# All Questions

**2**

votes

**0**answers

10 views

### Graphs with prescribed numbers of k-cliques

Let $(a_0,a_1,\dots, a_n)$ be a sequence of non-negative integers.
Q. When does there exists a graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_0$ vertices, $a_1$ edges, ...

**1**

vote

**0**answers

35 views

### Product of Bruhat Cells

Fix a $(B,N)$ pair (Tits system) of a semisimple Lie group $G$. Let $u$ and $v$ be two Weyl group elements such that $l(uv)=l(u)+l(v)$. It is known that $BuvB=(BuB)(BvB)$ (see for example Humphreys's ...

**0**

votes

**0**answers

14 views

### stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g.
$ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$
$ \nabla \cdot u = 0 \; \text{in} \; \Omega$
$ u = 0 \; \text{on} ...

**0**

votes

**0**answers

8 views

### symmetry of a square - is it possible pure geometric approach in didactics?

Consider a square : four points in a plane constructed with classical means (compass and straightedge). Since no point is different from others (no coordinates, no labels...) it seems that we can not ...

**1**

vote

**1**answer

98 views

### How many the distinct linear factors of $f(x)-f(y)$ can be for f in Q[x]?

Let $f \in \mathbb{Q}[x]$.
Let $S(f)$ denote the number of distinct linear factors
of $f(x)-f(y)$.
$S(f)$ is bounded by $\deg(f)$.
Q1 Is $S(f)$ bounded by constant?
Q2 Is it possible ...

**2**

votes

**0**answers

54 views

### Convergence in trace

Let $A$ and $B$ be two self-adjoint, positive definite Compact operators on a Hilbert space $\mathcal{H}$. Further, let $A$ be trace class. Define $C_n \equiv AB(\frac{I}{n} + BAB)^{-1}$. Does ...

**-1**

votes

**1**answer

87 views

### Prove this conjecture inequality 2 [on hold]

Let $n$ be postive integer,I conjecture
$$(1+2n)^n\ge 1^n+2^n+4^n+6^n+\cdots+(2n)^n \tag{1}$$
This problem when I solve this equation
$$(1+2n)^n=1^n+2^n+4^n+6^n+\cdots+(2n)^n\tag{2}$$
if this ...

**0**

votes

**1**answer

52 views

### $Ext$ functor over a product of groups

Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 ...

**0**

votes

**1**answer

132 views

### On the quadratic reciprocity law? [on hold]

In the Quadratic Reciprocity Law
$$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$.
Is there any relation between $x$ and $y$ ...

**2**

votes

**0**answers

57 views

### Ideals of $L^1(G)$

I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?

**-3**

votes

**0**answers

84 views

### Is the Poincare duality still non-degenerate when restricted to subgroups of algebraic cycles?

On smooth varieties over the complex numbers there are conjectures (1) the usual Hodge conjecture (generalized Hodge conj. of level 0) (2) generalized Hodge conjecture of level 1 (3) standard ...

**6**

votes

**1**answer

325 views

### Normalization of complete intersection

Let $A$ be an integral complete local ring over a field which is complete intersection.
Let $B$ be a normalization of $A$.
Q. Is $B$ Gorenstein?
I guess that even the normalization of Gorenstein ...

**3**

votes

**1**answer

121 views

### Alternative parallel paths

There are $n$ non-intersecting strings (with ends $x_1,\dots, x_n$ and $y_1,\dots, y_n$). An additional string intersects the first $n$ strings somehow. All the intersections are simple (vertices of ...

**12**

votes

**2**answers

511 views

### Lattice n-gons with ordered side lengths 1,2,3,…,n

Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8).
Are there other (infinitely many) polygons, such as this, lying entirely in ...

**7**

votes

**23**answers

3k views

### A search for theorems which appear to have very few, if any hypotheses [closed]

I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.
By "few hypotheses" I ...

**62**

votes

**15**answers

7k views

### Mathematical research published in the form of poems

The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...

**28**

votes

**5**answers

2k views

### (Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...

**17**

votes

**4**answers

1k views

### Splitting Pythagorean triples

Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would
be least surprised ...

**15**

votes

**1**answer

2k views

### What are the monomorphisms in the category of schemes?

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...

**22**

votes

**2**answers

2k views

### What is the algebraic closure of the field with one element?

If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the ...

**10**

votes

**2**answers

919 views

### Jacobi's equality between complementary minors of inverse matrices

What's a quick way to prove the following fact about adjugates of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only ...

**14**

votes

**2**answers

539 views

### Which groups have strictly rational representations?

It can be shown, via the construction of the representations of the symmetric group, that every representation of $S_n$ is equivalent to a representation with values in $\mathbb{Q}.$ Presumably, this ...

**12**

votes

**1**answer

1k views

### How much mathematics has been formally verified?

That's a vague question so allow me to tighten it up a bit.
I recently noticed that there is a formal machine verified proof of the Central Limit Theorem (CLT) implemented with Isabelle. This ...

**9**

votes

**1**answer

263 views

### Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...

**-2**

votes

**0**answers

27 views

### Graph Isomorphism of Graphs that are not Locally Triangle-Free

What is the computational complexity of graph isomorphism problem for graph-class $\mathcal{G}$ that excludes graphs which are locally triangle-free ?
Is this $\mathcal{G}$ class included in any ...

**6**

votes

**0**answers

38 views

### Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds

Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...

**30**

votes

**2**answers

3k views

### Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...

**9**

votes

**2**answers

625 views

### Folner sets and balls

Several related questions were asked before on MO, but it is not clear to me if the following was settled.
Given a finitely generated amenable group, is it always possible to find some finite ...

**17**

votes

**3**answers

787 views

### Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:
Sets and functions, due to Lawvere.
Modules over some ...

**14**

votes

**8**answers

6k views

### Suggestions for good books on class field theory

Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...

**18**

votes

**5**answers

1k views

### is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...

**27**

votes

**1**answer

3k views

### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...

**18**

votes

**3**answers

1k views

### Does Con(ZF) imply Con(ZF + Aut C = Z/2Z)?

How many field automorphisms does $\mathbf{C}$ have? If you assume the axiom of choice, there are tons of them -- $2^{2^{\aleph_0}}$ I believe. And what if you don't -- how essential is the axiom of ...

**3**

votes

**1**answer

235 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...

**60**

votes

**23**answers

26k views

### Open problems with monetary rewards

Since the old days, many mathematicians have been used to attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express ...

**44**

votes

**30**answers

6k views

### Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...

**1**

vote

**1**answer

292 views

### When can we write fundamental units explicitly

Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...

**31**

votes

**3**answers

2k views

### What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...

**25**

votes

**0**answers

720 views

### Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ ...

**25**

votes

**7**answers

3k views

### Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...

**12**

votes

**5**answers

1k views

### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...

**7**

votes

**1**answer

129 views

### Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...

**1**

vote

**1**answer

53 views

### Asymptotic formula for average geodesic length on graph?

If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length?
That is suppose I know two nodes $n_1$ and $n_2$, the number of edges in my graph and at those ...

**97**

votes

**16**answers

24k views

### How do you keep your research notes organized? [closed]

One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the ...

**7**

votes

**2**answers

796 views

### fundamental group and complete invariant of irreducible 3-manifolds

I heard that,by Perelman's work,we can get that the fundamental group is a
complete invariant of irreducible 3-manifolds (except for lens spaces).
can someone help explain this.Thank you!

**1**

vote

**1**answer

164 views

### harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where
What happens in the $p$-adic case? Is there sphere ...

**-1**

votes

**0**answers

70 views

### A question posed by Erdos and Mahler about continued fractions

In 1938, Erdos and Mahler raised the following question:
Let $\xi$ be a real number such that $(p_n/q_n)_n$ are convergents of its continued fraction. If there exists a subsequence ...

**20**

votes

**1**answer

882 views

### Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic,
$$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...

**212**

votes

**7**answers

108k views

### 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 ...

**127**

votes

**33**answers

75k views

### Best Algebraic Geometry text book? (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 ...