# All Questions

**36**

votes

**3**answers

3k views

### Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...

**9**

votes

**3**answers

649 views

### Fibered knot with periodic homological monodromy

It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act ...

**4**

votes

**1**answer

465 views

### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...

**16**

votes

**3**answers

2k views

### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

**3**

votes

**1**answer

1k views

### self-dual representations

Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of ...

**1**

vote

**1**answer

210 views

### Natural number properties as uninterpreted functions in first order logic

Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so ...

**1**

vote

**4**answers

2k views

### A differentiable approximation to the minimum function

Suppose we have a function $f : \Re^N \rightarrow \Re$ which, given a vector, returns the value of its smallest element. How can I approximate $f$ with a differentiable function(s)?

**9**

votes

**2**answers

319 views

### Monomial orderings in noncommutative setting

An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ...

**2**

votes

**3**answers

1k views

### Homology with Coefficients

We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in ...

**9**

votes

**0**answers

315 views

### Singular chains as an HZ-module spectrum

For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of ...

**0**

votes

**0**answers

42 views

### Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...

**5**

votes

**0**answers

93 views

### Semi-continuity of intersection numbers

I always trusted the following quite vague statement:
If you have a family of divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in ...

**0**

votes

**1**answer

159 views

### Approximation of real numbers

Is there any function $f(x)$, such that for all real $\alpha$ and rational $p/q$
$$\left|\alpha-\frac{p}{q}\right|>\frac{1}{f(q)}.$$
or at least
...

**1**

vote

**1**answer

175 views

### Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...

**3**

votes

**0**answers

115 views

+100

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**0**

votes

**1**answer

69 views

### Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform.
I know that if we perform a tetrad rotation, say of Class III:
$(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, ...

**9**

votes

**1**answer

120 views

### Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...

**47**

votes

**13**answers

10k views

### How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...

**33**

votes

**5**answers

19k views

### Advice for pure-math Phd students [closed]

Pursuing a Phd in pure math can be a daunting task. A number of students who begin a Phd in pure math don't complete it, and there are high-quality dissertations and those which are not so high ...

**4**

votes

**0**answers

98 views

### Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem:
Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where ...

**3**

votes

**1**answer

693 views

### Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus? Is $\lambda x$ the same as $\tau_x$?

Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus ? Whether $\lambda {x}$ is same as $\tau _{x}$ ? Are the rules of Meta-Mathematics (Criteria of Substitution, ...

**1**

vote

**1**answer

142 views

### Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-character variety

Does the following statement:
"Let $G$ be a finitely generated
group and let $X(G)$ be the
$SL(2,\mathbb{C})$-character variety
of $G$. Suppose $X(G)$ contains an
irreducible component ...

**8**

votes

**4**answers

394 views

### On the fixed point of automorphism of F_3[[T]]

Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq ...

**3**

votes

**0**answers

239 views

### Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?

This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...

**6**

votes

**2**answers

93 views

### Relationship between Multiplicative Ergodic Theorems

One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb ...

**7**

votes

**4**answers

388 views

### Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...

**1**

vote

**0**answers

60 views

### extension for a complex operator

Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...

**12**

votes

**1**answer

806 views

### Rationalised K-theory of number fields

Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...

**5**

votes

**2**answers

537 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...

**8**

votes

**3**answers

342 views

### References for Yang-Mills Theory

We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the ...

**4**

votes

**0**answers

42 views

### Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I ...

**10**

votes

**4**answers

2k views

### How to refer to a theorem that you have shown to be wrong

I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards ...

**8**

votes

**6**answers

3k views

### If Erdős is published as Erdös in a paper, which do I cite?

There seems to be a few papers around with Erdős written as Erdös. For example:
MR0987571 (90h:11090) Alladi, K.; Erdös, P.; Vaaler, J. D. Multiplicative functions and small divisors. II. J. Number ...

**1**

vote

**1**answer

284 views

### calculate function from its divizor

There is elliptic curve $C (y^2 = x^3 + Ax + B)$ over $GF(q)$.
There is algebraic function f on C.
We have div(f).
How calculate f as rational function ( $f = (f_1(x) + yf_2(x)) / (g_1(x) + ...

**4**

votes

**0**answers

1k views

### Intersection of subspaces

If you have two linear subspaces $V_1$ and $V_2$ of a vector space $V,$ both given by their bases, there is fairly heavy handed way of computing their intersection: write down the projection matrices ...

**14**

votes

**3**answers

1k views

### Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...

**18**

votes

**0**answers

356 views

### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all
$x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

**0**

votes

**0**answers

54 views

### Find a estimate for quasilinear parabolic equation

I am studying quasilinear parabolic operator:
$Pu=-u_t+u_{xx}+a(x;t;u,u_x)$ where $(x,t)\in \Omega=(0,\pi)\times(0,T)$ and $za(x,t,z,0)\le kz^2+b$
Suppose that $u$ satisfies: $P(u)=0$ in $\Omega.$
...

**1**

vote

**1**answer

100 views

### A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...

**16**

votes

**2**answers

1k views

### Status of Fontaine-Mazur conjecture

In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...

**4**

votes

**1**answer

148 views

### Classification of 2-dimensional Alexandrov spaces

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)?
If yes, a reference would be helpful.
EDIT: If the ...

**42**

votes

**4**answers

7k views

### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...

**31**

votes

**6**answers

3k views

### Why does one think to Steenrod squares and powers?

I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...

**11**

votes

**6**answers

2k views

### Different definitions of the dimension of an algebra

I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...

**3**

votes

**3**answers

1k views

### What is this restricted sum of multinomial coefficients?

It is relatively easy to show that
$$
\sum_{a_1 + \cdots + a_k=\ell} \binom{\ell}{a_1,\ldots,a_k} = k^\ell
$$
where $\binom{\ell}{a_1, \ldots, a_k} = \frac{\ell!}{a_1!\cdots a_k!}$. What can be said ...

**9**

votes

**3**answers

556 views

### What is the optimal size in the finite axiom of symmetry?

Freiling's axiom of symmetry states that if you assign to each real number $x$ a countable set $A_x\subset\mathbb{R}$, then there should be two reals $x,y$ for which $x\notin A_y$ and $y\notin A_x$.
...

**26**

votes

**1**answer

1k views

### Identifying the stacks in Devinatz-Hopkins-Smith

I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what ...

**-4**

votes

**0**answers

81 views

### Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial ...

**-7**

votes

**0**answers

39 views

### Anyone else here believe entangled particles create mini wormholes? [closed]

The properties of their interactions are a little too similar to ignore. Entangled objects ability to align regardless of distance. Their ability to align seemingly even before the other is viewed. ...

**30**

votes

**1**answer

818 views

### Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff ...