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most recent 30 from mathoverflow.net 2018-02-23T10:14:37Z

Exponential decay for the wave equation with a potential

Fri, 02/23/2018 - 02:30

Consider the wave equation with a potential: $$ E_{tt} = \Delta_x E - q(x) E, \quad (x,t) \in \mathbb R^3 \times (0,+\infty), \\ E(x,0) = 0, \quad E_t(x,0) = \delta_{x_0}(x), $$ under the assumption that $q \in L^\infty_\text{comp}(\mathbb R^3,\mathbb R)$. Then the solution $E$ can be represented in the form $$ E(x,x_0,t) = \frac{\delta(t-|x-x_0|)}{4\pi|x-x_0|} + \widetilde E(x,x_0,t), $$ where $\widetilde E(x,x_0,t) = 0 $ for $t<|x-x_0|$ and is sufficiently regular for $t>|x-x_0|$.

I was told that the function $\widetilde E(x,x_0,t)$ at fixed $x \neq x_0$ must decay exponentially as $t\to+\infty$, as well as its derivatives with respect to $x$ and $t$ up to order two. Is this result well-known?

Pointwise convergence in Lawvere metric spaces

Fri, 02/23/2018 - 01:45

In the formalism of Lawvere metric spaces, we have that the distance in the hom-space $[X,Y]$ is given by: $$ d(f,g) = \sup_{x\in X} d(f(x),g(x)) . $$ Therefore, a sequence of functions $f_n:X\to Y$ converges to $f:X\to Y$ in $[X,Y]$ if and only if it converges uniformly.

How can we talk about pointwise convergence in this setting, instead?

Logic Diagram in discrete mathematics [on hold]

Fri, 02/23/2018 - 00:38

I just wanted to ask about this problem "Design a combinational circuit with three inputs.The output will be the square of the input" I would like to ask on how will i get the output? is it the square of 3 ? or is it based on binary ?.. and do you know any free diagramming tool?

Expected value of maximal accumulation of functions $f:\{1,\ldots,n\} \to \{1,\ldots,n\}$

Fri, 02/23/2018 - 00:19

For any positive integer $n$, let $[n] = \{1,\ldots,n\}$. Let $[n]^{[n]}$ denote the set of functions $f:[n]\to [n]$. For $f\in[n]^{[n]}$, we define the maximum accumulation $\text{macc}(f)$ by $$\text{macc}(f) = \max\{|f^{-1}(\{k\})| : k\in[n]\}.$$ (Note that $f\in[n]^{[n]}$ is bijective if and only if $\text{macc}(f) = 1$.)

Let $$E_n = \frac{1}{n^n}\sum_{f\in [n]^{[n]}} \text{macc}(f).$$ This is the expected value of the maximum accumulation when randomly given a member of $[n]^{[n]}$. I want to get a feeling of how fast $E_n$ grows as $n\to \infty$.

Questions. Do we have $\lim_{n\to\infty}\frac{E_n}{\log n} \in \;]0,\infty[$? Or is $\lim_{n\to\infty}\frac{E_n}{\sqrt{n}} \in \;]0,\infty[$? (If any statement is true, then the actual value of the limit would also be interesting to know.)

Bound of complex power of the Laplacian on a complex Riemann surface at $s=-1$

Thu, 02/22/2018 - 23:46

Let $X$ be a compact Riemann surface and $\Delta_{\overline{\partial}}$ be the Dolbeault Laplacian. I am wondering if there is any non-trivial bound on $$ \frac{d}{ds}\Delta^{s}, s=0,-1 $$ (say) in terms of $g$ and the area of $X$. Normalizations via Gauss-Bonnet or other conventions are okay to me. Here the complex power is defined using either Mellin transform of the heat kernel or via cuts through the spectral plane. I assume the subject has been well-studied by experts, but I could not find any reference on it, especially for bounds of the value of the derivative of the zeta function at $s=-1$ (Ray-Singer's paper already gave closed form expression for $s=0$). I am not sure if it is because the subject is difficult.

Motivation: According to Soule's paper, any non-trivial bound on this under Arakelov metric would imply a bound on Faltings delta function $h_{\delta}(F)$, which controls the self-intersection number of the relative canonical class minus the contribution from the singularities.

Rationally connected blow-ups [on hold]

Thu, 02/22/2018 - 22:21

Are blow-ups of $\mathbf{P}^n$ rationally connected? unirational?

Is the blow-up of a rational variety rationally connected? unirational?

Picard group of a finite type $\mathbb{Z}$-algebra

Thu, 02/22/2018 - 21:56

Let $A$ be a finitely generated $\mathbb{Z}$-algebra. Is $\operatorname{Pic}(A)$ finitely generated (as an abelian group)?

Thoughts:

  1. We may assume that $A$ is reduced since $\operatorname{Pic}(A) = \operatorname{Pic}(A_{\mathrm{red}})$.
  2. If $A$ is reduced, then the group of units $A^{\times}$ is a finitely generated abelian group, see e.g. [1, Appendix 1, no. 3] or [4, Théorème 1] (which I learned about through this question).
  3. The case $A$ is normal is proved in [3, Chapter 2, Theorem 7.6].
  4. The following argument is from [2, Lemma 9.6]: Let $B$ be the normalization of $A$, set $X := \operatorname{Spec} A$ and $Y := \operatorname{Spec} B$ and let $\pi : Y \to X$ be the normalization morphism. We have the Leray spectral sequence $$ \mathrm{E}_{2}^{p,q} = \mathrm{H}^{p}(X,\mathbf{R}^{q}\pi_{\ast}\mathbb{G}_{m,Y}) \implies \mathrm{H}^{p+q}(Y,\mathbb{G}_{m,Y}) $$ with differentials $\mathrm{E}_{2}^{p,q} \to \mathrm{E}_{2}^{p+2,q-1}$. Since $\pi$ is a finite morphism (e.g. since $\mathbb{Z}$ is Nagata and [5, 030C]), every invertible sheaf on $Y$ can be trivialized on an open cover obtained as the preimage of an open cover of $X$ (e.g. [5, 0BUT]). Hence $\mathbf{R}^{1}\pi_{\ast}\mathbb{G}_{m,Y} = 0$, so we have $\operatorname{Pic}(Y) \simeq \mathrm{H}^{1}(X,\pi_{\ast}\mathbb{G}_{m,Y})$ from the Leray spectral sequence. Set $Q := \pi_{\ast}\mathbb{G}_{m,Y}/\mathbb{G}_{m,X}$; then the long exact sequence in cohomology associated to the sequence $1 \to \mathbb{G}_{m,X} \to \pi_{\ast}\mathbb{G}_{m,Y} \to Q \to 1$ gives an exact sequence $$ \Gamma(Y,\mathbb{G}_{m,Y}) \to \Gamma(X,Q) \stackrel{\partial}{\to} \operatorname{Pic}(X) \to \operatorname{Pic}(Y) $$ where the first and fourth terms are finitely generated. But what can I say about the sheaf $Q$? I know that it is $0$ on a dense open since $\pi$ is an isomorphism on a dense open (e.g. since $A$ is reduced, the regular locus is an open subset containing the generic points [5, 07R5]).
  5. I should also note that there is a Hartshorne exercise (II, Exercise 6.9) which relates the Picard group of a singular curve (over a field) to that of its normalization.

References:

  1. Bass, Introduction to Introduction to Some Methods of Algebraic K-Theory, Number 20 in CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1974.

  2. Jaffe, "Coherent functors, with application to torsion in the Picard group", Transactions of the American Mathematical Society, vol. 349, no. 2, 1997, pp. 481–527 link

  3. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag (1983)

  4. Samuel, "A propos du théorème des unités", Bulletin des Sciences Mathématiques, vol. 90, 1966, pp. 89–96

  5. Stacks Project link

Keywords: arithmetic scheme, Picard group, finite type $\mathbb{Z}$-algebra

Why should one subscribe to print Journals?

Thu, 02/22/2018 - 21:48

It seems obvious to me. Yes, Arxiv, MathSciNet, Overflow and most Mathematicians keeping preprints and reprints of their papers in their homepages, which changed everything in front of my eyes and I have been beneficiary of these.

But I still can't get myself agree to the opinion that one should stop subscribing to print copies of Journals. Is that the norm? So many times I chanced upon a result while browsing through the pages of a Journal; sometimes relevant to my own area, and sometimes totally unrelated but so exciting that it got me interested in that area.

I am asking this as Higher Ups in The University are of the opinion that we should stop subscribing to Print Version Of The Journals. How does one defend the case for need of Print Journals?

Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group

Thu, 02/22/2018 - 21:02

Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous homomorphism. Does $\chi$ extend to a continuous homomorphism into $\mathbb{C}^{\ast}$ defined on all of $G$?

If $\chi$ maps $H$ into the circle $S^1$, then $\chi$ does extend to a continuous homomorphism on all of $G$, also mapping into $S^1$. This follows from Pontryagin duality, and in fact this is true when $H$ is a closed, not necessarily open subgroup, and with no assumption about $G/H$.

Sufficient conditions for secondary invariants

Thu, 02/22/2018 - 20:45

Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension $n$ with ring of invariants $k[V]^G$. Suppose further that $k[V]^G$ is Cohen-Macaulay, so that there are primary invariants $f_1,\ldots,f_n$ and secondary invariants $h_1,\ldots,h_m$ such that $k[V]^G$ is a free module over $k[f_1,\ldots,f_n]$ with basis given by $1,h_1,\ldots,h_m$.

Given a list $f_1,\ldots,f_n$ of invariant functions on $V$, one may check to see if they form a set of primary invariants using the sufficient condition (which is also necessary) that the variety defined by the $f_i$ over $\overline{k}$ is $\{\bf{0}\}$.

Given primary invariants $f_1,\ldots,f_n$, I'm looking for sufficient conditions for a list of invariant functions $h_1,\ldots,h_m$ to be secondary invariants. Since $k[V]^G$ is Cohen-Macaulay, a necessary condition is that $$ m=\frac{\prod_{i=1}^n\operatorname{deg}(f_i)}{|G|}. $$ In the non-modular case, one could make use of Molien's formula for the Hilbert series to determine the degrees of the secondary invariants, and then show that $$ k[V]^G_{\operatorname{deg}(h_i)}\subset k[f_1,\ldots,f_n]\cdot 1\oplus k[f_1,\ldots,f_n]\cdot h_1\oplus\ldots\oplus k[f_1,\ldots,f_n]\cdot h_m $$ for all $i=1,\ldots,m$, where $k[V]^G_{\operatorname{deg}(h_i)}$ is the $k$-space of homogeneous invariant functions of degree $\operatorname{deg}(h_i)$ (which is presumably computable given $\rho$).

What can be said about the modular case? If I write down $h_1,\ldots,h_m$, how can I test that they are secondary invariants?

Can the wavelet bispectrum be normalised so that its integral "gives the right answer"?

Thu, 02/22/2018 - 20:08

Fix a rapidly decreasing function $\psi \in \mathcal{S}(\mathbb{R})$ with the properties that $\int_\mathbb{R} \psi = 0$, $\mathrm{Re}(\psi(\cdot))$ is an even function, and $\mathrm{Im}(\psi(\cdot))$ is an odd function.

Given any bounded integrable function $x \colon \mathbb{R} \to \mathbb{R}$, define \begin{align*} \mathcal{F}x(\xi) \ :=& \ \int_\mathbb{R} x(\tau)e^{-2\pi i\xi\tau} \, d\tau \\ \mathcal{W}x(\xi,t) \ :=& \ \int_\mathbb{R} x(\tau)\psi(\xi(\tau-t)) \, d\tau \end{align*} for all $\xi,t \in \mathbb{R}$. (Obviously $\mathcal{F}x$ is the Fourier transform; $\mathcal{W}x$ is a version of the wavelet transform.)

It is known that

$$ \hspace{20mm} \int_\mathbb{R} x(t)^2 \, dt \ = \ \int_\mathbb{R} |\mathcal{F}x(\xi)|^2 \, d\xi \ = \ C_\psi\!\iint_{\mathbb{R}^2} |\xi\,\mathcal{W}x(\xi,t)|^2 \, d\xi dt \hspace{18mm} (1) $$

where $C_\psi$ is a constant that depends only on $\psi$ and not on $x$, namely $$ C_\psi \ = \ \left( \int_\mathbb{R} \frac{\mathcal{F}\psi(r)^2}{|r|} \, dr \right)^{\!-1}\,. $$ It is also known that the first equality in $(1)$ can be extended to the 3rd moment by

$$ \int_\mathbb{R} x(t)^3 \, dt \ = \ \iint_{\mathbb{R}^2} \mathcal{F}x(\xi_1)\mathcal{F}x(\xi_2)\overline{\mathcal{F}x(\xi_1+\xi_2)} \, d\xi_1d\xi_2 $$

where the bar denotes complex conjugate. This naturally leads to the question of whether the second equality in $(1)$ can be extended to the third moment:

Does there exist a function $f_\psi \colon \mathbb{R}^2 \to \mathbb{C}$ such that for every bounded integrable function $x \colon \mathbb{R} \to \mathbb{R}$, $$ \int_\mathbb{R} x(t)^3 \, dt \ = \ \iiint_{\mathbb{R}^3} f_\psi(\xi_1,\xi_2) \mathcal{W}x(\xi_1,t)\mathcal{W}x(\xi_2,t)\overline{\mathcal{W}x(\xi_1+\xi_2,t)} \, d\xi_1d\xi_2dt \, ? $$ (If not, then [a more open-ended question] are there reasonable conditions on $\psi$ under which one can find a function $f_\psi$ such that the above equality "approximately" holds?)

A little bit of numerical experimentation by myself and a friend seems to suggest that the function $$ f_\psi(\xi_1,\xi_2) \ = \ D_\psi(\xi_1+\xi_2) $$ might work for some constant $D_\psi \in \mathbb{C}$.

Birational invariants in étale cohomology

Thu, 02/22/2018 - 19:29

If $k$ is a perfect field (not necessarily separably closed), $f : X\to Y$ a proper birational map of smooth projective $k$-varieties, $\ell$ a prime invertible in $k$, is the induced map, for some $j$

$$f^* : H^j_{ét}(Y, \mu_{\ell^n}^{\otimes i})\to H^j_{ét}(X, \mu_{\ell^n}^{\otimes i})$$ an isomorphism, for every $n\ge 1$?

For example, if $k$ is of characteristic zero and $i=1$, $j=2$, we have

$$0\to \text{CH}^1(X)/\ell^n\to H^2(X,\mu_{\ell^n})\to \text{Br}(X)[\ell^n]\to 0$$

and left and right sides are birational invariants.

identity involving generating sum over colored plane partitions

Thu, 02/22/2018 - 19:01

Some notation. Introduce 3 variables $q_1$, $q_2$, $q_3$ and $N$ variables $u_1,\ldots,u_N$ for some fixed positive integer $N$. Let $\pi$ denote a plane partition (see Wikipedia for definition), and $\vec\pi=(\pi_1,\ldots,\pi_N)$ a $N$ tuple of plane partitions, namely a colored plane partition. Its dimension $|\vec \pi|$ is given by the sum of dimensions of its components. Let $q^{r+i}=q_1^{r_1+i} q_2^{r_2+i} q_3^{r_3+i}$ for $r \in \pi$, $i \in \mathbb Z$, $u_{km}=u_k u_m^{-1}$, $Q=q_1 q_2 q_3$. Finally, $[x]=x^{1/2}-x^{-1/2}$.

It seems plausible that the following identity holds: $$ \sum_{k=0}^\infty ((-1)^N p)^k \sum_{|\vec{\pi}|=k} \mu_{\vec \pi} ( \{q_i\}, \{ u_i\} ) = \exp \sum_{m=1}^\infty \frac 1m F_N (q_1^m,q_2^m,q_3^m,p^m) $$ where $$ F_ N(q_1, q_2, q_3, p) = - \frac {[q_1 q_2] [q_2 q_3] [q_1 q_3]}{[q_1] [q_2] [q_3]} \frac { [Q^N] } {[Q] } \frac p {\left( 1- p Q ^{\frac N2} \right) \left( 1- p Q ^{-\frac N2} \right)} $$ and (probably with many cancellations) $$ \mu_{\vec \pi} ( \{q_i\}, \{ u_i\} )= \prod_{k,m=1}^N \frac { \displaystyle \prod_{s \in \pi_k} [q^s u_{km}] \prod_ {s' \in \pi_m} [q^{s-s'} u_{km}] \prod_{i=1}^3 [ q_i^{-1} q^{s-s'+1} u_{km}] } { \displaystyle \prod_{s' \in \pi_m} [q^{1-s'} u_{km}] \prod_{s \in \pi_k} [q^{s-s'+1} u_{km}] \prod_{i=1}^3 [q_i q^{s-s'} u_{km}] } $$

Is there an elementary proof of that? (by elementary I mean not involving Hilbert schemes of points etc)

Connes-Chern pairing, compatibility with periodicity operator in the odd case

Thu, 02/22/2018 - 17:57

Let $A$ be an algebra (say unital). For an odd (say $2n-1$) cyclic cocycle $\varphi$ and a class in $K_1(A)$ represented by invertible $u$ we define $$\langle [\varphi],[u] \rangle:=\frac{2^{-(2n+1)}}{(n-\frac12)(n-\frac32) \cdot \ldots \cdot \frac32 \cdot \frac12}\varphi(u^{-1}-1,u-1,...,u^{-1}-1,u-1).$$ Let $S$ be a periodicity operator in cyclic cohomology: $S: HC^n(A) \to HC^{n+2}(A)$. It is defined as the composition of two connecting homomorphisms in two long exact sequence: it can be also defined directly at the level of cocycles as $S=-b(1+2\lambda+3\lambda^2+...+(n+2)\lambda^{n+1})b'$ where $\lambda$ is the cyclic operator, $b$ is a boundary in Hochschild (or cyclic) cohomology and $b'$ is $b$ without the last term.

How to prove that $\langle [\varphi],[u] \rangle=\langle S[\varphi],[u] \rangle$?

It is left as an exercise in Connes book or in the IHES paper: it is also left as an exercise in Khalkhali ,,Basic Noncommutative Geometry'' book. In this book the even case is treated using the formula above for the $S$ operator. Connes works with the rather different formula for operator $S$ using the closed graded trace $\hat{\varphi}$ defined on the universal differential graded algebra (on $\Omega^n(A)$ precisely) by the formula $$\widehat{\varphi}(a_0da_1...da_n)=\varphi(a_0,...,a_n).$$

I haven't found any place in the literature where it is carefully verified.

On certain number theoretic sextuples?

Thu, 02/22/2018 - 17:36

Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,c,d,e,f$ with $a,b$ coprime and $n^{1+\epsilon}<a,b<2n^{1+\epsilon}$ with $a$ even and $b,e,f$ odd, $p/2<c,d<p$ and $p^{\epsilon'}<e,f<2p^{\epsilon'}$ such that $$p|(ad-bc)\mbox{ and }p|(cf-de)$$ holds?

$$\mbox{Note }ad\equiv bc\bmod p,\quad cf\equiv de\bmod p\implies cd^{-1}\equiv ab^{-1}\equiv ef^{-1}\bmod p\mbox{ holds}.$$

If $e,f$ odd are fixed with $p^{\epsilon'}<e,f<2p^{\epsilon'}$ then there are $\frac{\frac{n^{1+\epsilon}}2n^{1+\epsilon}}{\zeta(2)}=\frac{n^{2(1+\epsilon)}}{2\zeta(2)}$ choices of coprime $a,b$ with $a$ even and so at least one of these choices should give right $a,b$ with $ab^{-1}\equiv ef^{-1}\bmod p$ (we should expect $\frac{n^{2\epsilon}}{2\zeta(2)}$ choices of coprime $a,b$ with $a$ even since $p$ is of size $O(n^2)$). A similar argument holds for $c,d$.

Do such sextuples really exist? The argument indicates each of the $O(p^{2\epsilon'})$ different pairs of $e,f$ have at least one $a,b,c,d$ associated with them.

Polynomial convexity of a multi-circular set (not domain)

Thu, 02/22/2018 - 16:36

It is well-known that there is a simple criteria for a (bounded) multi-circular domain (Reinhardt domain) $\Omega$ in $\mathbb{C}^n$ to be polynomially convex: namely, if the image of $\Omega\cap(\mathbb{C}^{*})^{n}$ under the tropicalisation map $$Log:(\mathbb{C}^{*})^{n}\to (\mathbb{R})^n, \text{ }(z_1,\dots, z_n)\mapsto (\log|z_1|, \dots, \log|z_n|)$$ is linearly convex, then $\Omega$ is polynomially convex. Now I would like to know what happens for polynomial convexity of multi-circular sets which are not domains.

For example, let $$\Omega=\{0\}\times \overline{\mathbb{D}}(0,2)\text{ }\cup\text{ } \overline{\mathbb{D}}(0,1)\times \overline{\mathbb{D}}(0,1)\subset \mathbb{C}^2$$ be a multi-circular compact set, where $\overline{\mathbb{D}}(0,1)$ is the closed disc of center $0$ and radius $1$. This set is a union of a 1-dimensional disc and a 2-dimensional poly-disc. By constructing a holomorphic disc whose boundary lies in $\Omega$ but with some interior point not lying in $\Omega$, one knows that by maximum principle, $\Omega$ is not polynomially convex. The construction is to build two holomorphic functions on $\overline{\mathbb{D}}(0,1)$ with certain properties, and use them as coordinates for the map of the holomorphic disc.

Note that the construction of holomorphic functions here needs Cauchy integral to build certain Riemann map. To show the (non) polynomial convexity, is there a simpler proof just via some analysis of convergence of series? Is there a criteria in terms of the image of generalized/compactified tropicalisation map which maps $\Omega$ into $(\{-\infty\}\cup\mathbb{R})^n$? The need for such robustness of proof comes from the hope that a simpler proof may be generalized to the non-archimedean situation, namely one would like to show the NON-polynomial-convexity of $$\Omega'=\{0\}\times \overline{\mathbb{D}}(0,2)\text{ }\cup\text{ } \overline{\mathbb{D}}(0,1)\times \overline{\mathbb{D}}(0,1)\subset (\mathbb{A}_k^2)^{an}$$ where $(\mathbb{A}_k^2)^{an}$ is the Berkovich analytification of affine space over a complete ultra-metric field $(k,|\cdot|)$.

For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?

Thu, 02/22/2018 - 16:17

Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ necessarily extend to a smooth (resp. continuous) homomorphism on $T(k)$?

Even in the special case where $T$ is split over $k$, I am not sure of the answer. One can find a complentary $k$-subtorus $S'$ of $T$ such that $T$ is the direct product of $S$ and $S'$. Then $S(k) \times S'(k)$ is isomorphic to a subgroup of finite index in $T(k)$, so one is reduced to the case of considering finite index subgroups of a finite product of copies of $k^{\ast}$.

We do have a homomorphism $H_T: T(k) \rightarrow \textrm{Hom}_{\mathbb{Z}}(X(T)_k,\mathbb{Z})$ defined by

$$H_T(t)(\chi) = \log |\chi(t)|$$

whose kernel is the unique maximal open compact subgroup of $T(k)$, see for example my previous question. The same for $S(k)$. It might be possible to restrict a given character to $\textrm{Ker } H_S$, which is then necessarily unitary, and look at the fact that $S(k)/\textrm{Ker } H_S$ is a discrete finite rank free abelian group, and do something there.

If there is a good notion of an Ext functor in the category of locally compact abelian Hausdorff groups, I was also thinking it might be possible to look at a sequence like

$$0 \rightarrow \operatorname{Hom}_{\textrm{top-grp}}(T(k)/S(k),\mathbb{C}^{\ast}) \rightarrow \operatorname{Hom}_{\textrm{top-grp}}(T(k),\mathbb{C}^{\ast})$$ $$\rightarrow \operatorname{Hom}_{\textrm{top-grp}}(S(k),\mathbb{C}^{\ast}) \rightarrow \operatorname{Ext}^1_{\textrm{top-grp}}(T(k)/S(k),\mathbb{C}^{\ast})$$

and look at $\operatorname{Ext}^1_{\textrm{top-grp}}(T(k)/S(k),\mathbb{C}^{\ast})$ when $\mathbb{C}^{\ast}$ is alternatively viewed in its usual topology and the discrete topology.

Confusion about Teichmuller curves and $SL_2$ action

Thu, 02/22/2018 - 16:13

Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $SL_2(\mathbb{R})$. When the image $\pi(SL_2(\mathbb{R})\cdot x)\subset M_g$ of an orbit is a closed algebraic curve of $M_g$ it's called a Teichmuller curve. This is the most common definition of Teichmuller curve. On the other hand sometimes they are also referred to as affine invariant submanifolds (i.e. complex submanifolds of a stratum of $\Omega M_g$ which are $SL_2(\mathbb{R})$-invariant and linear in the period coordinates). My confusion stems from the following fact: if $(X,\omega)$ generates a Teichmuller curve then its stabiliser under the action of $SL_2(\mathbb{R})$ is a lattice and the orbit would have real dimension equal to $dim SL_2(\mathbb{R})=3$, so I don't see how to interpret a Teichmuller curve as an affine invariant submanifold.

Matrix exponential, containing a thermal state

Thu, 02/22/2018 - 11:37

This question was originally posted on MSE, and I'm cross posting it here.

Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 & \cdots \\ 0 & 2 & 0 & -3 & \cdots \\ 0 & 0 & 3 & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}$$

Numerically, I've found that the first column of $\exp(M)$ is given by $\alpha(1,e^{-\lambda},e^{-2\lambda},e^{-3\lambda},\dots)^T$, where $\lambda \approx 0.27$ and $\alpha = \sqrt{1-e^{-2\lambda}} \approx 0.65$.

Question: How can we prove analytically that the first column of $\exp(M)$ has the stated form?

Context: This question comes from a quantum mechanical model of two harmonic oscillators coupled by a Hamiltonian of the form $\hat{H} \propto \hat{a}_1^\dagger \hat{a}_2^\dagger - \hat{a}_1 \hat{a}_2$, where $\hat{a}_i$ is the lowering operator for oscillator $i$. The matrix $M$ is essentially this Hamiltonian on the subspace spanned by $\{\left|nn\right>\}$, and so $\exp(M)$ is a time evolution operator. The exponentially decaying coefficients in the first column indicates that either of the individual harmonic oscillators (after tracing out the other one) is in a thermal state.

I've tried several tricks with the BCH formula, but was discouraged by the fact that the commutators are not very cooperative.

How singular is the metric on an orbifold

Thu, 02/22/2018 - 08:08

I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the group of rotation by $\pi$. This orbifold $\mathbb{S}^2/\mathbb{Z}_2$ inherits the metric from $\mathbb{S}^2$, which we call them $g_{AF}$ and $g_0$ respectively. If we pull them both back to $\mathbb{R}^2$, one can prove that $$g_{AF}=\frac{1+O(r)}{4r}g_0$$ where $r=\sqrt{x^2+y^2}$. To prove this, we must use some complex analysis technique. Clearly one can see the metic has cone singularity at origin.

Now suppose we have an orbifold of higher dimension $n\geq 3$ with isolated singularity. As we all know a neighborhood of the orbifold tip is diffeomorphic to a ball in $\mathbb{R}^n$ quotient by some finite group $\Gamma$. For simplicity, let us assume the orbifold inherits the manifold of $\mathbb{R}^n$. If we pull back the metric on this manifold, how does it look like? Still be conic?

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