# Tagged Questions

**0**

votes

**0**answers

27 views

### integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting.
Let $M$ be a smooth curve over an algebraically closed field. Let ...

**0**

votes

**1**answer

82 views

### Dimension of Inverse image

Suppose we have a smooth map between two smooth manifolds $f:M→N$ such that $\dim M>\dim N$, and let $p∈N$ be a critical value. Suppose we know that $X:=f^{-1}(p)$ is a differentiable manifold (or ...

**8**

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**0**answers

109 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere $S^4$ cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_0)\to(\mathbb{R}^4,\omega_0)$. A ...

**-3**

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**0**answers

113 views

### Learning math from the very beginning with no previous knowledge [on hold]

I didn't do any math like calculus, functions, vectors, etc, not even in high school. I want to build my math knowledge up from the ground up. A friend recommended that I start with Principia ...

**1**

vote

**0**answers

37 views

### on reductive monoids which are gorenstein

Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore ...

**2**

votes

**2**answers

77 views

### Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...

**13**

votes

**1**answer

391 views

### Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$?
Any ...

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**0**answers

173 views

### Geometric meaning of the black hole horizon

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...

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**0**answers

32 views

### Sign of Laplacian in a N-Dimensional Space Under Other Specific Conditions [on hold]

Consider that $f: \mathbb{R}^N \to \mathbb{R} $ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$:
$$
...

**5**

votes

**2**answers

152 views

### Are Anderson $T$-motives motives for the function field analogy?

this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question.
Let $\mathbb{C}_{\infty}$ be the function field analog of ...

**-1**

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**0**answers

12 views

### 3D Vector projection on a Plane [migrated]

I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ...

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**0**answers

112 views

### excess intersection theory

Can the excess intersection theory be applied to the following problem:
I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, ...

**2**

votes

**1**answer

273 views

### Paper by Mumford

In the paper of "The spectrum of difference operators and algebraic curves",
by P. van Moerbeke and D. Mumford, Acta Mathematics, vol.143, 1979,
(link: ...

**3**

votes

**2**answers

151 views

### A question on certain elliptic PDE

Consider the elliptic PDE "CR"
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are similar to the Cauchi ...

**2**

votes

**1**answer

138 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

**8**

votes

**1**answer

180 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**7**

votes

**1**answer

80 views

### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...

**9**

votes

**1**answer

436 views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...

**4**

votes

**1**answer

274 views

### Differential Geometric Aspects of Rubber Bands

What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only ...

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vote

**0**answers

92 views

### Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...

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votes

**3**answers

240 views

### Can the Einstein Field Equations be written as Difference Equations? [on hold]

Does anyone know if the Einstein Field equations have ever been written as Difference Equations, and if so does that simplify anything or produce solutions not available in the usual Differential ...

**6**

votes

**0**answers

183 views

### Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...

**4**

votes

**1**answer

191 views

### Realizing algebraic curves as complete intersections

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).
Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in ...

**8**

votes

**1**answer

472 views

### Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...

**1**

vote

**0**answers

139 views

### varieties whose canonical bundle has finite order in Pic?

Is there a structure theorem for such varieties?
If X is a smooth and proper/projective variety whose canonical bundle $\omega_X$ has finite order in the Picard group, do we know anything about X?
...

**5**

votes

**1**answer

143 views

### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...

**5**

votes

**1**answer

137 views

### Existence of affine hulls

(This question is inspired by Matthieu Romagny's answer to my previous question about base change properties of affine hulls.)
Given a scheme $S$, it is well-known (cf. EGA I.9.1.21) that the ...

**-1**

votes

**1**answer

35 views

### Parallel transport along a reparametrized geodesic

Let $M$ denote a Riemannian manifold, $\gamma$ a geodesic of $M$ defined on $\mathbb{R}$. Let $t_{0} \in \mathbb{R}$ and $(\alpha,\beta) \in \mathbb{R}^{2}$. I define the reparametrized geodesic ...

**1**

vote

**0**answers

53 views

### Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...

**12**

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**0**answers

213 views

### Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...

**0**

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**0**answers

36 views

### Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m - 1 }
$$
on variables $X_1,\dots,X_N$. Consider the analytic manifold $V(I)$ defined by the ideal $I$ in
...

**1**

vote

**0**answers

77 views

### DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...

**4**

votes

**2**answers

350 views

### Are linear algebraic groups rigid?

The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's ...

**0**

votes

**1**answer

201 views

### irreducible etale cover of a blowup

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.
a.Does there exist a collection of smooth morphisms ...

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votes

**0**answers

210 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...

**-3**

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**0**answers

64 views

### Euclidean geometry problems for thirs year students [closed]

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...

**2**

votes

**2**answers

208 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**9**

votes

**0**answers

117 views

### Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
...

**5**

votes

**0**answers

242 views

### Counterexamples to Elkik's theorem in the non-Noetherian case

Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that:
Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$.
That is, if ...

**9**

votes

**4**answers

459 views

### A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
Families of similar shapes, such as squares, are closed in this metric. Also more general ...

**1**

vote

**2**answers

74 views

### Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...

**2**

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**0**answers

82 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

**1**

vote

**1**answer

107 views

### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where ...

**2**

votes

**0**answers

84 views

### Focal points for the exponential map and Jacobi fields

It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...

**3**

votes

**1**answer

231 views

### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...

**0**

votes

**0**answers

122 views

### Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...

**0**

votes

**0**answers

7 views

### Minimum height of convex area, with minimum area dependant on number of fixed length sides [migrated]

I've come across a problem while coding that can be solved simply, to an adequate standard, but I'm sure has a very interesting perfect solution.
The problem is as follows:
Given N lines of ...

**0**

votes

**0**answers

96 views

### Galois descent of stacks

Suppose $X$ is a algebraic stack over a field $F$. Consider a Galois finite extension $F'$ of $F$. Let $X_{F'}:=X\times_F F'$ be the base extension.
Suppose $X_{F'}$ is isomorphic to an algebraic ...

**0**

votes

**1**answer

87 views

### linear section of codimension $k+1$ of a variety of dimension $k$

Let $X$ be a variety of dimension $k$ and degree $d$. If $L$ is a linear subspace of codimension $k+1$ such that $|L\cap X|$ consists of a finite number of points, is there a way to find the maximum ...

**0**

votes

**1**answer

71 views

### Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.
(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)
Assume we ...