Math Overflow Recent Questions

Subscribe to Math Overflow Recent Questions feed
most recent 30 from mathoverflow.net 2018-10-17T02:59:20Z

Fraction fields of strict henselizations of DVRs

Tue, 10/16/2018 - 20:42

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\mathrm{sh}}$. Are $K_{1},K_{2}$ isomorphic?

Convergence of the intertwining operator as a vector valued integral

Tue, 10/16/2018 - 20:35

Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \mathfrak a_{M,\mathbb C}^{\ast}$, we have the induced representation

$$I(\nu,\pi) = \operatorname{Ind}_P^G \pi q^{\langle \nu+\rho,H_M(-)\rangle}$$ of $G$. For $w$ in the Weyl group sending $\theta$ to $\theta' \subset \Delta$, and $P' = M'N'$ corresponding to $\theta'$, we have the intertwining operator $A = A(\nu,\sigma,w): I(\nu,\pi) \rightarrow I(w(\nu),w(\pi))$ defined by

$$A(f)(g) = \int\limits_{N_w} f(w^{-1}ng)dn$$

where $N_w$ is generated by the root subgroups of positive roots made negative by $w^{-1}$. The given integration takes place in the vector space $V$, and I am trying to understand:

  • What is the meaning of this vector valued integral?

  • Why does the integral converge (whatever that means, depending on the answer to my first question) for $\nu$ in a suitable cone?

I had asked a question about the meaning of the integral before, but I am sorry to say that after all this time I still do not understand what is going on. Paul Garrett provided an answer in which he suggested that we should not think of $V$ as having the discrete topology, but having a locally convex, quasi-complete topological vector space structure (coming as a colimit of its f.d. subspaces) in which one could make sense of the integral as a Pettis integral. That is, we should show that there exists a vector $v = A(f)(g)$ in $V$ with the property that for all $v^{\ast}$ in the algebraic dual of $V$,

$$\langle v^{\ast},v \rangle \rangle = \int\limits_N \langle v^{\ast}, f(w^{-1}ng)\rangle dn$$

He also suggested that taking a good maximal compact subgroup $K$ of $G$, so that we have $G = PK = P'K$, we could use the fact that elements of the induced representation are determined by their effect on $K$ to reduce to the case where the vector valued integrals are just finite sums. I still have not figured out how to do this, and wanted to ask math overflow again for help.

These intertwining operators are unfortunately still very much a mystery to me, and I have not seen any reference explain them rigorously.

Are the intersection of proximinal sets in a Hilbert Space proximinal?

Tue, 10/16/2018 - 20:24

Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is proximinal if $P_C(x) \neq \emptyset$ for all $x \in X$.

I'm wondering if the intersection of two proximinal subsets of $X$ must be proximinal? To avoid trivial counterexamples, I need their intersection to be non-empty.

The answer is most likely "no", or at least, something that is difficult to prove in the affirmative if true. Asplund proved that, given any non-convex Chebyshev subset $C$ (meaning $|P_C(x)| = 1$ for all $x$), there exists a closed half-space $H$ such that $X \cap H$ is not proximinal. Therefore, were proximinal sets closed under intersection, this would prove the Chebyshev conjecture: that all Chebyshev subsets are convex.

I was wondering if someone had an explicit example of two proximinal sets whose intersection is not proximinal?

Find an open subset in the variety of affine lines where fibers inherit rational connectedness

Tue, 10/16/2018 - 20:00

Let $V$ be an irreducible and smooth variety over a number field $k$ and let $f:V\rightarrow\mathbb{A}_k^n\backslash F$ be a surjective morphism with rationally connected generic fiber, where $F$ is a closed subset of codimension at least $2$. Let $\mathcal{A}$ be the variety of affine lines in $\mathbb{A}_k^n$. Can we find a non-empty open subset $\mathcal{U}\subseteq\mathcal{A}$ such that $f^{-1}(l)$ is rationally connected for every $l\in\mathcal{U}$?

This question came to my mind when I was reading D. Harari's paper "Flèches de spécialisations en cohomologie étale et applications arithmétiques", in which a question of similar kind was solved by Theorem of Bertini.

Cokernel of map of ètale sheaves

Tue, 10/16/2018 - 18:26

This is probably a very dumb question. Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ canonically isomorphic to the cocharacter lattice $L$ of $T$?

If $\operatorname{Spec}A$ is affine scheme, it seems to me that $p^*T=T\times \mathbb{G}_m$, so $$p_*p^*T(A)=p_*((T\times \mathbb{G}_m))(A)=T(A[t^{\pm 1}])\times \mathbb{G}_m(A[t^{\pm 1}]).$$ Say that $A=K$ is a field. Then $$p_*p^*T(K)=L\oplus\mathbb{Z},$$ but why is the image of $T(K)$ equal to $\mathbb{Z}$?

Proof of Littman-Stampacchia-Weinberger theorm on the fundamental solution for elliptic PDEs

Tue, 10/16/2018 - 17:06

Where can I find a (readable and self-contained) proof of the following result?

Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\mathrm{div}(A(x)\nabla u) = \delta_0,$$ $$u|_{\partial \Omega} = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$). Then on $B_{1/2}$, we have $$\frac{C_2}{|x|^{n-2}} \le u(x) \le \frac{C_1}{|x|^{n-2}}.$$

The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

Tue, 10/16/2018 - 16:41

In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with (i.e. have the same parameter space as) the instantons in 2D for the theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$."

Question (1): What is the precise restriction of $G$ in Atiyah's on Instantons in 2D and in 4D? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

In Atiyah's work [Ref. 1], he also mentions that Donaldson's work [Ref. 2] gives the proof only for the classical groups but it seems likely that his result holds for all $G$.

Question (2): What is the precise restriction of $G$ in Donaldson's work here? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

Question (3): The instanton study here in 2D for Atiyah's theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$.) How is this story of $CP_n$ v.s. $\Omega G=\Omega SU(n)$ here related to the $G=SU(n)$-Yang Mills theory? Here $CP_n$ is finite $n$-dimensional complex manifold, while $\Omega G$ is said to be infinite-dimensional.

Refs:

  1. Instantons In Two-dimensions And Four-dimensions, 1984 - 15 pages Commun.Math.Phys. 93 (1984) 437-451, M.F. Atiyah

  2. Instantons and geometric invariant theory, Comm. Math. Phys. Volume 93, Number 4 (1984), 453-460. S. K. Donaldson

Note: Classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces

DeGiorgi oscillation lemma

Tue, 10/16/2018 - 16:26

Where can I find a proof of the following result?

Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$) in $B(0,1)$ such that $u \le 1$ and $|\{ u \le 0\}| =a>0$. Then $$\sup_{B_{1/2}}u \le \mu(a) < 1.$$

Asymptotic Constancy of solutions of delay/integro differential equations

Tue, 10/16/2018 - 15:59

I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference).

I am however most interested in methods for finding the constant to which these solutions converge. My question is for references which provide such a method or some type of review paper or book where I can read more about asymptotic constancy in general.

Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Tue, 10/16/2018 - 15:47

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-X$ is still clear. What is known about the action of $W$ on $L$? Is anything known about the orbit structure? References would be most welcome.

Projective dimensions of the terms in a minimal injective resolution of the regular module

Tue, 10/16/2018 - 15:03

Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$.

The study of the projective dimensions of the $I_i$ is an important tool to test whether certain subcategories are extension closed or closed under submodules, see for example the article "Homolocial theory of noetherian rings" by Idun Reiten and https://www.sciencedirect.com/science/article/pii/0022404994900442 .

Questions:

  1. Is there an easy example with $pd(I_i)=1$ for some $i>1$? (probably yes, but im too blind at the moment to construct an example. It necessarily has to have global dimension at least 3.)

  2. Can we have $pd(I_i)=1$ for some $i>1$ in case $pd(I_0)=0$?

  3. Can we have $pd(I_i)=1$ for some $i>1$ in case $A$ is a Nakayama algebra?

To my surprise my computer found no such example for a Nakayama algebra.

(of course this question has the danger that I oversee something obvious)

edit: The reason might be as follows when $pd(I_0)=0$:

We have $0 \rightarrow A \rightarrow I_0 \rightarrow \Omega^{-1}(A) \rightarrow 0$ and thus all indecomposable injective modules of projective dimension one appear in $\Omega^{-1}(A)$ and thus also in $I_1$. Now it is probably easy to see that they cant appear later again, but Im not sure why at the moment.

Obtaining a Relation Between Positive Matrices

Tue, 10/16/2018 - 15:01

Consider $n \times n$ non-negative binary matrices ${\bf A}_i$ with $1\leq i \leq m$ over $\mathbb{R}$.

Assume that $1\leq k \leq n$ is selected as a fixed number.

Suppose that a subset of size $k$ of the set $\{1,2,\cdots , m\}$ is denoted with ${\bf I}_k=\{a_1,a_2,\cdots ,a_k\}$ and the related matrix to ${\bf I}_k$ is defined by ${\bf H}=(h_{i,j})=\prod_{t=1}^k\, A_{a_t}$.

My Question: With which conditions over ${\bf A}_i$'s we have the following property.

If for an ${\bf I}_k$ the matrix ${\bf H}$ be a positive matrix over $\mathbb{R}$ ($h_{i,j}>0$), then the value of $\sum_{i=1}^n \sum_{j=1}^n h_{i,j}$ be a fixed number (not depends on the value of ${\bf I}_k$)

How to construct $ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $?

Tue, 10/16/2018 - 14:38

Set : $$ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $$

I would like to know if there exists a projective variety $ H ( \mathbb{P}^n ) $ whose points are in a natural one-to-one correspondance with smooth projective varieties of $ \mathbb{P}^n $ ( i.e : with elements of $ \mathcal{H} ( \mathbb{P}^n ) $ ).

If it's the case, how to construct $ H ( \mathbb{P}^n ) $ ?

If there is a way to construct $ H ( \mathbb{P}^n ) $, is there any book treating in detail this subject of how to construct $ H ( \mathbb{P}^n ) $ and its properties ?

Thanks in advance for your help.

Reflection-invariant monomial ideals and Alexander duality

Tue, 10/16/2018 - 13:23

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally we provide an example that seems to contradict that claim. The question is

Question: Is the example given below a counterexample to the claim? And if not, why not?

Definitions

Fix an Artinian monomial ideal $I$ of a polynomial ring $K[\mathbf{x}] = K[x_1, \dots, x_n]$. A monomial $\mathbf{x}^{\mathbf{b}}$ is a socle monomial of $I$ if $\mathbf{x}^{\mathbf{b}} \notin I$ and $x_i\mathbf{x}^{\mathbf{b}} \in I$ for all $i$. Let $\mathrm{MonSoc}(I)$ be the set of all socle monomials of $I$.

Def: $I$ is reflection invariant if there is a canonical monomial $\mathbf{x}^{\mathbf{K}}$ such that the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$.

Following these definitions the authors note the following.

The Claim

Claim: $I$ is reflection invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$ if and only if the monomial ideal generated by $\mathrm{MonSoc}(I)$ equals the Alexander dual $I^{[\mathbf{K} + \mathbf{e}]}$ where $\mathbf{e} = (1,1,\dots, 1)$.

The (Counter?) Example

Let $I = \langle a^4,~ab^2,~b^3,~a^3c,~abc,~c^3 \rangle \subset K[a,b,c]$ and let $\mathbf{K} = (3,2,2)$. Then

$$\mathrm{MonSoc}(I) = \left\{a^{3}b,~a^{2}c^{2},~b^{2}c^{2}\right\}.$$

By (the constructive proof of) Proposition 5.2 in this paper the ideal $J = \langle a^4, a^2b, b^3, ac, b^2c, c^3 \rangle$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(J) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}} \mid \mathbf{x}^{\mathbf{b}} \in \mathrm{MonSoc}(I)\right\}.$$

Moreover, the same algorithm can be used to show that $I$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(I) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{c}} \mid \mathbf{x}^{\mathbf{c}} \in \mathrm{MonSoc}(J)\right\}.$$

In particular, the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$, so $I$ is reflection-invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$. We now get a contradiction to the claim above by computing the Alexander dual (in Macaulay2, for example) and noting that the minimal generators of $I^{[(4,3,3)]}$ are $\{a^4bc,~a^2b^3c,~ab^2c^3\} \neq \mathrm{MonSoc}(I)$.

Again, the question is

Question: Is the example just given a counterexample to the above claim? And if not, why not?

Is there any significance to Bousfield localization in the non-derived context?

Tue, 10/16/2018 - 12:23

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:

  1. There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically just means a reflective subcategory of $C$.

  2. There is also a more restrictive usage (as when talking about spectra), which requires $C$ to be monoidal, and means a reflective subcategory where the class of maps being localized at is of the form $\{X \mid E \otimes X = 0\}$ for some fixed $E \in C$.

In this question, I'm interested in the more restrictive usage (2).

In this sense, Bousfield localization makes sense in either an ordinary monoidal category or in a monoidal $\infty$-category (for that matter, the more general usage makes sense in either an ordinary category or in an $\infty$-category). But it's typically only discussed in an $\infty$-categorical setting (e.g. in model categories or triangulated categories).

Question 0: Is there a good reason why Bousfield localization for ordinary categories (in sense (2)) is rarely discussed?

I think the answer may be "yes" because the behavior of Bousfield localization may be quite different in the two settings, and it seems somehow "better" in the $\infty$-categorical setting. But I'm not sure how to articulate this.

Here are two examples of what I mean:

  1. $E = \mathbb Z/p$:

    • When $C = Ab$ is the (ordinary) category of abelian groups and $E = \mathbb Z / p$, the Bousfield localization consists of the abelian groups which have no nonzero infinitely $p$-divisible elements.

    • But when $C = D(Ab)$ is the $\infty$-category of chain complexes of abelian groups (localized at the quasi-isomorphisms) and $E = \mathbb Z/p$, the Bousfield localization consists of chain complexes whose homology is $p$-complete.

  2. $E = \mathbb Z_{(p)}$:

    • When $C = Ab$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of abelian groups which are $\ell$-torsionfree for $\ell\neq p$.

    • When $C = D(Ab)$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of chain complexes whose homology is a $\mathbb Z_{(p)}$-module.

By "different behavior", I mean, to a first approximation, that even though $D(Ab)$ is "the natural $\infty$-categorical counterpart to $Ab$", in these cases it's not the case that the restriction of the $E$-Bousfield localization in $D(Ab)$ to $Ab$ coincides with the $E$-Bousfield localization in $Ab$ itself.

Part of the problem is that I'm not exactly sure what qualifies as "being in the $\infty$-categorical setting". After all, an ordinary category is in particular an $\infty$-category. But maybe for concreteness, I'll ask a slightly less vague version of the question:

Question 1: If $T$ is a tensor triangulated category with a $t$-structure, and $E \in T^{heart}$, is there any reason to think about the Bousfield localization of $T^{heart}$ at $E$ rather than the Bousfield localization of $T$ at $E$?

Number of self avoiding paths on a grid graph?

Tue, 10/16/2018 - 11:56

Let $G$ be an $n \times n$ grid graph. Is there anything known about the asymptotic growth rate of the number of self avoiding paths from $(0,0)$ to $(n,b)$ (from the lower left corner to some arbitrary point on the right hand side)?

For example, is there anything known about the $b$ that maximizes this number? A limiting probability distribution on $[0,1]$ for the value of $b$ (if we pick a self avoiding walk from $(0,0)$ to $x = n$ uniformly at random)?

Some similar questions go unanswered:

Any approximation algorithms for self-avoiding walks?

How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Thank you!

Question about Jacobson rings

Tue, 10/16/2018 - 10:55

I am studying theorem (1.11)* from this article: https://sci-hub.tw/https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-9.2.337

Theorem (1.11)*. Let $R$ be a Jacobson ring, then $S = R[x,\alpha]$ is a Jacobson ring.

In the proof, when $x \notin P$, the author takes the isomorphism $\dfrac{S}{(P\cap R)S} \cong \dfrac{R}{P \cap R} [x,\alpha]$ and says to replace $R$ by $R/(P \cap R)$ and reduce to the case when $S$ is a prime ring and $P$ is a prime ideal which satisfies $P \cap R = 0$.

My question is: I don't understand why we can only consider the case when $P \cap R = 0$.

Any help would be great.

The ample cone of a surface with an algebraic $\mathbb C^*$ action

Tue, 10/16/2018 - 10:53

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder if this is statement is correct and whether it is written down in some book/article.

$X$-rays of permutations

Tue, 10/16/2018 - 09:54

Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.

There has been some study (e.g. see this paper) on the notion of diagonal and antidigonal $X$-ray sequences. Given a permutation matrix $\pi\in\mathfrak{S}_n$, starting from the bottom, construct the $k$-th diagonal sum $y_k$ of its entries for $k=1,2,\dots,2n-1$. Then, the sequence or word $y(\pi)=y_1y_2\cdots y_{2n-1}$ is called the diagonal $X$-ray of $\pi$.

For example, if $\pi=231\in\mathfrak{S}_3$ then $y(\pi)=10020$.

It is easy to check that the number $N(\mathfrak{S}_n)$ of distinct $X$-ray sequences of the set $\mathfrak{S}_n$ is less than $n!$ while the exact value is unknown.

QUESTION. Is there at least a $1^{st}$-order asymptotic estimate on the growth rate of $N(\mathfrak{S}_n)$?

A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Mon, 10/15/2018 - 22:28

Is there an analytical solution satisfying these 3 equations with non-constant z?

$$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - y\cdot\sin(\omega t)$$

Pick a specific non-zero $\omega$ (e.g., 1 or $\pi$) if you must do so.

Pages