>0$. Does there always exist a variety $Y$ and a ...
68
votes
8answers
8k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
48
votes
11answers
5k views

What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
10
votes
3answers
953 views

When did coordinate plane “as we know it” come into play?

This is a historical question that needs some background to make sense. Let me start with the longer version of the question: When did negative numbers, algebra and coordinate plane come together? ...
9
votes
2answers
404 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)

Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian ...
6
votes
1answer
238 views

Intersections in almost complex manifolds

Question: Suppose $(M,J)$ is an almost complex manifold, and $X$ and $Y$ are two almost complex submanifolds (i.e. $J(T(X)) \subset T(X)$ and $J(T(Y)) \subset T(Y)$). Then must $X \cap Y$ also be an ...
3
votes
0answers
162 views

Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies: $X$ has a finite number of singularities on $S^{2}$ The equator is invariant under flow of $X$ 3.$g_{*}X=\pm X$ where $g$ ...
0
votes
0answers
103 views

Nontrivial norm values in rings of integers

Let $L=Q(\sqrt{d})$ for some SQF integer $d\equiv_4 3$ (the same can be asked for $d\equiv_4 1$). In this case the ring of integers of $L$ is $O_L=\{x+\sqrt{d}y \mid x,y\in Z\}$ so the norms of $O_L$ ...
18
votes
2answers
2k views

Where are Georg Cantor's Original Manuscripts?

Georg Cantor is famous for introducing transfinite numbers and set theory. A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...
15
votes
1answer
459 views

Lower-Hölder embeddings of the sphere

My question is very simple: Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that $$ |f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r, $$ for ...

15 30 50 per page