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

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32 views

Counting number of points in a lattice with bounded length

I am interested in counting number of lattices using the following theorem. The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, ...
2
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1answer
35 views

Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture)

What is a good source for Silver's proof (or a more modern version) that Con($\exists \omega_1$-Erdos cardinal) implies Con(Chang's Conjecture)? Silver's original proof seems to have never been ...
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0answers
32 views

Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?

It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e. $$ ...
8
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1answer
112 views

Thales' semicircle theorem in higher dimensions

Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle. Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle ...
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0answers
57 views

Square-free sieve over number fields

I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
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0answers
46 views

orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$. I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1})$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates. For ...
6
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2answers
322 views

Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...
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0answers
56 views

Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
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1answer
111 views

Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...
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5answers
981 views

Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...
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0answers
37 views

Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
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48 views

Cyclic structure on a balanced (or ribbon) monoidal category

As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of ...
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47 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
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0answers
72 views

References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...
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1answer
69 views

Source for equations involving congruences of Fibonacci and Lucas numbers

In a paper of Cohn (see here), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and ...
4
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1answer
210 views

Differential forms along the fiber

Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...
0
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1answer
136 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
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1answer
88 views

Formal definition of arithmetic transfinite recursion

Recently I have been trying to find the definition of the subsystem $ATR_0$ of second-order arithmetic. Only "definitions" I have found were quite vague, like informal definition on Wikipedia which ...
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0answers
54 views

Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...
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0answers
47 views

$\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup with Langlands ...
1
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1answer
46 views

Nested convex optimization

Suppose I have a convex optimization problem of the form $$\min_x f(x) ~~s.t.\\x\in X$$. Say that $f(x)$ and its (sub)gradient are not given in a closed form, but are determined by solving a convex ...
3
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1answer
279 views

Notation: Categories of measur(abl)e spaces

Is there a common notation in the literature for the category of measurable spaces and measurable maps? the category of measure spaces and measure-preserving maps? The nlab suggests ...
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0answers
46 views

Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
2
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1answer
159 views

Mal'cev “rational equivalence” and model theory

In Universal Algebra, it is possible to say that two presentations denote the "same" kind of algebraic structures, if the two corresponding varieties are "rationally equivalent" (Mal'cev 1958). In ...
3
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1answer
193 views

Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$

Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
2
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1answer
128 views

What is the “type” of a contact vector field?

Let $(M,\theta)$ be a $(2n+1)$-dimensional contact manifold, $\mathcal{C}=\ker\theta$ the contact distribution, and $X\in\mathcal{C}$ a vector field belonging to $\mathcal{C}$. In a couple of minor ...
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0answers
138 views

When do we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?

Let $(X,\mathcal{O}_X)$ be a scheme (or more generally a ringed space). We know that in general the derived category of complexes of quasi-coherent modules $D(\text{Qcoh}(X))$ is not equivalent to the ...
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0answers
228 views

Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange, but I did not receive any responses, but a short comment. So, I decided to post it here, hoping to receive answers from experts. ...
2
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0answers
73 views

Residual scheme to local complete intersection schemes in the projective space

Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection ...
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0answers
24 views

On a remark regarding to initial-boundary elliptic estimate

In the paper "On the motion of the free surface of a liquid" see [CL], the authors proved the initial boundary elliptic estimate (proposition 5.28) $$ ||\nabla^r q||_{L^{2}(\Omega)}+||\nabla^{r-1} ...
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0answers
43 views

Control of Hessian by its trace in a bounded domain

We know if $u:\mathbb{R}^n \to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by parts, that $$ \|\Delta u\|=\|\nabla^2u\| \tag 1 $$ where $||\cdot||$ is ...
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1answer
236 views

$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$

Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$? These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
2
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1answer
92 views

An alternative definition of pseudo-coherent complex

Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...
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0answers
23 views

Orbifold metric and ample line bundle

Let $X$ be a complex projective variety with only orbifold singularities, and $L$ be an ample line bundle on $X$. Is there a K\"{a}hler metric in the orbifold sense representing the first Chern class ...
2
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1answer
39 views

Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma\in (0,1)$

I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold. Any references are appreciated. PS I am ...
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1answer
314 views

Is $\mathcal M _{g,n}$ anabelian?

Are the moduli spaces $\mathcal M _{g,n}$ expected to be anabelian? Is there anything known in that direction? Thank you very much for your help in any case!
4
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1answer
121 views

Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...
4
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1answer
206 views

Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of ...
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28 views

Riemannian simplicial complex and quasi-conformal complex

In this paper by Robert Young, the author defines We define a riemannian simplicial complex to be a simplicial complex with a metric which gives each simplex the structure of a riemannian ...
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1answer
117 views

Vector Bundles of small rank

I recently started the study of vector bundles on $\mathbb{P}^n$, and started to read Rao's article 'A family of vector bundles on $\mathbb{P}^3$'. There, there is a notion of spectrum of a vector ...
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1answer
46 views

Reference for multivariate orthogonal polynomials

I want to learn about multivariate orthogonal polynomials. Is there a good textbook/survey that you could suggest? I need to see common examples like Jack's polynomials etc .. and also general ...
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0answers
137 views

Morgan Shalen compactification of $\mathbb C^2$

I'm reading the Otal's survey on the compactification of Morgan Shalen. (available here) He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...
2
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1answer
116 views

Frey's Formula and utilisation of the Hasse Invariant in “Links between Stable elliptic curves and Diophantine equations.”

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then ...
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0answers
31 views

Tail inequality for orthomartingales/martingale difference random fields

It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale, then for each $ \beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the ...
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1answer
130 views

Is the Poincaré metric continuous with respect to the domain?

Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ ...
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1answer
80 views

global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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0answers
83 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
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1answer
79 views

Samuel multiplicity

Let $X$ be the hyper-surface defined by $$f:=\sum_{i=1}^k x_i^n=0$$ in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal $$I=(x_1^{n-1},\dots , x_k^{n-1}) $$ What is ...
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1answer
224 views

'Unitary' charts on odd-dimensional spheres

Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific ...
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1answer
76 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...