**4**

votes

**1**answer

345 views

### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...

**5**

votes

**0**answers

75 views

### What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory

I would like a reference/argument for the truth/falsity of the following statement:
The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification ...

**2**

votes

**0**answers

19 views

### Maximizing a convex bounded function of a PSD matrix

Let $f(X)$ be convex and continuous function , with $X$ a PSD matrix.
Assume that under the affine set of constraints $\mathcal{A}(X)=b$ and the convex constraint $f(X)\le1$ there is an optimal, ...

**8**

votes

**1**answer

154 views

### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
...

**64**

votes

**48**answers

9k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**0**

votes

**0**answers

29 views

### Systems of linear modular equations with unknowns in the moduli

I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be:
$A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$
where A ...

**5**

votes

**1**answer

403 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...

**0**

votes

**0**answers

19 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**2**

votes

**0**answers

50 views

### Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...

**6**

votes

**1**answer

170 views

### Minimal immersions of the 2-sphere

Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his ...

**0**

votes

**0**answers

19 views

### References on Lorentzian geometry with non-vanishing torsion [on hold]

For my thesis I have to study Lorentzian geometry with non-vanishing torsion. Do you know any references on this? 'Riemannian geometry' with non-vanishing torsion will also be usefull.

**1**

vote

**1**answer

353 views

### When does $R [x]/I $ has infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...

**1**

vote

**0**answers

126 views

### Constructing special holomorphic functions

I would appreciate any help with this question as I am not sure how I should approach it.
Suppose $ D$ is the unit disk and that $A(x)$ is a positive continuous function on $D$.
Does there exist a ...

**7**

votes

**2**answers

360 views

### asymptotic for the number of involutions in GL(n,2)

Is it known how the number of involutions in $GL_n(2)$, the group of $n\times n$ matrices over $\mathbb{Z}/2\mathbb{Z}$, behaves as $n\to\infty$ ?
Equivalently, one may ask this for the number of ...

**1**

vote

**0**answers

147 views

### the first chern class of complex vector bundles [on hold]

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$).
Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class
$c_1(\xi^\mathbb{C})$ is zero ...

**2**

votes

**1**answer

185 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**2**

votes

**1**answer

162 views

### A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow.
Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...

**8**

votes

**2**answers

201 views

### Is this a functor on the category of $C^{*}$ algebras?

The category of $C^{*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...

**7**

votes

**0**answers

33 views

### Does non-stablity imply that there is a difference between non-forking and coheir extension

Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$.
Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.
Also ...

**96**

votes

**17**answers

23k views

### Mathematical software wish list

Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has ...

**3**

votes

**1**answer

112 views

### about transverse complete intersection

There are several questions about transverse complete intersection arising from L. Guth's paper:
http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html
We say a polynomial ...

**8**

votes

**0**answers

199 views

+150

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

**3**

votes

**0**answers

111 views

### Deformation and Representations

Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...

**2**

votes

**0**answers

60 views

### How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?

For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...

**20**

votes

**2**answers

622 views

### Antirandom reals

This is a crossposting of http://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...

**3**

votes

**0**answers

76 views

### Is this limit of a sequence of sets correct? [closed]

The following calculation of a limit of a sequence of sets according to https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior has been claimed to be wrong in MSE ...

**10**

votes

**0**answers

224 views

### When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...

**4**

votes

**1**answer

171 views

### Zeta-Determinant Theorem

Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...

**194**

votes

**97**answers

32k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**6**

votes

**0**answers

89 views

### Cutting a piece of cake that $n$ people value as exactly $w$

Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...

**232**

votes

**72**answers

89k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**5**

votes

**0**answers

227 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...

**206**

votes

**67**answers

100k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**3**

votes

**1**answer

211 views

### variation on an exact sequence of logarithmic differentials

Let $X$ be a smooth projective complex variety and $D$ a divisor with simple normal crossings on $X$, with irreducible components $D_i$. If $D_1$ is one of these (smooth) irreducible components, then ...

**85**

votes

**7**answers

4k views

### 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 ...

**2**

votes

**0**answers

178 views

### One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ...

**0**

votes

**2**answers

88 views

### Ergodic automorphisms of a compact metric abelian group are Bernoulli

In the literature, such as in this article, it is proved that every ergodic automorphism of a compact metric abelian group is Bernoulli. A rotation is not isomorphic to a Bernoulli shift because it ...

**4**

votes

**2**answers

388 views

### $C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.
I ...

**114**

votes

**69**answers

22k views

### Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...

**43**

votes

**4**answers

2k views

### How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...

**5**

votes

**1**answer

174 views

### Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...

**158**

votes

**41**answers

60k views

### A single paper everyone should read? [closed]

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.
Do ...

**153**

votes

**35**answers

35k views

### Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...

**129**

votes

**36**answers

23k views

### Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...

**8**

votes

**1**answer

230 views

### Iterated sumset inequalities in cancellative semigroups

This question is motivated by the following well-known theorems:
Thm (Plünnecke): If $A$ is a finite nonempty subset of an abelian group, then for every $n$ we have $|A^n| \le ...

**1**

vote

**1**answer

176 views

### Counting Roots of Unit

Let $p\left( x\right) =%
%TCIMACRO{\tprod \limits_{k=1}^{m}}%
%BeginExpansion
{\textstyle\prod\limits_{k=1}^{m}}
%EndExpansion
\left( x^{e_{k}}-\omega_{k}^{e_{k}}\right) $ be a polynomial with
...

**137**

votes

**27**answers

17k views

### What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose A is an abelian group such that every ...

**88**

votes

**20**answers

12k views

### Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public.
Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...

**123**

votes

**43**answers

69k views

### Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...

**33**

votes

**7**answers

4k views

### Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...