Tagged Questions
This tag is used if a reference is needed in a paper or textbook on a specific result.
10
votes
3answers
544 views
Maryam Mirzakhani's works
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...
1
vote
0answers
35 views
K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?
For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the ...
2
votes
1answer
65 views
Exact reference for Liouville theorem
It seems hard for me to find that the solution of the following equation
$$
\Delta u+e^u=0
$$
defined on a simply-connected domain $D\subset R^2$ must be of form
$$
...
8
votes
4answers
163 views
Application of Fraïssé construction in set theory
As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...
2
votes
0answers
23 views
Reference that contains examples of absolutely indecomposable representations of quivers over a finite field
Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
3
votes
3answers
349 views
Category of Gödel Codings? [Reference Request]
Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...
2
votes
0answers
59 views
Errata for Getzler-Kapranov “Cyclic operads and Cyclic homology”
Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...
3
votes
1answer
160 views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
1
vote
0answers
66 views
Integral Cohomology of Symmetric Groups
Does anybody know a reference for the explicit description of the integral cohomology ring of $S_5$ and $S_6$. I can not find them anywhere in the internet. For $S_4$, I found C. B. Thomas's nice ...
1
vote
0answers
131 views
On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and ...
0
votes
0answers
59 views
In search of a preprint by Litherland
I've seen the following citation a lot: "R. LITHERLAND: A formula for the Casson-Gordon invariants of a knot, preprint." I can't seem to find a corresponding publication. [Added in edit: apparently ...
3
votes
0answers
52 views
Replacing functors by topologically or simplicially enriched functors
I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, ...
0
votes
0answers
103 views
Motivation of $a_p$ for non-CM elliptic curves [on hold]
For an elliptic curve $E$ without CM let $\overline{E}$ be the good reduction of $E$ modulo $p$ prime. The value $a_p = p+ 1 - \#\overline{E}(\mathbb{F}_p)$ is referenced by DDT on p.19 and Ribet ...
2
votes
0answers
72 views
Density of smooth functions in Sobolev space, respecting nonnegative traces
I am looking for a reference for the following result (if it is true):
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be sufficiently regular.
Let ...
8
votes
1answer
227 views
A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves
Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
2
votes
0answers
103 views
Fixed area, largest mass — is there a name?
Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as
...
14
votes
2answers
487 views
What other books are like these?
A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident ...
-2
votes
0answers
80 views
Reference Request for Finite Axiom Of Choice [on hold]
I am looking for a book that would contain the kind of proofs that were given in the answers to this question: Finite axiom of choice: how do you prove it from just ZF? because I want to quote them in ...
-1
votes
2answers
211 views
Serre's Theorem for Coherent Sheaves
I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...
1
vote
0answers
104 views
Characterization of the Riemann curvature tensor
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...
1
vote
0answers
79 views
Strichartz estimates for the wave equation
Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
7
votes
2answers
227 views
Definition of internal field objects
Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...
2
votes
0answers
61 views
Uniqueness of solution of elliptic equation with exponential nonlinearity
Consider the following equation
$$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$
where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...
6
votes
0answers
63 views
Vanishing of Dolbeault cohomologies and Steinness
That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
5
votes
0answers
156 views
Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?
A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for this result?
For the sake of being definite, let's say that
“citeable” means ...
4
votes
1answer
182 views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...
2
votes
0answers
53 views
Potentiality classes and Borel reductions
In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ ...
2
votes
1answer
164 views
Blowing-up the Grassmannian at a point
Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl_V(G)$ should mean.
Of course for affine ...
7
votes
1answer
158 views
On the coherence theorem for bicategories
The coherence theorem for bicategories, as usually stated, reads
Any bicategory $B$ is biequivalent to a (strict) 2-category.
It is possible to give an explicit construction of the ...
5
votes
1answer
149 views
Uncountably categorical theories which are interpretable in a strongly minimal
Definition: Let $\lambda$ be a cardinal. An $\mathcal{L}$-theory $T$ is called $\lambda$-categorical whenever every two models of $T$ of cardinality $\lambda$ are isomorphic.
Definition: An ...
3
votes
0answers
196 views
Galois correspondence subgroups/subsystems
In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
By applying this result to finite groups, we get a Galois correspondence ...
0
votes
0answers
19 views
exit time of a non degenerate diffusion
Let $n, d \geq 1$, $b : \mathbb{R}^n \to \mathbb{R}^n$ and $\sigma: \mathbb{R}^n \to \mathbb{R}^{n \times d}$ two Lipschitz functions.
We assume that
\begin{equation}
\exists \mu >0, \xi^T ...
6
votes
0answers
148 views
Tree property and singular strong limit cardinals
I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:
Theorem. It is consistent, relative to the existence of large cardinals, that ...
1
vote
0answers
119 views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...
9
votes
2answers
413 views
Non-Forcing and Independence
I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here.
Do there exists sentences which are independent of ZFC, cannot be shown to ...
2
votes
1answer
111 views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ...
8
votes
1answer
207 views
3D models of the unfoldings of the hypercube?
There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the
tesseract, into 3D.1
These unfoldings (or "nets") are analogous to the 11 unfoldings of
the 3D cube into the plane.2
...
3
votes
1answer
209 views
Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
2
votes
2answers
171 views
boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$
Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
5
votes
1answer
120 views
Symmetry type of non-cohomological automorphic forms
By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...
0
votes
0answers
79 views
When an integral with respect to a Poisson point process is finite?
Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
2
votes
1answer
214 views
What happens to the cohomology ring after a “flip-flop”?
I've been trying to understand what happens to the cohomology ring (say with coefficients in $\mathbb{R}$) of a smooth complex projective manifold after blowing up along a smooth complex submanifold. ...
8
votes
3answers
214 views
How do small central extensions drop the dimension of a faithful representation?
Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...
8
votes
2answers
247 views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...
5
votes
1answer
148 views
Analytic perturbation of eigenfunctions
Consider a domain $\Omega_0 \subset \mathbb{R}^n$, and deformations of $\Omega_0$, called $\Omega_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is ...
7
votes
1answer
225 views
A proper smooth surface is projective
My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this well-known ...
7
votes
2answers
509 views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
1
vote
0answers
39 views
How often can a single length occur as a boundary distance?
Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$?
We can make any assumptions about the ...
3
votes
0answers
120 views
Prime zeta zeros - reference
Is there an online repository for zeros of the prime zeta function? I looked at the Yahoo group
Prime numbers and primality testing listed on the MathWorld notebook for the prime zeta function, but ...
1
vote
0answers
80 views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...
