9
votes
2answers
133 views
Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?
Assume for this question that ZF set theory is sound.
Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF p …
1
vote
0answers
11 views
In what rigorous sense are Sperner’s Lemma and the Brouwer Fixed Point Theorem equivalent?
I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which a …
0
votes
0answers
35 views
Finite rank free modules over PIDs
I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules
$\phi:M\r …
2
votes
2answers
51 views
A measure of closure under sumset?
Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = …
5
votes
1answer
82 views
Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R …
5
votes
1answer
67 views
Constructing Polynomial Count Varieties
I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$:
Are all reductive algebraic groups strongly polynomial-count?
Are products of strongly poly …
3
votes
0answers
41 views
Iterated Tangent Category Construction
We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions …
1
vote
1answer
114 views
About the curvature of a connection?
In "Lectures on gauge theory and integrable systems" of M.Audin, she identifies the space of conections $\mathcal{A}$ on the trivial bundle $G\times S$ ($G$ Lie group, $S$ surface …
1
vote
1answer
54 views
Equivariant versus retractive spaces: a reference request
Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G. …
17
votes
14answers
803 views
objects which can’t be defined without making choices but which end up independent of the choice
It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure …
17
votes
1answer
371 views
How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?
This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question wi …
-2
votes
0answers
112 views
Are there any precise results about the intuition behind Morse functions?
A Morse funnction on a smooth manifold is usually intuitively interpreted as follows: Imagine the manifold to be a mountainous landscape and the Morse function as the elevation of …
1
vote
0answers
26 views
identity for number of monomials
Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$.
Let b(n,k) denote the number of monomials of degree kd-(n+1) in n+1 variables x_0,..x_n with …
2
votes
0answers
31 views
Categorical notions involving $\ell_p$ spaces.
First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\ …
7
votes
1answer
119 views
Are small knots generic?
A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ cross …
0
votes
2answers
56 views
Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something a …
0
votes
1answer
87 views
enumerative Gromov-Witten invariants
Let $X$ be a sympletic manifold and $A\in H_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers.
Assume that $$\mathcal{M}_{g,k}(X;A)$$
is dense in
$$\overline{\mathcal{M}} …
2
votes
2answers
133 views
Help with this system of Diophantine equations
A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm tr …
1
vote
3answers
242 views
Help with this Diophantine equation
Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at http://mathoverflow.net/questions/131353/help-with-this-sys …
0
votes
0answers
57 views
Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?
I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously: …
0
votes
2answers
122 views
What does a singular simplex with real coefficient mean
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which repre …
2
votes
1answer
308 views
Probability $k$ bins are non-empty.
The following problem arises in the analysis of Bloom filters.
Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at …
42
votes
2answers
4k views
Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture
Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville:
“The big experts in the field had
already tried to make this approach
w …
0
votes
2answers
78 views
Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?
Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain c …
1
vote
0answers
17 views
h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the …
2
votes
0answers
38 views
Duality between K-theory and K-homology in the non-compact, spin$^c$ case
Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms …
1
vote
1answer
65 views
Non-(stable)-triviality of the tautological bundles
This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann m …
1
vote
0answers
37 views
Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Her …
1
vote
1answer
98 views
fundamental class is the sum of simplices of triangulation of the manifold?
M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable tr …
0
votes
0answers
44 views
Circle segment of exact length [closed]
I need to find:
a spherical line
that passes through the points 0,0 and 8,8
and the distance of the line between those two points must be exactly 12
I imagine the answer will b …
