This tag is used if a reference is needed in a paper or textbook on a specific result.

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6
votes
3answers
472 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
70 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
200 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
88 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
154 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
62 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 ...
8
votes
0answers
306 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, ...
4
votes
1answer
153 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, which I would expect): Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be ...
8
votes
1answer
300 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
115 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 ...
22
votes
4answers
844 views

Communal problem books

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 ...
0
votes
2answers
226 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
128 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
100 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 ...
8
votes
2answers
278 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
80 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 ...
12
votes
1answer
185 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
189 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
210 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
61 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
180 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
167 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
163 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
215 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
25 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
178 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
122 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
451 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
119 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 ...
14
votes
1answer
293 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
220 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
180 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
171 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
86 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
220 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
230 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
263 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
164 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
240 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
536 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
42 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
127 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 ...
2
votes
0answers
86 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 ...
5
votes
1answer
368 views

Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
2
votes
2answers
175 views

Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...
5
votes
1answer
346 views

Boardman-Vogt tensor product

Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category ...
1
vote
0answers
114 views

The fundamental group of the complement of codimension 2 submanifold

Suppose that $M^n$, $V^{n+2}$ are connected, compact smooth manifolds. Let $f\colon M^n\to V^{n+2}$ is a smooth embedding. Let $K_f$ be the kernel of the inclusion induced homomorphism ...
5
votes
1answer
158 views

Did Lucas discover Lucas circles?

MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online ...
2
votes
0answers
129 views

How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc. Where is there written a corresponding formula incorporating the differential of a dg Lie algebra and module?
2
votes
1answer
117 views

Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...