It is known that $\alpha$-Hölderness impose restrictions on the bad behavior of a graph in the sense of dimension. To put it precisely, let $I=[0,1]$ and $f:[0,1] \to \mathbb{R}$. Then, if $f$ is $\alpha$-Hölder, its graph $ \Gamma_f = \{(x,f(x)):x\in I\}$ has Hausdorff dimension at most $2-\alpha$. Equivalently, if $\operatorname{dim}_H(\Gamma_f) > 2 - \alpha$, then $f$ can't be $\alpha$-Hölder.

The previous example is an instance of the following principle: if the graph of $f$ is bad enough in the sense of dimension, then the regularity of $f$ is bounded above. By "bouded above" I mean "less than $C^\alpha(I)$" as in the previous example.

Are there reverse implications in the spirit of "upper bound on dimension of $\Gamma_f$ implies some regularity of $f$"? I'm not necessarily talking about being $C^\alpha$, it can be some weaker notion of regularity: being in some Sobolev space, agreeing with a $C^\alpha$ function on some big set, being approximable in a particularly good/efficient way, etc...

How should I go about proving that K3,3 is not planar using the Jordan curve theorem?

I am intending to know some of the open problems on integrals,series and number theory with rewards. Also if there is any site to earn small monetary rewards by solving mathematical problems.

Can it be shown, on the assumption that $ZF$ is consistent, that there is a model of $ZF$ in which the reals cannot be well-ordered but there does exist a set of reals which is not Lebesgue measurable?

Let $\overline {\mathcal{M}_{g_{i}, n_{i}}}, \ i \in \{1, 2\},$ be a moduli stack of pointed stable curves of type $(g_{i}, n_{i})$ over a finite field $\mathbb{F}_{p}$. For any algebraic stack $\mathcal{X}$, we write $|\mathcal{X}|$ for the set of points of $\mathcal{X}$ (cf. Stack projects, properites of algebraic stacks, Definition 4.2). My question is as follows:

Is the natural surjective morphism of the sets of points of $$|\overline {\mathcal{M_{g_{1}, n_{1}}}}\times_{\mathbb{F}_{p}}\overline {\mathcal{M_{g_{2}, n_{2}}}}| \rightarrow |\overline {\mathcal{M_{g_{1}, n_{1}}}}|\times_{|\text{Spec} \ \mathbb{F}_{p}|}|\overline {\mathcal{M_{g_{2}, n_{2}}}}|$$ a homeomorphism?

I know that the morphism between the set of points of the fiber product of two algebraic stacks and the fiber product of the sets of points of two algebraic stacks is not a homeomorphism in general. Does the special case mentioned above hold?

I am writing a paper on the topological structure of the Golomb space (defined here) and arrived to the following question:

**Question 1.** Is it true that for a number $a\in\mathbb N$ the equation $x^2+x=a$ has an integer solution $x$ if and only if for any number $b\in\mathbb N$, coprime with $a$, the equation $x^2+x=a \mod b$ has a solution $x$ (i.e., a solution in the ring $\mathbb Z/b\mathbb Z$).

In fact, I need a more general fact.

**Question $2^n$.** Is it true that for any number $n\ge 0$ and any number $a\in\mathbb N$ the equation $(x^2+x)^{2^n}=a$ has an integer solution $x$ if and only if for any number $p\in\mathbb N$, coprime with $a$, the equation $(x^2+x)^{2^n}=a \mod p$ has a solution $x$ (i.e., a solution in the ring $\mathbb Z/p\mathbb Z$).

We can also ask a more general

**Problem.** For which monic polynomials $f\in\mathbb Z[x]$ the following local-to-global principle holds:

$(*)$: for every $b\in\mathbb N$ the equation $f(x)=a$ has a solution in $\mathbb Z$ if and only if for any $b\in\mathbb N$, relatively prime with $a$ the equation $f(x)=a \mod b$ has a solution in the ring $\mathbb Z/b\mathbb Z$?

**Remark.** I have edited the second question a little bit.

Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form

$G/H\times D^n$, where $D^n$ has trivial $G$-action, or

$G/H\times D(V)$, where $D(V)$ is the unit disk of a $G$-representation $V$, or

$G\times_H D(V)$, where $D(V)$ is the unit disk of an $H$-representation $V$

(The subgroup $H$ and representation $V$ is allowed to vary for different cells. Cells of type 1 are what are used in the standard definition of $G$-CW complex.)

Megan Shulman, on p.47 of her thesis (link goes directly there), says regarding standard $G$-CW complexes that

If we are interested in putting CW structures on spaces found in nature, this is a very restrictive definition...

However, I seem to remember that even for a $G$-cell complex $X$ constructed with cells of type 3, we can always rearrange (triangulate?) somehow to give $X$ a standard $G$-CW complex structure. Is that true or false? Which papers look into this?

Ferland and Lewis, on p.23 of their paper (link goes directly there), say that cells of type 3

... are of interest because they arise naturally from equivariant Morse theory (see, for example, [**21**]). Further, if $G$ is a finite abelian
group, then the usual Schubert cell structure on Grassmannian manifolds generalizes in an obvious way to a generalized $G$-cell structure on the Grassmannian manifold $G(V, k)$ of $k$-planes in some $G$-representation $V$ (see Chapter 7).

[**21**] A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150.

However, skimming over at that section of Wasserman's paper (link goes directly there), I must admit I don't see where anything like cells of type 3 show up, much less demonstrate their usefulness. Could anyone explain their relationship to equivariant Morse theory?

Also, in their Chapter 7 (link goes directly there; look at the bottom of p.83 & top of p.84 in particular), I'm not sure I see where cells of type 3 are necessary, because it seems like $W$ is inherently a $G$-representation, and hence they are only obtaining cells of type 2. Could anyone clarify this?

So to summarize, I'd greatly appreciate any of the following:

simple / natural examples that illustrate the additional flexibility of each generalized cell:

spaces which can easily be seen to have a cell structure when cells of type 3 are allowed, but not when only cells of type 2 are allowed

spaces which can easily be seen to have a cell structure when cells of type 2 are allowed, but not when only cells of type 1 are allowed

any other reasons why cells of type 2 and 3 are reasonable

a citation / proof / counterexample for this half-remembered "fact" that cells of type 1 suffice

Let $C$ be a small category: we define its *nerve* $(N(C)_k)_k$ as the following simplicial set: $N(C)_0=Ob(C)$ (the set of objects), $N(C)_1=Mor(C)$ (the set of all morphisms) and $N(C)_k$ to be a set of all $k$-tuples of compasable morhpisms $(f_1,...,f_k)$. This is equipped with the face maps $d_i$ defined by $d_i(f_1,...,f_k):=(f_1,...,f_{i-1},f_i \circ f_{i+1},f_{i+2},...,f_k)$ except of $d_0$ and $d_k$ which omit the first and the last morphism. Degeneracies act as follows $s_j(f_1,...,f_k):=(f_1,...,f_{j-1},id,f_j,...,f_k)$.

When $G$ is a group then $G$ can be viewed as a small category with only one object and the set of morphisms being $G$ itself. Thus everything is composable and $N(G)_k=G^k$.

One thus arrives at the well defined simplicial set. Therefore one can speak about its geometric realization $|N(G)|$: there are in fact two possible versions, the standard one, sometimes called ,,thin realization'' and another version, which is called ,,thick'' realization which is denoted by $\| N(G) \|$ and which is defined using only relations involving the face maps. See also this discussion.

I was told that this geometric realization is in fact (possible version of) the classifying space $BG$. I wonder why it is true. This question is motivated by this discussion. Since my original motivation was the statement that the group cohomology coincides with the (singular for example) cohomology of the classifying space and with the help of the cited discussion I was able to understand why is it true directly (not invoking the fact that $BG$ is geometric realization of $N(G)$) I'm posting this question as another topic. So my question is:

Why the geometric realization of $N(G)$ is (homotopy equivalent to) $BG$?

Some ideas can be found in Hatcher's book ,,Algebraic Topology'' however, as I understood correctly, the face maps $d_i$ from this book would act rather as $(g_1,...,g_k) \mapsto (g_1,...,g_{i-1},g_{i+1},...,g_k)$ which is not the same as for $N(G)$.

EDIT: The classifying space is not unique up to homeomorphism, but it is unique up to homotopy: the definition which is most familiar for me is the abstrack definition as the quotient of contractible principial $G$-bundle $EG$. There is also specyfic *construction* due to Milnor using infinite join which is also familiar to me.

EDIT 2: As I understood correctly the answer below: each element from $\mathcal{E}G_n$ being of the form $(g_0,g_1,...,g_n)$ may be presented in the form $(g_0',g_0'g_1',...,g_0'g_1'...g_n')$. Solving in $g_0',...,g_n'$ gives $g_0'=g_0,g_1'=g_0^{-1}g_1,...,g_n'=g_{n-1}^{-1}g_n$. Thus we obtain a bijective map $\alpha:\mathcal{E}G_n \to \mathcal{E}G_n$ given by $\alpha(g_0,...,g_n)=(g_0,g_0^{-1}g_1,...,g_{n-1}^{-1}g_n)$. Let $d_i,s_j$ be the face maps and degeneracies which you described (omitting $i$-th vertex and repeating) and $d_i',s_j'$ be faces and degeneracies as in the definition of the nerve of the category. Then $\alpha s_j=s_j' \alpha$ and $\alpha d_i=d_i' \alpha$ for all $j$ and for all $i$ *except* $i=0$ where we obtain the difference on the zeroth coordinate $(\alpha d_0)(g_0,...,g_n)=(g_1,...)$ and $(d'_0 \alpha)(g_0,...,g_n)=(g_0^{-1}g_1,...)$ (the remaining entries are the same). Therefore if we pass to the quotient we get an equality. This suggest that there is something correct in this procedure: however I'm little bit confused with the following: $\mathcal{E}G$ is a simplicial set. Its geometric realization is just topological space: the bar notation which you are using refers to $EG/G$ which is also topological space. However you described face and degeneracies maps for $EG/G$ which looks like the relevant maps for $NG$ but the former is just topological space while the latter is (abstract) simplicial set.

In the 2010 paper Quantum Money from Knots Farhi, Gosset, Hassidim, Lutomirski, and Shor give a doubly stochastic Markov chain acting on grid diagrams. Transitions in the Markov chain are permutations of the configuration space of grid diagrams, given by random Cromwell moves. They imply that the security of their quantum money system is dependent on the rapid *polynomial-time* mixing of their Markov chain. They comment that, at least at the time it was written, even for the unknot it wasn't known if there were two equivalent grid diagrams requiring a superpolynomial number of moves to go from one to the other.

In the 2013 paper A polynomial upper bound on Reidemeister moves Lackenby shows that the number of Reidemeister moves needed to untangle a diagram of the unknot with $c$ crossings is, at most, $(236c)^{11}$. From other comments, it appears that Lackenby has extended the above to show that arbitrary knots can be converted to one another with a polynomial number of Reidemeister moves.

Is Lackenby's result strong enough to show, or lend credence to, Farhi, Gosset, Hassidim, Lutomirski, and Shor's conjectured polynomial-time mixing?

I envision Lackenby's result as putting a polynomial upper bound on the diameter/God's number of the graph of Reidemeister moves - similar to the graph of Cromwell moves that can be randomly walked with Farhi, Gosset, Hassidim, Lutomirski, and Shor's Markov chain.

I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems strange. Is there a way to verify the correctness of these computations without relying on numerical simulations, perhaps ?

**Some known results**

**The Girsanov formula for fBm**
Let $B^H \left(t\right)$ denote fractional brownian motion with mean $0$ and variance $t^{2H}$, for all $H \in \left(0, 1\right)$ defined on $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$. Let $a$ be a scalar. Define a new probability measure $\mathbb{Q}$ on $\left(\Omega, \mathcal{F}\right)$ via the Radon Nikodym derivative with respect to $\mathbb{P}$ \begin{equation}
\frac{\mbox{d}{\mathbb{Q}}}{\mbox{d}{\mathbb{P}}} = \exp\left\{-\mu M_T - \frac{1}{2}{\mu}^2 \langle M, M\rangle_T \right\}\label{girsanovRadonNikodymDerivativeWithInnerProduct}
\end{equation}
where
\begin{equation}
M_T = \frac{1}{2 H \Gamma \left(\frac{3}{2} - H\right) \Gamma \left(H + \frac{1}{2}\right)} \int_0^T \left(s\left(T - s\right)\right)^{\frac{1}{2} - H} \mbox{d}{{B_s}^H}.
\end{equation}
The process $M_T$ is a martingale with independent increments, zero mean and variance function $c^2 T^{2 - 2H}$ where
\begin{equation}\label{c}
c = \sqrt{\frac{\Gamma\left(\frac{3}{2} - H\right)}{2 H \left(2 - 2H\right)\Gamma\left(H + \frac{1}{2}\right)\Gamma\left(2 - 2H\right)}}.
\end{equation}
Then the process defined, for all $t \in \left[0, T\right]$, by $B^H \left(t\right) + \mu t$ is the standard $\mathbb{Q}$-fractional Brownian motion on $\left[0, T\right]$. In other words, under probability measure $\mathbb{Q}$,
$B^H \left(t\right)$ restricted to $t \in \left[0, T\right]$ is distributed as an arithmetic fractional Brownian motion with drift $\mu$.

A proof of the Girsanov formula for fractional Brownian motion can be found in Norros's paper, where the term the fundamental martingale is also coined for the process $M_t$. It's noteworthy that using the variance of $M_t$, Radon Nikodym Derivative can also be re-written as \begin{equation} \frac{\mbox{d}{\mathbb{Q}}}{\mbox{d}{\mathbb{P}}} = \exp\left\{-\mu M_T - \frac{1}{2}{\mu}^2 c^2 T^{2 - 2H} \right\}.\label{girsanovRadonNikodymDerivative} \end{equation}

**corollary**
Let $X\left(t\right) = b B^H \left(t\right)$ be an arithmetic fractional brownian motion with volatility $b$, for all $H \in \left(0, 1\right)$ defined on $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$. Let $a$ be a scalar. Define a new probability measure $\mathbb{Q}$ on $\left(\Omega, \mathcal{F}\right)$ via the Radon Nikodym derivative with respect to $\mathbb{P}$
\begin{equation}
\frac{\mbox{d}{\mathbb{Q}}}{\mbox{d}{\mathbb{P}}} = \exp\left\{-\frac{a}{b} M_T - \frac{1}{2}\frac{a^2}{b^2} c^2 T^{2 - 2H}\right\}.\label{scaledGirsanovRadonNikodymDerivative}
\end{equation}
Then the process defined, for all $t \in \left[0, T\right]$, by $Z \left(t\right) = X \left(t\right) + a t$ is an arithmetic $\mathbb{Q}$-fractional Brownian motion process on $\left[0, T\right]$ with volatility $b$. In other words, under probability measure $\mathbb{Q}$, $X\left(t\right)$ restricted to $t \in \left[0, T\right]$ is distributed as an arithmetic fractional Brownian motion with drift $a$ and volatility $b$.

**Proposition**
Let $B^H \left(t\right)$ denote scaled fractional brownian motion with
mean $0$ and variance $b^2 t^{2H}$, for all $H \in \left(0, 1\right)$
with respect to measure $\mathbb{P}$. Define $\tau_k = \inf \left\{t \ge
0 : B^H \left(t\right) = k\right\}$ for $k > 0$. Then the conditional
mean and variance of $M_t$ given $B_t$ are
\begin{equation}
E\left(M_t | B^H \left(t\right) = k \right) = {t^{1-2H} k \over b}
\end{equation}
and
\begin{equation}
\hbox{Var}\left(M_t | B^H \left(t\right) = k\right) = t^{2-2H}\left(c^2-1\right).
\end{equation}

**Proof**
Both $B^H \left(t\right)$ and $M_t$ have mean zero, the variance
of $B^H \left(t\right)$ is $b^2 t^{2H}$, the variance of $M_t$ is $c^2 T^{2 - 2H}$,
and their covariance $bt$, can be derived
similarly to, as in Proposition 3.2, in Norros's paper. Hence
the correlation coefficient $\rho$ between $M_t$, $B^H \left(t\right)$
is $1/c$. Therefore, using elementary results for the bivariate normal
distribution, we find
\begin{eqnarray}
E\left(M_t | B^H \left(t\right) = k \right) &=& {\rho \sigma_{M_t}\over \sigma_{B^H \left(t\right)}}k\\\nonumber
&=& {t^{1-2H} k \over b}\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\hbox{Var}\left(M_t | B^H \left(t\right) = k\right) &=& {\sigma^2}_{M_t}\left(1 - \rho^2\right) \\\nonumber
&=& t^{2-2H}\left(c^2-1\right).\nonumber
\end{eqnarray}

**The Derivation Attempt**
By Doob's optional sampling theorem (justified by the uniform integrability of the martingale
\begin{equation}
\exp\left\{\frac{a}{b} M_T - \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_T\right\}\nonumber
\end{equation}
on $\left[0, T\right]$
and the fact that $\left\{\tau_k \le T \right\} \in \mathcal{F}_{\tau_k}
\cap \mathcal{F}_{T} = \mathcal{F}_{\tau_k \bigwedge T} \subseteq \mathcal{F}_T$ ),
\begin{equation}
\mathrm{E}\left[\left.\exp\left\{\frac{a}{b} M_T - \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_T\right\} \right| \mathcal{F}_{\tau_k \bigwedge T}\right]
= \exp\left\{\frac{a}{b} M_{\tau_k \bigwedge T} - \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_{\tau_k \bigwedge T}\right\}.
\end{equation}
Therefore,
\begin{eqnarray}
\mathbb{P}^{a,T}\left[\tau_k \in (t,t+dt)\right] &=& \mathbb{E}^{a,T} \left[ 1_{\left\{\tau_k \in (t,t+dt)\right\}} \right]\\
\nonumber
&=& \mathbb{E}\left[\exp\left\{\frac{a}{b} M_T - \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_T\right\}1_{\left\{\tau_k \in (t,t+dt)\right\}}\right]
\end{eqnarray}
by Corollary above
\begin{eqnarray}
\nonumber
&=& \mathbb{E}\left[\left.\mathbb{E}\left[\exp\left\{\frac{a}{b} M_{\tau_k \bigwedge T}
- \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_{\tau_k \bigwedge T}\right\} \right|{\cal F}^{B^H}_{\tau_k\wedge T}\right]1_{\left\{\tau_k \le T\right\}}\right]\\
\nonumber
&=& \mathbb{E}\left[\exp\left\{\frac{a}{b} M_{\tau_k} - \frac{1}{2}\frac{a^2}{b^2} \langle M, M\rangle_{\tau_k}\right\}1_{\left\{\tau_k \le T\right\}}\right]\\
\end{eqnarray}
by Doob's sampling theorem
\begin{eqnarray}
\nonumber
&=& \mathbb{E}\left[\exp\left\{\frac{a k}{b^2}{\tau_k}^{1-2H} - \frac{1}{2}\frac{a^2}{b^2} {\tau_k}^{2-2H}\left(c^2 - 1\right)\right\}1_{\left\{\tau_k \le T\right\}}\right]\\
\end{eqnarray}
by propostion above
\begin{eqnarray}
\nonumber
&=& \int_0^T \exp\left[\frac{a k t^{1-2H}}{b^2} - \frac{1}{2}\frac{a^2 t^{2 - 2H}}{b^2} \left(c^2 - 1\right)\right] \mathrm{P}\left[\tau_k \in \mbox{d}{t}\right].
\end{eqnarray}
On the other hand, $\left\{B^H \left(t\right) + a t\right\}_{t \in \left[0, T\right]}$ is a scaled fractional Brownian motion under $\mathbb{P}^{a,T}$, so
\begin{equation}
\mathbb{P}^{a,T} \left[\tau_k \le T\right] = \mathbb{P}\left[\hat{\tau_k} \le T\right],
\end{equation}
where $\hat{\tau_k}$ is the first hitting time of the level $k$ of the scaled fractional Brownian motion
with drift $a$. Using the formula for asymptote of the first passage density of fBM without drift due to Molchan, it follows immediately that the long-time form of the first passage time density for fBm with drift is given by
\begin{eqnarray}
f\left(t\right) &=& \exp\left[\frac{a k t^{1-2H}}{b^2} - \frac{1}{2}\frac{a^2 t^{2 - 2H}}{b^2} \left(c^2 - 1\right)\right] t^{H-2}.\label{firstPassageDensityArithmeticFbm}
\end{eqnarray}

One of the things which really worries me about this asymptotic density formula is that it seems $c^2 < 1$ for most values of $H$.

Is there a way to verify the correctness of these computations without numerical simulations, perhaps ?

I'm trying to read the proof of the Lefschetz hyperplane theorem from Griffiths-Harris. They prove the theorem (on pages 156-157) using the Kodaira vanishing theorem. I have a basic question regarding their strategy of proof.

They begin by noting that we have Hodge decompositions for both the Kähler manifold $M$ and the positive hypersurface $V$, and so, they claim that to prove the statement about the restriction map $H^r(M,\mathbb C)\to H^r(V,\mathbb C)$, it is enough to prove the corresponding statement for the restriction maps $H^q(M,\Omega^p_M)\to H^q(V,\Omega_V^p)$.

It is not clear to me why this is sufficient, i.e., why the restriction map from the Dolbeault cohomology of $M$ to that of $V$, and the restriction map on de Rham cohomology are compatible with the Hodge decompositions of $M$ and $V$. Could someone please explain why this is the case? I think this would follow if the restrictions of harmonic forms on $M$ are harmonic on $V$, but I don't see how to prove that either.

For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following $$ \bar{1} \cdot (z,w) = (z e^{\frac{2\pi i}{m}}, w e^{\frac{-2\pi i}{m}}), $$ where $\bar{1} \in \mathbb{Z}_m$. For $m=2$, in other words $(z,w) \sim (-z,-w)$, there is a resolution $\mathcal{O}(-2)$ for $\mathbb{C}^2/\mathbb{Z}_2$ with the holomorphic map $\pi:\mathcal{O}(-2) \rightarrow \mathbb{C}^2/\mathbb{Z}_2$ defined as $$ \pi:(z, \xi) \mapsto [z\sqrt{\xi}, \sqrt{\xi}] $$ on $\mathcal{O}(-2)|_{U_1}$ where $U_1 = \{[z,w] \in \mathbb{CP}^1|w\not = 0\}$, and $$ \pi:(w, \eta) \mapsto [\sqrt{\eta}, w\sqrt{\eta}] $$ on $U_2 = \{[z,w] \in \mathbb{CP}^1|z\not = 0\}$ similarily.

However, for $m=3$ I have no idea to write down the similar holomorphic map and the resolution. Is there any reference point out the resolution of $A_3$-singularity?

Let $K$ be a global field and $\upsilon$ a place of $K$. Let $K_{\upsilon}$ denote the completion of $K$ at $\upsilon$ and $K_{(\upsilon)}:=K^{sep}\cap K_{\upsilon}$ the henselization (which is the quotient field of the henselization of the discrete valuation ring in case of a non-archimedean valuation).

Let $V$ be a $K_{(\upsilon)}$ variety with a point $\operatorname{Spec}(K_{\upsilon}) \to V$. I have seen used in a paper that in this case $V$ has to have a $K_{(\upsilon)}$-point. Why is this true?

I am thankful for any thoughts and ideas as I am not so familiar with algebraic geometry and maybe missed a result that would be helpful here.

Thoughts so far:

- If $\upsilon$ is complex, then $K_{(\upsilon)}$ is separably closed and the statement is true.
- If $\upsilon$ is non-archimedian, let $R^h$ denote the henselization of the corresponding discrete valuation ring and let $t$ be a generator of the maximal ideal. By a result of Greenberg a $R^h$-variety has a $R^h$-point iff it has a $R^h/t^n$-point for all $n \geq 1$. I fail to see how a point $\operatorname{Spec}(K_{\upsilon})=\operatorname{Spec}(\operatorname{Quot}(\varprojlim R^h/t^n)) \to V$ gives a point $\operatorname{Spec}(\varprojlim R^h/t^n) \to V$ (in the case that $V$ is not proper), such that the result can be used.

Let $M,N$ be topological manifolds such that $M$ does not admit a $PL$ structure and $N$ does. Is $M\#N$ still a triangulable manifold?

Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, *Nonnegative functions as squares or sums of squares*, J. Funct. Anal. **232** (2006) 137-147 (see also this MO question and this math.SE question), not every such an $f$ is of the form $f=g^2$ for $g\in\mathscr{D}(\mathbb{R}^n)$ or even a finite sum of such. On the other hand, as mentioned in the paper above, C. Fefferman and D. H. Phong sketched a proof (*On positivity of pseudo-differential operators*, Proc. Natl. Acad. Sci. U.S.A. **75** (1978) 4673-4674) of the fact that any $0\leq f\in\mathscr{C}^\infty(\mathbb{R}^n)$ can be written as a sum of squares $$f=\sum^k_{j=1}g_j^2$$ with $g_j\in\mathscr{C}^{1,1}(\mathbb{R}^n)$ (i.e. $g_j$ is a differentiable function whose derivatives are locally Lipschitz) for all $1\leq j\leq k$ for some $k\in\mathbb{N}$. This fact was a key ingredient of the proof of the important inequality for scalar pseudodifferential operators with non-negative symbols that bears their name. For a modern, more detailed proof of the above formula, see N. Lerner, *Some Facts About the Wick Calculus*, in L. Rodino, M.W. Wong (eds.), *Pseudodifferential Operators: Quantization and Signals*, Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 19–24, 2006 (Springer Lecture Notes in Mathematics **1949**, 2008), pp. 135-174, particularly Theorem 5.2, pp. 167-172, and the discussion right after Theorem 1.1 in the paper by Bony *et alii* above on page 139. It immediately follows by multiplication by squares of smooth bump functions that the $g_j$'s may be chosen to be compactly supported if $f\in\mathscr{D}(\mathbb{R}^n)$. Bony *et alii* showed above that this regularity for $n\geq 4$ is sharp.

All this leads naturally to the following

**Question:** Is every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ the *limit of a
sequence* of sums of squares *in* $\mathscr{D}(\mathbb{R}^n)$ in the latter's topology?

In other words, is the cone of non-negative elements of $\mathscr{D}(\mathbb{R}^n)$ the (sequential) closure of the cone of sums of squares in $\mathscr{D}(\mathbb{R}^n)$?

I am particularly interested in arguments that do not rely on the result by Fefferman and Phong.

**EDIT - Follow-up subquestion:** (suggested by André Henriques) Is every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ the limit of an *increasing* sequence of sums of squares in $\mathscr{D}(\mathbb{R}^n)$ in the latter's topology? Particularly, can $f$ be written as $$f=\sum^\infty_{j=1}g_j^2$$ with $g_j\in\mathscr{D}(\mathbb{R}^n)$ for all $j$ and convergence in $\mathscr{D}(\mathbb{R}^n)$?

In Penrose's construction of the twistor space of Minkowski spacetime $\mathbb R^{1,3}$, we first complexify $\mathbb R^{1,3}$ to $\mathbb C^4$ and then think of points in it as matrices acting on $\mathbb C^2$ via a tensor decomposition $\mathbb C^4 = \mathbb C^2\otimes \mathbb C^2=\mathrm{End}(\mathbb C^2)$. Such a matrix can be depicted as a graph in $\mathbb C^2\oplus \mathbb C^2$, which Penrose calls the twistor space. It makes sense to quotient out by a $\mathbb C^\times$ action, following which we get the projective twistor space. In particular, points in the complexified Minkowski space correspond to embeddings of $\mathbb{CP}^1$ into the projective twistor space.

Salamon, on the other hand, defines the twistor space of a general (possibly Ricci flat) quaternionic (pseudo-)Kähler manifold $M$ as the bundle of almost complex structures in the quaternionic subbundle of $\mathrm{End}(TM)$. The fibre over a point is a sphere which may be identified with $\mathbb {CP}^1$. So there is a $\mathbb{CP}^1$ for every point in $M$.

Particularly, in Penrose's construction, these spheres are supposed to intersect whenever two points in Minkowski spacetime are null-separated, whereas in Salamon's consruction, the spheres are all separate fibres over the points of $M$.

[EDIT: My question was originally about what happens when we take $M$ to be the Minkowski spacetime, but as Deane Yang pointed out in the comments, this is not possible.]

**Question.** Is there a general definition of a twistor space on a manifold $M$ which specialises to the Penrose (resp. Salamon) twistor space in the case that $M$ is the Minkowski spacetime (resp. quaternionic (pseudo-)Kähler)? In particular, why are Hermitian structures the natural analogues of null rays?

A tensor is en element of a vector space contructed by tensor product(s). So every tensor is a vector. But every vector space can be weived as a tensor product of itself with its field. Also something is said to be a tensor if its linear in all arguments. Is for example a real valued function also a tensor?

A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you have a symmetric monoidal category of quasicoherent sheaves, you have forms and differentials, you have pullbacks, (several) notions of locality, an étale theory, etc.

There is a generalization to infinity-stacks, which mix the notions of algebraic space and infinity-groupoid (equivalently, $(\infty, 0)$-category, or topological space).

I want a generalization of this to a notion of "semi-stack", which mixes the concepts of algebraic space and semigroup -- and more generally, in the derived context, a notion of $(\infty, 1)$-stacks which interpolates between spaces and $(\infty, 1)$-categories. Probably because I'm not familiar with higher algebra, I haven't been able to find a definition. Here are some properties I would like from the category of "derived semistacks", $Stck_{(\infty,1)}$, roughly in decreasing order of importance (non-derived versions to be supplied by the reader):

(1) Given any derived semistack $X\in Stck_{(\infty, 1)}$ I want defined a symmetric monoidal dg category (at least in characteristic zero, with some infinity-categorical modifications in characteristic p) of (quasi-)coherent sheaves $QCoh(X)$.

(2) It should be true that any suitably finite $(\infty, 1)$-category $c$ has a corresponding object $c\times \text{pt}\in Stck_{(\infty,1)}.$ The category $QCoh(c\times \text{pt})$ is the symmetric monoidal representation category $Rep_k(c)$. (Tensor product of representation of categories is defined object-wise, like for quivers.)

(3) The category $Stck_{(\infty, 1)}$ is fibered over the category of small categories (i.e. $Hom(X, Y)$ is a small, "discrete" category for X, Y derived semistacks, with composition given by functors in a functorial way).

(4) The category $Stck_{(\infty)}$ (viewed as fibered in $(\infty)$-groupoids) is fully faithfully contained in $Stck_{(\infty,1)}$. Moreover, I want a "stackification" functor $X\mapsto X^\times$ from $Stck_{(\infty,1)}\to Stck_{\infty}$ which satisfies the following adjunction property: $Hom(X, Y)^\times = Hom(X, Y^\times)$, for $X\in Stck_\infty$ and $Y\in Stck_{\infty, 1}$. Here $Hom(X, Y)^\times$ is the $(\infty, 0)$-category underlying $Hom(X, Y)$ (consisting of the same objects and all invertible morphisms).

(5) Let $\overline{G}_m$ be the semigroup scheme given by $\mathbb{A}^1$ with product given by $x\times y:\, = xy$, extending the group scheme $G_m = \mathbb{A}^1\setminus \{0\}$. I want an object $\text{pt}/\overline{G}_m$ (one point with "endomorphisms $\overline{G}_m$") such that $(\text{pt}/\overline{G}_m)^\times = BG_m$ is the classifying space of the multiplicative group, and such that for an ordinary scheme (or stack) $X$, we have $$Hom(X, \text{pt}/\overline{G}_m)$$ given by the category with objects line bundles $L$ over $X$ and morphisms $$\pi_0Hom(L, M) \cong Hom_{Coh}(L, M)$$ ("morphisms between line bundles which are allowed to go to zero".)

Does anyone know of such a construction? Would it be sufficient to consider infinity-categories (in Lurie's simplicial set formulation) over, say, the fppf site on a point?

It would be nice if this category (perhaps under some additional conditions, like smoothness of some sort) were to exhibit some of the good behavior of stacks, viz. good notions of locality, an étale homology theory, other motivic invariants, etc...

[Apologies for a subjective and very naive question.]

If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an example that comes up in other areas of mathematics? Particularly if one can comment on its natural domain (either a subset of $\mathbb C$ or an extension, i.e. a Riemann surface), with special mention of any pole structure, conjectural or not, without getting into details of that field.

Number theory of course offers a whole slew of examples that go under $L$-functions and modular forms (not to forget elliptic functions), where they encapsulate a lot of very deep mathematics. Would be great to be more specific. Of course the Riemann zeta function would be high on everyone's list.

What about "special functions" that solve differential equations (on the complex domain)? I'd also love to see a layman's example of a Painlevé transcendent.

I seem to recall Weierstrass's nowhere-differentiable function was discovered as the boundary value of an analytic function. Anyone know what it was?

Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = id$ and $XX^{\star} \neq id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, the ${\rm C}^{\star}$-algebra generated by $X$, is the Toeplitz algebra.

We are interested in an extension of Coburn's theorem. Consider $Y \in B(H)$ such that $Y^{\star}Y = id+p$, with $p \in B(H)$ a projection, and $YY^{\star} \neq id$.

*Question*: Is ${\rm C}^{\star}(Y)$ always the Toeplitz algebra? If not, is it still a ${\rm C}^{\star}$-algebra of type ${\rm I}$?

*Motivation*: I'm mainly interested in a shift on a trivalent directed tree with one parent and one or two children for each vertex, such that the set of vertices with one child and the set of vertices with two children are both infinite (it is a generalization of the operator $S^{\star}$ of this post).