Consider configurations consisting of 4 distinct circles on the sphere. Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/classify such configurations?

Equivalent problem: classify the arrangements of 4 hyperbolic planes in the hyperbolic space, up to homeomorphisms of the space.

Before voting to close this question as trivial, you may look at the classification of generic configurations which we obtained by brute force:

Each region bounded by more than 3 sides is labeled by the number of its boundary sides. This is used to show that all configurations are non-equivalent.

Questions: Is this new? Is there a scientific method to obtain this? Is there any structure on these 35 configurations?

There is a large research area about hyperplane arrangements in a Euclidean space. How about hyperbolic space? There is also a large body of research on hyperbolic tetrahedra. But it is always assumed that the tetrahedron is compact (or has only vertices at infinity).

We encountered this question in our studies of the Heun and Painlevé VI equations with real coefficients. (See Appendix II). Projective monodromy groups associated to these equations are generated by 4 reflections in circles.

For theories with well known proof-theoretic-ordinals, (what) is there a correspondence between their proof-theoretic-ordinal and (ordinal indexed families of?) fast growing functions provable total in a given theory ?

Recently I came across a paper which I really need to read through and which uses language of Morava $E$-theory. Since I'm not comfortable with this cohomology theory, I've been looking for quite a while some source to read (at least the basics) but wasn't able to find out something concrete. Does anyone know where can someone learn about Morava $E$-theory? I highly doubt that some self-contained textbook exists that covers this particular material, therefore any instructive paper or preprint is what am looking for most likely!

P.S. The same question had been asked yesterday on MSE, where I got no answer or comment (I deleted today). I didn't know if it is or not suitable for MSE from the beginning, therefore I ask here!

Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely generated (thus $FP_0$ means finitely generated and $FP_1$ means finitely presented). Consider an extension of modules

$$0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0.$$

Assume that $M$ is of type $FP_n$ and that $M''$ is of type $FP_m$. What can we say about $M'$? I suppose that I am also interested in the other possibilities (where finiteness properties of $M'$ and $M$, or of $M'$ and $M''$, are given), but the one I indicated above is the most relevant one for what I am doing. Any references for these kinds of finiteness conditions in homological algebra are also welcome (the only one I know is Brown's book on group cohomology, and of course it focuses on examples coming from group cohomology).

There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \omega_1)\simeq \mathrm{Ham}(M_2, \omega_2)$ then there exists a diffeomorphism $f:M_1\rightarrow M_2$ such that $f^*\omega_2=\lambda \omega_1$ for some $\lambda\in\mathbb{R}^{*}$. In other words, a symplectic structure on a closed manifold is determined (up to rescaling) by the group of Hamiltonian diffeomorphisms.

The result of Banyaga tells us that, in principle, it should be possible to understand the topology of $M$ only from $\mathrm{Ham}(M, \omega)$. However, as far as I understand, it's pretty hard to actually give an algorithm for doing that.

So my question is: are there any explicit procedures for recovery of algebro-topological invariants of $M$ from $\mathrm{Ham}(M, \omega)$? I am particularly interested in recovery of the fundamental group $\pi_1(M)$.

A weaker question would be: is there any explicit condition on $\mathrm{Ham}(M, \omega)$ which guarantees that $M$ is simply-connected?

I am aware of one comparatively weak result in this direction. Banyaga has constructed a map (called flux map) $f:\pi_1(\mathrm{Symp}_0(M, \omega))\rightarrow H^1(M, \mathbb{R})$. He has also shown that there is an isomorphism $$ \mathrm{Symp}_0(M, \omega)/\mathrm{Ham}(M, \omega) \simeq H^1(M, \mathbb{R})/\mathrm{ker}\:f. $$ Therefore, if we know both $\mathrm{Symp}^0(M, \omega)$ and $\mathrm{Ham}(M, \omega)$ we can at least recover a quotient of $H^1(M, \mathbb{R})$ (so we have a lower bound for its rank, for example).

P.S.: A clear exposition of some results on the group of Hamiltonian diffeomorphisms can be found in Polterovich's book "The geometry of the group of symplectic diffeomorphisms" (chapter 14 is particularly relevant to the question).

Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional $$ F(\pi) = \mathbb{E}_\pi |x-y| $$ It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the entropy of $\pi$: $$ \mathbb{E}_\pi \left[ \ln \pi \right] \geq C $$

Is there an analytical solution to $\min_\pi F(\pi)$ subject to such an entropy constraint? In case of multiple solutions, I'd like the one(s) closest to some given $\pi_0$ (in the $L_p$ sense for a convenient $p$).

In the absence of analytical solutions, numerical methods would be useful. The entropy constraint can be addressed by a Lagrange multiplier, but perhaps there is some elegant way to deal with its non-linearity.

Now, the functional I'd really like to minimize over $L_1$ is $$ F_\alpha(\pi) = \mathbb{E_\pi} \left[ |x-y| \cdot (\pi^\alpha (x) + \pi(y)^\alpha) \right] $$ for $1 \leq \alpha \leq 2$. Perhaps, start with $\alpha=1$.

The entropy constraint isn't critical, but I will start with some $\pi_{init}$ and would like to prevent unnecessary entropy loss if possible.

Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some even built toy models for a quantum computer in the lab. For instance, see IBM's 50-qubit quantum computer.

However, some scientists are not that optimistic when it comes to the predicted potential advantages of quantum computers in comparison with the classical ones. They believe there are *theoretical obstacles* and *fundamental limitations* that significantly reduce the efficiency of quantum computing.

One mathematical argument against quantum computing (and the only one that I am aware of) is based on the Gil Kalai's idea concerning the sensitivity of the quantum computation process to noise, which he believes may essentially affect the computational efficiency of quantum computers.

**Question.** I look for some references on similar theoretical (rather than practical) mathematical arguments against quantum computing — if there are any. Papers and lectures on potential theoretical flaws of quantum computing as a concept are welcome.

**Remark.** The theoretical arguments against quantum computing may remind the so-called *Goedelian arguments* against the artificial intelligence, particularly the famous Lucas-Penrose's idea based on the Goedel's incompleteness theorems. Maybe there could be some connections (and common flaws) between these two subjects, particularly when one considers the recent innovations in QAI such as the Quantum Artificial Intelligence Lab.

In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof.

Given a quadratic surd $\sqrt d$ where $d$ is a natural number and not a perfect square. $(c_i)_{i=1}^\infty$ is the sequence of convergents of the continued fraction of $x$. Let $r_i:=\frac{c_{i+1}-\sqrt d}{c_i-\sqrt d},\,\forall i\in\mathbf N$. Let $n$ be the period of the continued fraction. It has been shown $\exists \,l_r:=\lim_\limits{i\rightarrow\infty}r_{in+r},\, \forall r\in\{0,1,\cdots,n-1\}$. Is the following statement, suggested by a numerical experiment, true?

For $n\ge 3$, there exists at least two distinct $l_r$'s. For $n\le 2$, $l_0=l_1$.

Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(A)|$, where $\lambda_i(A)$'s are eigenvalues of $A$.

Note that, by minimizing $\sum_{i=1}^n |\lambda_i|$ over two constraints $\sum_{i=1}^n \lambda_i = 0$ and $\sum_{i=1}^n \lambda_i^2= n(n-1)$, one can obtain $\sqrt{2n(n-1)}$ as a lower bound. But it seems that isn't tight.

On the other hand, if $A := J - I$ (all ones matrix minus identity), then $\sum_i |\lambda_i(A)| = 2(n-1)$.

Is it true that $2(n-1)$ is actually a lower bound (for large enough matrices, say $n \geq 10$) ?

**Added:**

As Alex's answer below, the minimum of trace norm of such matrices may be less than $2(n-1)$, even for arbitrarily large matrices.

@fedja, in a comment below, has been said that the minimum is $(2+o(1))n$ as $n\to\infty$. Could someone give a proof for that?

Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let $X_1$ be supported on $[-1,1]$. Can $(X_1,X_2)$ be supported on the unit disc centered at the origin?

(Under the stronger condition that the sequence is i.i.d., the joint distribution of $(X_1,X_2)$ must be supported on the square $[-1,1]^2$.)

$\bf{Edit:}$ In response to @Nate Eldredge. I will use the following definition of ergodic:

Let $\mu$ be the (shift-invariant) measure induced on $\left(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}})\right)$ by the stationary stochastic process $X:=\{X_{t}\}_{t\in\mathbb{N}}$. Let $T:\,\mathbb{R}^{\mathbb{N}}\rightarrow\mathbb{R}^{\mathbb{N}}$ be the left shift operator mapping sequences $\{x_{t}\}_{t\in\mathbb{N}}$ onto $\{x_{t+1}\}_{t\in\mathbb{N}}$.

I will say $\{X_{t}\}_{t\in\mathbb{N}}$ is ergodic if for any measurable $f\in L^{1}(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}),\mu)$, the averages $\frac{1}{T}\sum_{t=1}^{T}f(T^{t-1}X)$ converge pointwise almost everywhere to $\int_{\mathbb{R}^{\mathbb{N}}}f(x)d\mu$.

My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.

** Q1**. Is it the case that the maximum

Let $C$ be a smooth curve in $\mathbb{R}^3$, whose maximum curvature at any point $x \in C$ is $\le 1$. Now consider a tubular neighborhood of $C$— (used also in Light rays bouncing in twisted tubes)— width of $r<1$.

** Q2**. Let the curvature of the smooth $C \in \mathbb{R}^3$ be bound by $\le 1$.
What is (a description of) the maximum volume convex shape that could move
(via rigid motions) through

A smooth curve $C$ with curvature everywhere $\le 1$. Tube of radius $r < 1$.

I presume the optimal shape is convex. I suspect this question has been considered previously...?

Related: Sofa in a snaky 3D corridor.

Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as $$ Rad_G(g) = \left\{r \in G \mid r^a = g^b \mbox{ for some } a,b \in \mathbb{Z}\setminus\{0\}\right\} \cup \{1_G\}. $$

For which torsion-free groups can we show that $Rad_G(g)$ is an infinite cyclic subgroup of $G$ for every nontrivial element? So far I have been able to establish it for the following classes:

- residually finitely generated torsion-free nilpotent groups,
- torsion-free hyperbolic groups,
- relatively hyperbolic groups (in the sense of Bowdich), where the associated subgroups already have the property (this includes for example toral relatively-hyperbolic groups).

Are there any easy examples I am missing?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my limited understanding, $J^{\#}$ is a sort of a TQFT (with $\mathbb{F}_2$ coefficient), or a functor from the category of webs and foams to the category of $\mathbb{F}_2$ vector spaces and morphisms.

There is a familiar TQFT in one lower dimension which is clearly related to the number of Tait colorings : namely, the (Chern-Simons) quantum invariant for the second symmetric representation of $SU(2)$. In the classical limit $q=1$, it is exactly the number of Tait colorings.

I know this is probably just a far-fetched speculation, but I am curious if these two TQFTs can be somehow related. More precisely, **is it possible to categorify the latter such that it becomes $J^{\#}$ when reduced to the $\mathbb{F}_2$ coefficient?**

Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=X\times X/\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts by $(x,y)\mapsto (y,x)$. Then there is a birational map $X^{[2]}\to X^{(2)}$. Let $E$ denote the exceptional divisor if this map. Or $E$ can be described as the divisor whose locus is the set of non-reduced subschemes. Can someone point a nice reference where it is explained that there is a line bundle, whose square is the line bundle corresponding to $E$.

Alternatively, the question can be posed as follows. Let $Y$ denote the blow up of the diagonal of $X\times X$. Then the action of $\mathbb{Z}_2$ extends to $Y$, (I think this action is trivial when restricted to the exceptional divisor). The quotient is $\pi:Y\to X^{[2]}$. How do I see that there is a divisor $F$ on $X^{[2]}$ such that $2F=\pi_*(E)$?

I am looking for a reference or explanation of this fact.

Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm looking for an elementary proof that there are infinitely many distinct (unitary) characters $\chi$ of $E^\times$ such that ${}^\sigma \chi = \chi \circ \sigma \neq \chi$.

Here's a sketch: the character $\chi$ is Galois invariant, i.e., ${}^\sigma \chi = \chi$ if and only if $\chi$ is trivial on the kernel of the norm map $N_{E/F}: E^\times \rightarrow F^\times$. Let $K = \ker N_{E/F}$. The group $K$ is a closed subgroup of the compact group $U_E$ of units in $E^\times$. We can extend any non-trivial character $\tilde \chi$ of $K$ to $E^\times$ to obtain a nontrivial character $\chi$ of $E^\times$ such that ${}^\sigma \chi \neq \chi$.

The part of the argument that is missing is to show that either:

(a) the character group $\widehat K = Hom(K,S^1)$ of $K$ is infinite,

or, if (a) is false (?),

(b) if $\widehat K$ is finite, then we need to show that there are infinitely many distinct extensions of at least one $\tilde \chi \in \widehat K$ to $E^\times$

I expect that (a) is true. Any suggestions to address this fact (or a reference) would be greatly appreciated.

In recent question we have discussed a coeffcients $A_{m,j}$, such that for every integer $n\geq0$ we have identity \begin{equation} n^{2m+1}=\sum\limits_{1\leq k \leq n}D_{m}(n,k) \end{equation} where $D_m(n,k)$ is defined by \begin{equation*} D_{m}(n,k):=A_{m,m}k^m(n-k)^m+A_{m,m-1}k^{m-1}(n-k)^{m-1}+\cdots+A_{m,0} \end{equation*} Coefficients $A_{m,j}$ in above definition are terms of OEIS sequences A302971 and A304042 for numerators and denominators of $A_{m,j}$, respectively. Consider a few examples for some positive integers $m,n$. Let be $m=3, \ n=4$, then \begin{eqnarray*}\label{gen_22} 4^{2\cdot3+1} &=&1-14\cdot3^1+0\cdot3^2+140\cdot3^3 \\ &+&1-14\cdot4^1+0\cdot4^2+140\cdot4^3\\ &+&1-14\cdot3^1+0\cdot3^2+140\cdot3^3\\ &+&1-14\cdot0^1+0\cdot0^2+140\cdot0^3\\ &=&3739+8905+3739+1=16384 \end{eqnarray*} Where coefficients $\{A_{3,j}\}_{j=0}^{3}=\{1,-14,0,140\}$ are terms of third row of A302971 and $\{3,4,3,0\}$ are terms of forth row of triangle A094053. Similarly, let show example for $m=4, \ n=5$, we get \begin{eqnarray*}\label{gen_24} 5^{2\cdot4+1} &=&1-120\cdot4^1+0\cdot4^2+0\cdot4^3+630\cdot4^4 \\ &+&1-120\cdot6^1+0\cdot6^2+0\cdot6^3+630\cdot6^4\\ &+&1-120\cdot6^1+0\cdot6^2+0\cdot6^3+630\cdot6^4\\ &+&1-120\cdot4^1+0\cdot4^2+0\cdot4^3+630\cdot4^4\\ &+&1-120\cdot0^1+0\cdot0^2+0\cdot0^3+630\cdot0^4\\ &=&160801+815761+815761+160801+1=1953125 \end{eqnarray*} As it in previous example, the terms are of sequences A302971 and A094053 with respect to $m,n$. We can observe that the number of lines in each example is $n$ and the number of terms in each line is $m+1$, therefore, we can assume that result of each example could be reached by operations on certain $n,m-1$ dimension matrices. Hence, the question stated

**Question**: Is it possible to implement above method in terms of certain matrices? And, if yes, how?

Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ defined as, $F_1 = \oplus_{k=0}^{\infty} Sym^k(H_n)$ and $F_{-1}= \oplus_{k=0}^{\infty} \Lambda^k(H_n)$.

Physically for a Quantum Field Theory one "defines" its so-called Hilbert space as the dual of an implicit vector space over $\mathbb{C}$ whose basis is in bijective correspondence to the set of all possible values for all the classical fields that occur in the underlying Lagrangian.

Now my question is two fold,

Does this physical notion of a "Hilbert space of a QFT" correspond to the $H_n$ or some ``total Fock space" that can be defined from the first mathematical definition as, $\otimes_{i \in Fields} F^i_{p_i}$ where $p_i=$1 if the $i^{th}$ field is Bosonic or $-1$ if it is Fermionic? (..I guess this tensoring is needed because the QFT can have both Fermionic as well as Bosonic fields..)

If we agree as above that the states of a QFT live in such a "total Fock space" and not in the Hilbert space defined in the second paragraph then shouldn't the "Quantum Field" be mapping into a space of Hermitian operators on this total Fock space and not just the Hilbert space?

We are studying the behavior of families of curves inside stable families of surfaces. The non-existance of the following configurations of curves in a non-normal surface would be sufficient to prove our result.

Let $X$ be a non-normal surface over $\mathbb{C}$ with non-normal locus (i.e. double locus) $X_{dl}$.

Let $D \subset X$ be a nodal curve, with node $p$ so that $p \subset X_{dl}$, the pair $(X,D)$ has semi-log canonical singularities, and $K_X + D$ is ample.

Let $C = C_g \cup C_0$ be a genus $g(C) = g$ irreducible curve, which is the union of a genus $g$ curve $C_g$ and a rational curve $C_0$. Further suppose that $C_g \cap C_0 = p$ (the node of $D$), and that $C_0 \subseteq X_{dl}$.

Finally, suppose that $C_0 \cap D = p$ and the normalization of $X$ is irreducible.

Is such a configuration possible? For instance, is $C_0 \cap D$ forced to be more than one point?

I am looking at the first proof of the existence of a fundamental solution for Linear partial differential equations with constant coefficient (The 3.1.1, Linear Partial Differential Operators, Hormander, Springer Verlag 1964), using Hahn-Banach. In order to construct a fundamental solution such that $$ P(D) E = \delta $$ Hormander uses a lemma giving for $u$ a smooth function with compact support : $$| u(0) | \leq C || P(D)u || .$$ He constructs a function $E \ast \delta_0$ which is defined as $P(D) u \mapsto u(0)$ on the vector space of functions $\{g | g = P(D)u \text{ for some } u \}$ and then extended to $E \ast \delta_0 : \mathcal{D}(\mathbb{R}^n) \to \mathbb{R} $ by Hahn-Banach. My naive question is how is $E \ast \delta_0$ well defined in the first place, as when considering a function $f = Du$, $u$ and thus $u(0)$ may vary ? I agree that once $E$ is fixed then $u$ and $u(0)$ are unique, but I don't see how this can be at the beginning of the proof. Is any of you familiar with this proof ? Am I missing something obvious ?

Thanks.