It is a standard exercise in embedding theory to show that $S^3 \to \mathbb{R}^4$ given by $(x,y,z) \mapsto (x^2-y^2,xy,xz,yz)$ induces an embedding $\mathbb{R}P^2 \to \mathbb{R}^4$. Since $\mathbb{R}P^2/\,\mathbb{R}P^1 \cong \mathbb{R}P^2$, the previous map gives an embedding of $\mathbb{R}P^2/\,\mathbb{R}P^1$ into $\mathbb{R}^4$.

Is there a nice embedding of $\mathbb{C}P^2/\,\mathbb{C}P^1$ into $\mathbb{R}^8$?

Is there a simple way to construct such a graph? For example a fully connected graph obviously has degree of separation between every vertex of 1 but has maximal total degree. If we only wanted to minimise the total degree then I think the answer would be a star graph. But I want the average degree to be smallest rather than just relying on a single high degree vertex to be the common neighbour for all vertices. I can sort of see an algorithm starting with a cycle5 graph and adding nodes until the degree of separation between each pair of nodes is <= 2, but not sure if this would be optimal.

Let A, B $\in S^{n}_{+}$, means that A and B are symmetric semidefinite matrix. Can we prove that $tr(AB) = 0$ if and only if $AB = 0$ ?

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{pmatrix}. $$ There is one famous application in physics. Which is that a symmetric bilinear $K_{E_8}$-Chern-Simons theory describes the low energy physics of a so-called $E_8$-quantum Hall state (with a $U(1)^8$ gauge group). The field theory partition function $Z$ is given by $$ Z= \int[DA]\exp(\frac{(K_{E_8})_{IJ}}{2 \pi}\int A_I dA_J). $$ This $E_8$-quantum Hall state occurs in a 2-dimensional spatial condensed matter system.

The above is what I am familiar already. Now I am asking a different question about the **application of $E_8$ manifold**.

My question here is that: Is there some **real-world application of $E_8$ manifold**, such as in physics or in any branch of science, or in the engineer?

The $E_8$ manifold is the unique compact, simply connected topological 4-manifold with intersection form the $E_8$ lattice.

The $E_8$ manifold has no smooth structure.

The $E_8$ manifold is not triangulable as a simplicial complex.

The $E_8$ manifold is constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for $E_8$. This results in P$E_8$, a 4-manifold with boundary equal to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the $E_8$ manifold.

Some refs and introductory level of explanations are welcome! Thanks.

A while ago over at our sister site, there was an interesting question [not by me] with next to no answers which I feel is, fleshed out in a more precise fashion, appropriate for MathOverflow.

**The question.** Which objects and concepts would you regard as *mathematical phantoms*? I'm especially interested in examples from applied mathematics, since I already know a couple of examples from pure mathematics (listed below).

**Mathematical phantoms.** Following John Baez, Gavin Wraith and surely others, a *mathematical phantom* is "an object that doesn't exist within a given mathematical framework, but nonetheless 'obtrudes its effects so convincingly that one is forced to concede a broader notion of existence'.
Like a genie that talks its way out of a bottle, a sufficiently powerful mathematical phantom can talk us into letting it exist by promising to work wonders for us. Great examples include the number zero, irrational numbers, negative numbers, imaginary numbers, and quaternions. At one point all these were considered highly dubious entities. Now they're widely accepted. They 'exist'."

**Being more precise.** The demarcation from what are merely very fruitful abstractions is obviously a bit blurry; perhaps useful criteria are:

- There should be a statement which was held to be obviously true before the discovery of the phantom, but which is false in view of the new concept. (Such as the statement "obviously no number squares to $-1$".)
- It should have required a nontrivial effort to make it precise (to "help it come into being").
- It should have great explanatory power and vast consequences.

**Examples.** Phantoms in this stricter sense could include:

- The irrational numbers. (Running counter to the basic tenet "all is number" by the Pythagorean school, where by "number" they meant "rational number".)
- The complex numbers.
- The $p$-adic numbers.
- Actual infinity, together with the flexible notion of sets we have nowadays (vastly surpassing recursive subsets of $\mathbb{N}$) and the axiom of choice.
- Sobolev function spaces.
- Infinitesimal numbers (as in the hyperreal numbers, where $\varepsilon$ is invertible, or as in synthetic differential geometry, where $\varepsilon^2 = 0$).

Phantoms in a broader sense (where I can't think of any held-to-be-obviously-true statement falsified by them) could include:

- Symmetries of zeros of polynomials, or more generally groups.
- The field with one element.
- Ideals in number theory.
- Motives.
- ∞-categories.
- Toposes. (Generalizing and unifying various cohomology theories.)
- Nonclassical logics. (Born in the foundational crisis, nowadays with applications in mainstream mathematics.)

For the purposes of this question, I'm interested in both kinds of phantoms.

Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).

Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes n} \neq 0 \}$ and $W_A:=D(A)^{\otimes \psi_A}$.

It seems horribly complicated, but in special cases those definitions might have nice properties. Were they studied before?

Note that Nakayama algebras with a linear quiver and $n$ simple modules are counted by the Catalan nubmers $C_{n-1}$ and thus are in bijection with 321-avoiding permutations on $n-1$-symbols. (Nakayama algebras with a linear quiver are exactly the quotient algebras of the ring of upper triangular matrices over the field $K$ by an admissible ideal)

Here some observations/guesses with the computer for Nakayama algebras with a linear quiver:

$W_A$ is injective.

There are exactly $2^{n-2}$ algebras with $\psi_A = 2$ and all of those algebras seem to have global dimension at most 3.

The generating function of the statistic $A \rightarrow \psi_A$ (http://www.findstat.org/StatisticsDatabase/St001290/) seems to coincide with the generating function on 321-avoiding permutations given by $ \pi \rightarrow f(g(\pi))$, where $g$ is the first fundamental transformation on permutations (http://www.findstat.org/MapsDatabase/Mp00086) and $f$ number of right-to-left minima of a permutation (http://www.findstat.org/StatisticsDatabase/St000991).

Now it looks rather horribly to try to prove those things by direct computations but maybe there is a trick or a nice interpretation of $\psi_A$ and $W_A$ for Nakayama algebras and maybe even more general algebras (maybe QF-3 algebras)?

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal $\infty$-category with unit object $\mathbf{Vect}_k$.

The forgetful functor $\operatorname{CAlg}(\mathcal{C}) \to \mathcal{C}$ admits a left adjoint $Sym^*: \mathcal{C} \to \operatorname{CAlg}(\mathcal{C})$. Now let $\mathcal{C}[z]=Sym^*(\mathbf{Vect}_k)$ - i.e. $\mathcal{C}[z]$ is the free stable symmetric monoidal category generated by $\mathbf{Vect}_k$.

My question is: Is the $\infty$-category $\mathcal{C}[z]$ compactly generated in $\mathcal{C}$? (Recall that a category $\mathcal{A}$ is compactly generated if there exists a subcategory $\mathcal{A}_0 \subseteq \mathcal{A}$ and an equivalence $\mathcal{A} \simeq Ind(\mathcal{A}_0)$.)

More generally: If $\mathcal{D} \in \mathcal{C}$ is compactly generated, is $Sym^*(\mathcal{D})$ also compactly generated?

My idea was to show that the map $\mathbf{Vect}_k \to \mathcal{C}[z]$ corresponding to the identity:$ \mathcal{C}[z] \to \mathcal{C}[z]$ under the adjunction above identifies the image of $k$ with a compact generator of $\mathcal{C}[z]$, but I'm not sure if this would work.

Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley group. Let $G=\textbf{G}(F)$ and $K=\textbf{G}(\mathcal{O})$. Let $B=TU$ be a fixed Borel subgroup with torus $T$. Let $\hat G$ be the Langlands dual group of $G$ and $\hat T$ be the dual torus. Let $W_F$ be the Weil group of $F$. Let $\varphi: W_F\rightarrow \hat T\rightarrow \hat G$ be an unramified Langlands parameter. Then associated with $\varphi$, by Satake isomorphism, there is an irreducible spherical representation $\pi(\varphi)$ of $G$. It is known that $\pi(\varphi)$ is a sub-quotient of a principal series representation $\textrm{Ind}_{TU}^G(\mu)$.

$\textbf{Question}$: How to determine $\mu$ in terms of $\varphi$?

If $\textbf{G}=\textrm{GL}_n$, my understanding is as follows. An unramified Langlands parameter $\varphi:W_F\rightarrow \hat T$ is determined by its image of $\textrm{Fr}$. Suppose that $\varphi(\textrm{Fr})=\textrm{diag}(a_1,\dots,a_r)\in \hat T$ with $a_i\in \mathbb{C}^\times$. Then the corresponding $\mu:T\rightarrow \mathbb{C}^\times$ is determined by $\mu(1,\dots,1,\varpi,1,\dots,1)=a_i$, where $\varpi$ is a uniformizer of $F$ in the $i$-th position.

Here is one example I am interested in. Define $J_n\in \textrm{GL}_n$ inductively by $$J_n=\begin{pmatrix} &1\\ J_{n-1}&\end{pmatrix}, J_1=(1).$$ Let $G$ be the split $\textrm{SO}_5$, which is defined by the matrix $J_5$. Then $\hat G=\textrm{Sp}_4(\mathbb{C})$, which is realized by the matrix $$\begin{pmatrix} &J_2\\ -J_2&\end{pmatrix}.$$ Let $\varphi:W_F\rightarrow \hat T$ by $\varphi(w)=\textrm{diag}(1,|w|,|w|^{-1},1)$. Then what is the corresponding $\mu$?

Any comments and references about the general case or the above special example are appreciated.

$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$ I recall Herbrand's theorem about class groups of cyclotomic fields: Let $p$ be an odd prime, let $\zeta$ be a primitive $p$-th root of $1$ and let $K = \QQ(\zeta)$, so $\mathrm{Gal}(K/\QQ)$ is canonically isomorphic to $(\ZZ/p \ZZ)^{\times}$. Let $A$ be the class group of $K$ and let $V = A/p A$, an $\mathbb{F}_p$ vector space. Then $\mathrm{Gal}(K/\QQ)$ acts on $V$ and $V$ splits accordingly into characters of $(\ZZ/p \ZZ)^{\times}$; let $V = \bigoplus V_r$ where $a \in (\ZZ/p \ZZ)^{\times}$ acts by $a^r$ on $V_r$.

For $1 \leq r \leq p-2$ odd, Herbrand's theorem tells us that, if $V_r \neq 0$, then $p$ divides the numerator of the Bernoulli number $B_{p-r}$. For example, since $B_2 = \frac{1}{6}$, we always have $V_{p-2}=0$. First question:

Is there a way to see that $V_{p-2}=0$ without understanding Herbrand's proof?

As an example of what I'm hoping for, it is straightforward to see that $V_{p-1}=V_0=0$. If $V_{p-1}$ were nonzero, class field theory would give an unramified extension $K/\mathbb{Q}(\zeta_p)$ so that $K/\QQ$ is Galois with Galois group $(\ZZ/p) \times (\ZZ/(p-1))$. But then the fixed field of $\ZZ/(p-1)$ is an unramified degree $p$ extension of $\QQ$, violating Minkowski's theorem.

I ask because I am still thinking about this very challenging question. If $V_{p-2}=V_{-1}$ were nonzero, I believe I could show that the ring of integers in the corresponding $(\ZZ/p) \rtimes (\ZZ/(p-1))$ extension of $\QQ$ would give a counter-example to this question. With similar motivation, I ask:

Is there a straightforward way to see that the eigenspace $V_1$ is zero?

This last occurs as Proposition 6.16 in Washington's Introduction to Cyclotomic Fields, but I can't figure out whether it is straightforward or whether it needs the 6 chapters that precede it.

During my research I come accross, on this result :

**Proposition :** Let $E$ be a finite set. If $f,g$ binary laws on $E$, with :

$$\forall a,b,c \in E,\; f(a,f(b,c))=f(g(a,b),c)$$

then $$\forall a,b,c,x \in E, \; f(g(a,g(b,c)),x)=f(g(g(a,b),c),x)$$

>

**Definition :** In this case, we say $g$ is pseudo associative.

**Remark :** This question is very important for calculus $h^{N}(x)$ with $N>2^{10000}$

where we choose a good fonction $h$ and it's very important in cryptanalyse **(1)**.

**Question :** Have there been studies of the notion of pseudo associativity in the literature?

>

>

**(1)** : for example and
roughly, if $h([x,y])=[x+1,x\times y]$ then $h^N([1,1])=[N+1,N!]$
This function can help to calculus the factoriel on set $\mathbb Z/ (p\times q)\mathbb Z$ and factorise $p \times q$

the question is choose the good function $h$ for break the security.

It's possible to do that for the discret log.

Let $V$ be a topological $k$-vector space.

Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all *continuous* linear functionals.

The weak (Mackey) topology is the weakest (strongest) topology on the vector space $V$, such that all continuous functionals $v' \in V'$ remain continuous, and all discontinuous functionals $v' \in V^{\star} \setminus V'$ remain discontinuous w.r.t. the new weak (Mackey) topology.

Let $V_{\sigma} (V_{\tau})$ denote the underlying vector space $V$ equipped with its weak (Mackey) topology. In my opinion its clear that the weak topology on $V$ is an initial topology w.r.t. the family V', i.e. a function $f: U \rightarrow V_{\sigma}$ is continuous iff $v' \circ f$ is continuous for all $v' \in V'$.

By construction, if $f: U \rightarrow V_{\tau}$ is continuous, also $v' \circ f$ is continuous for all $v' \in V'$.

**Does the inverse also hold? I.e. is the Mackey topology also initial? (No its not final)**

Thank you.

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:

Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold $M$ with fiber $F$, compact structure group $G$ and base $B$. Suppose that: i) $B$ has a metric of positive Ricci curvature; ii) $F$ has a $G$-invariant metric of positive Ricci curvature. Then $M$ carries a metric of positive Ricci curvature.

Both the referee and us are not sure if it is in literature or not. It happens that the conjecture is easily proved using classical arguments. Therefore, we would like to ask if someone knows a reference, or if the conjecture is well known to be true among specialists.

Partial results we could find are:

1) (Nash) https://projecteuclid.org/euclid.jdg/1214434973

2) W. A. Poor, Some exotic spheres with positive Ricci curvature, Math. Ann. 216 (1975) 245-252.

Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?

It is known that every surface group can be embedded into any semisimple connected Lie group.

I would be interested in similar results for fundamental groups of closed nonpositively curved manifolds of any dimension, but this is probably much harder. All I know is how to embed the fundamental groups of tori, closed surfaces, and their products.

Is there any Liouville type theorem for the half space problem \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\ v & =0 &&\text{in } \mathbb R^N \setminus \mathbb R^N_+\end{aligned} \right. \end{equation} where $s\in (0, 1)$ and $v$ changes sign. If $v$ is bounded, does it imply $v \equiv 0.$

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$.

For a paper, I need the result that $$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\varepsilon x^{\varepsilon} \tag{$*$},$$ for all $\varepsilon > 0$. I have a proof of this using complex analysis and Perron's formula, but this seems a bit overkill given that I'm looking for a weak upper bound for a problem in elementary number theory.

Does anyone know of a short elementary proof of the bound $(*)$? Or better yet, a reference?

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that preserve the fiber structure of the projection $\mathbb T\times\mathbb R\to\mathbb T$ in the sense that $h(\{z\}\times\mathbb R)=\{h(z)\}\times\mathbb R$ for any $z\in\mathbb T$ where $\mathbb T:=\{z\in \mathbb C:|z|=1\}$ is the unit circle on the complex plane.

As I understand, the group $G$ is a semidirect product $H(\mathbb T)\rtimes C(\mathbb T,H(\mathbb R))$ of the homeomorphism group $H(\mathbb T)$ of the circle and the group $C(\mathbb T,H(\mathbb T))$ of continuous maps form the circle to the homeomorphism group $H(\mathbb R)$ of the real line.

Do you know any paper that studies this automorphism group $G$ (from algebraic or topological point of view)? Maybe in a more general context of automorphism groups of fiber bundles or foliations?

In Question 315259 (cf. Primes arising from permutations) I asked a question on primes arising from permutations which looks quite challenging.

Here I pose a new question in this direction which does not involve upper bounds for the least prime in an arithmetic progression with common difference $n$.

QUESTION: Is my following conjecture true?

**Conjecture**. (i) For each $n=1,2,3,\ldots$, there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that $k^2+k\pi_n(k)+\pi_n(k)^2$ is prime for every $k=1,\ldots,n$.

(ii) For any positive integer $n\not=7$, there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that $k^2+\pi_n(k)^2$ is prime for every $k=1,\ldots,n$.

(iii) For each $n=1,2,3,\ldots$, the number of permutations $\pi_n$ of $\{1,\ldots,n\}$ with $k^2+\pi_n(k)^2$ prime for all $k=1,\ldots,n$, is always a square.

I have checked this conjecture for $n$ up to $11$. For example, $(6,3,2,5,4,1)$ is the unique permutation of $\{1,\ldots,6\}$ meeting the requirement in part (i) with $n=6$, and $(1,3,2,5,4)$ is the unique permutation of $\{1,\ldots,5\}$ meeting the requirement in part (ii) with $n=5$. Part (iii) of the conjecture looks quite mysterious!

Let $r(n)$ be the number of permutations $\pi_n$ of $\{1,\ldots,n\}$ meeting the requirement in part (i), and let $s(n)$ be the number of permutations $\pi_n$ of $\{1,\ldots,n\}$ meeting the requirement in part (ii). Then $$(r(1),\ldots,r(11))=(1,1,3,1,5,1,17,9,21,16,196)$$ and $$(s(1),\ldots,s(11))=(1,1,1,1,1,4,0,16,4,144,64).$$

Let $k$ be a positive real number. Which probability distribution over $\mathbb R$ maximizes $P(|x-E(x)|>k\cdot \operatorname{std}(x))$?

Let $\bar{X}$ be a complete smooth variety over $\mathbb{C}$ and $D$ be a simple normal crossing divisor. Denote $X:=\bar{X}\backslash D$. Then it is known that $H^\ast(X,\mathbb{C})$ admits a canonical mixed Hodge structure.

Denote $j:X\to\bar{X}$ be the open immersion and $A_X^\ast$ be the complex of sheaves of differential forms on $X$. Let $F^p$ be the subcomplex of $A_X^\ast$ consists of $(i,j)$-forms such that $i\geq p$. My question is

Dose $(j_\ast A_X^\ast,j_\ast F^\bullet,\tau^{\leq\bullet} j_\ast A_X^\ast)$ induces the canonical mixed Hodge structure on $H^\ast(X)\simeq H^\ast(\bar{X},j_\ast A_X^\ast)$?

Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.

The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $

It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form $ ln(z)e^z $ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set $$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$