All Questions
153,396
questions
4
votes
1
answer
222
views
Characteristic cycle
Is there a good introduction to characteristic cycle of D modules(or constructive sheaves)? I just encountered this concept recently and I really would like to see some examples of calculations using ...
8
votes
1
answer
650
views
Surgery along an embedded surface in a 4-manifold
Let $X$ be a 4-manifold and $\Sigma_g\subset X$ be an embedded closed orientable genus = $g$ surface. Suppose $\Sigma_g\subset X$ has a trivial closed normal bundle $N(\Sigma_g) = \Sigma_g\times D^2$. ...
9
votes
0
answers
252
views
Almost Poincaré duality
Let $M^n$ be a connected, closed manifold. It has Poincaré duality with $\mathbb{Z}/2$ coefficients $H^k(M;\mathbb{Z}/2)\cong H_{n-k}(M;\mathbb{Z}/2)$, induced by cap product with the fundamental ...
5
votes
1
answer
190
views
Equivariant cohomology defined by restrictions?
Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point.
Let $\alpha \in H^{2}_{S^1}(M)$ be such ...
2
votes
0
answers
74
views
What is the real locus of moduli space of stable weighted points?
A configuration of $n$ weighted points on $\mathbb{C}P^1$ is called \emph{stable} if the sum of the weights of equal points is strictly less than half the total weight. The moduli space in this case ...
3
votes
1
answer
430
views
Differential map of a dominant morphism in char zero
Let $k$ be a field of characteristic zero and $X,Y$ be integral schemes of finite type. Assume we have a dominant morphism $\pi\colon X\to Y$.
Then we know that $\pi$ is generically smooth (i.e. on ...
3
votes
1
answer
175
views
Reference for a folklore result about $T^1(B/k;M)$
If $B$ is a $k$-algebra, let $T^1(B/k;M)$ denote the first cotangent functor. It classifies first order deformations of the scheme $\mathrm{Spec} B$.
Now, if $X \subset \mathbb P^n$ is a smooth ...
2
votes
2
answers
366
views
A polynomial in multiple variables with nice properties
After answering another question (The number of values of $f(x)/x$ when $f$ is a linearized polynomial), I stumbled upon an interesting polynomial in multiple variables. Let $\mathbb{F}_q$ be the ...
0
votes
1
answer
296
views
Vector field with Harmonic flow
Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle_g$...
6
votes
1
answer
285
views
Reference request: Reduced reflection length in Coxeter groups
I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
2
votes
1
answer
129
views
Complete minors and minimal degree
Is there for every positive integer $n\in\mathbb{N}$ a finite, simple, undirected graph $G=(V,E)$ with the following property?
$G$ does not have a complete minor with more than $\frac{\delta(G)}{n}$...
9
votes
1
answer
813
views
Maximizing a ratio of determinants
Let $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix s.t. $D\leq I$ ($I$ denotes the $n$-dim. identity matrix) and let $\alpha$ be a strictly positive real number. Consider the ...
2
votes
0
answers
144
views
Do Alexandrov spaces with non-empty boundary satisfy $RCD^*$ condition?
Let $M$ be an $n-$ dim compact Alexandrov space with curvature $\geq k$ with non-empty boundary $\partial M$.
Recently, a notion of generalized lower Ricci curvature bound on metric measure spaces ...
16
votes
4
answers
1k
views
A labelling of the vertices of the Petersen graph with integers
The vertices of the Petersen graph (or any other simple graph) can be labelled in infinitely many ways with positive integers so that two vertices are joined by an edge if, and only if, the ...
2
votes
0
answers
193
views
The extension of vector bundle over Hermitian manifold
Bando-Siu's paper Stable sheaves and Einstein-Hermitian metrics
Theorem 2 Let $(E,h)$ be a holomorphic vector bundle with a Hermitian metric $h$ defined on a Kähler manifold $(Y,\omega)$ outside a ...
3
votes
1
answer
129
views
Continuity in the roots of an algebraic variety with respect to the coordinates
If I'm slightly misusing definitions forgive me I'm not an algebraist.
I have $N$ polynomials $f_n(x)$, $n=1,\ldots,N$ where $x\in \mathbb{R}^N$ and the set $\{x:f_n(x)=0\text{ for all }n\}$ is ...
1
vote
1
answer
212
views
Infinitesimal generators and conserved quantities (Schrodinger type evolution)
First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of $\psi: \mathbb{R}^{1+1}\to \mathbb{...
4
votes
0
answers
1k
views
Reach of manifold vs. $C^k$-manifold
The reach $\tau_M$ of a manifold $M$ is the largest number such that any point at distance less than $\tau_M$ from $M$ has a unique nearest point on $M$.
This concept seems quite related to the local ...
4
votes
0
answers
125
views
Bounded self-adjoint perturbation of a p-summable spectral triple
I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book 'Elements of Noncommutative Geometry' by Gracia-Bondía,Várilly and Figueroa.Now,after ...
3
votes
1
answer
266
views
Friedman's SCG function
Today i discovered the Friedman's subcubic graph function and i have some question.
1) Why the valence of the gaphs $G_{n}$ is limited to 3? Can we make a faster growing variation of the function by ...
5
votes
0
answers
141
views
On the relation between Lipschitz free-spaces
Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point,
$Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
3
votes
0
answers
149
views
The isotopy class of a Boothby-Wang contact structure
Let $p:\Sigma\to M$ be a non-trivial principal $S^1$-bundle over a closed orientable surface $M$. Let's call $V$ the vector field generating the $S^1$-action. It is known that there exists a Boothby-...
15
votes
1
answer
718
views
Is there anything significant about GAP's SmallGroup(512,2045)?
Here's the output of the GAP command "SmallGroupsInformation(512)"
There are 10494213 groups of order 512.
...
7
votes
0
answers
221
views
Phantom category with trivial Hochschild cohomology
An admissible subcategory $C\subset D$ of a triangulated category is called phantom if $K_0(C)=0$. Such categories may be detected by their Hochschild cohomology (but usually have trivial Hochschild ...
2
votes
1
answer
456
views
The flow of Harmonic vector fields
A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.
Motivated by conversations on this questions we ask:
...
2
votes
1
answer
109
views
Does this q-analogue have a nice closed form? [closed]
Let $[n]_q=1+q+\cdots+q^{n-1}$. Is there a nice closed form of $\sum_{s=1}^i[s]_{q}$? One would expect that the answer will be some q-analog of $\frac{i(i+1)}{2}$, since $\sum_{s=1}^i s=\frac{i(i+1)...
2
votes
1
answer
309
views
Questions about Levy measure in the canonical representation of infinitely divisible distributions
Let $X$ be a random variable with infinitely divisible and symmetric distribution $F$ distributed on $\mathbb{R}$.
It is well known that the characteristic function of $X$ has a canonical ...
12
votes
3
answers
1k
views
A "quantum" identity: in search of a proof -Part II
As usual, denote $[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q}$ and $[n]_q!=[1]_q[2]_q\cdots[n]_q$. Furthermore, we write
$$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}.$$
As a follow up on this ...
6
votes
1
answer
531
views
Analogue of j-invariant for CM fields
For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic ...
7
votes
2
answers
730
views
How can Wall's theorem be generalised to non-simply connected manifolds?
In a sense, this is a follow-up to this question.
By work of Freedman and Wall, it is known that if two simply-connected 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ ...
1
vote
1
answer
167
views
Averaging the Jacobi symbol over an ellipse
Let $f(x,y) = a^2 x^2 - (b^2 - 2ac)xy + c^2 y^2$ be a positive definite binary quadratic form with co-prime integer coefficients such that $b \ne 0$. For a given pair of integers $(u,v)$ and a prime $...
12
votes
2
answers
1k
views
An interesting identity: in search of a proof -Part I
I like the following binomial identity in that the RHS extracts the indeterminate $w$ from the LHS.
Question. Can you show that
$$\sum_{k=0}^n\binom{x+kw}k\binom{y-kw}{n-k}=\sum_{k=0}^n\binom{x+y-...
3
votes
1
answer
117
views
Least model over a certain configuration
I am interested in conditions under which there is a least model inside a (not saturated in general) model $N$ over certain configurations such as $M_0 \subseteq M_1, M_2$ with $M_1 \overset{\vert}{\...
2
votes
2
answers
115
views
Solving numerically an equation involving exponentials [closed]
I met an equation of the following form:
$$\sum_{i=1}^nk_ip_i e^{-k_i\lambda}~~=~~b,$$
where $p_i\ge 0$, $k_i$ and $b$ are known for $i=1,\cdots, n$. I'd like to know how to find the solution $\...
1
vote
1
answer
106
views
When an ideal is locally comaximal with idempotents(restated)
I saw the following question at mathstackexchang < https://math.stackexchange.com/questions/2282194/when-an-ideal-is-locally-comaximal-with-idempotents>. It seems to be a nice question and I need ...
2
votes
1
answer
217
views
Does the unit index divide the degree of an extension of number fields?
Let $L/K$ be a finite extension of algebraic number fields of degree prime $p$. Is it true that the index $(U_K:\text{Norm}(U_L))$ divides $[L:K]$,
where $U_K$ denotes the unit group and Norm denotes ...
4
votes
0
answers
93
views
Local systems and locally free sheaves in families
Let $X_{t}$ be a family of compact complex manifolds over the disk $\mathbb{D}\subset \mathbb{C}.$ Formally, $X_{t}$ is the fiber over $t\in\mathbb{D}$ of a proper, holomorphic submersion of a ...
3
votes
1
answer
226
views
Endomorphismensatz for Lie superalgebras
For semisimple complex Lie algebras there is Soergel's Endomorphismensatz
$$C = \operatorname{End}(P(w_0)) \cong \mathbf C[\mathfrak h]/\mathbf C[\mathfrak h]^W$$
for $w_0$ the longest element in ...
5
votes
1
answer
613
views
Do Riemann-Weil formulas exist for functions other than the Mangoldt function $ \Lambda (n) $
Are there formulas similar to the Riemann-Weil formula for other arithmetical functions like $ \mu (n) $ or $ \lambda (n) $, for example a sum of the form $ \sum_{n=1}^{\infty}a(n) f(n) $ with this ...
4
votes
1
answer
300
views
Functors between categories of equivariant sheaves are equivariant sheaves on the product?
This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be ...
10
votes
2
answers
560
views
Schematic locus of algebraic spaces in fibers
Let $S$ be a scheme and let $X$ be a quasi-separated algebraic space over $S$. Does there exist an open subspace of $X$ which is a scheme and which is dense in each fiber $X_s$, $s \in S$?
I am happy ...
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
0
votes
1
answer
236
views
Visualizing the 4th dimension [closed]
In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as shadows of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can atleast see ...
35
votes
7
answers
6k
views
Why is conformal invariance only possible for massless theories?
I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories ...
2
votes
1
answer
297
views
Papers on distribution of high order elements over $\mathbb{F}_p$
I am interested in knowing about the distribution of exponentially high order elements in $\mathbb{F}_p$. To be precise let $s$ be of the order $\frac{p}{\log^{k}(p)}$ for some fixed $k$ and integer. ...
6
votes
0
answers
188
views
How can I get access to Hijikata's mimeographed notes?
In several papers on buildings, Hijikata's mimeographed notes from Yale ("Maximal compact subgroups of some p-adic classical groups") are often cited as a precursor to the work of Bruhat-Tits. While ...
2
votes
1
answer
216
views
$C=A \cdot B$ matrices and exact sequence of $DVR$-modules
I'm looking for a proof or a reference for the following statement:
Let $R$ be a DVR (Discrete Valuation Ring) and $p$ a prime element, and let $\mathfrak a$, $\mathfrak b$ and $\mathfrak c$ be ...
4
votes
1
answer
488
views
Inequality with ratio of Normal CDFs
Let $\theta>0$, $0<\lambda<1$, and $\Phi$ be the standard normal cumulative distribution function. Is it true that
$$\frac{\lambda \Phi(-\theta)}{\Phi(-\lambda \theta) }< e^{\frac{\theta^...
9
votes
0
answers
188
views
Littlewood-Richardson sequences and Littlewood-Richardson coefficients
I'm looking for a proof or a reference for the following statement, I give the definitions below:
There exists a Littlewood-Richardson sequence of type $(\alpha, \beta, \lambda)$ if and only if $...
4
votes
1
answer
130
views
Criterion for homogeneity
Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and let $G=Aut(\Omega)$ be the full group of self-biholomorphisms of $\Omega$. Assume that there is $z\in \Omega$, such that the orbit of $z$ is ...