I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.

The basic statement of the theorem is $$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$ "in the Gromov–Hausdorff sense" as $n \to \infty$, where the convergence is in distribution.

Here $(m_n,d_n)$ and $(m_{\infty},d_{\infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.

For every compact metric space $(X,d)$ and $R > 0$, we have $$(*) \, \, \, \mathbb{P} \left[ d_{GH}[ (m_n,d_n), (X,d) ] < R \right] \to \mathbb{P} \left[ d_{GH}[ (m_{\infty},d_{\infty}), (X,d) ] < R \right]$$ as $n \to \infty$.

But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?

I recently thought about the following game (has it been considered before?).

Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an integer $a$. Similarly, Bob observes a sequence of independent unbiased random bits $(B_n)$, independent from $(A_n)$, and then chooses an integer $b$. Alice and Bob are not allowed to communicate. They win the game if $A_b=B_a=1$.

What is the optimal winning probability $p_{opt}$? A strategy for each player is a (Borel) function $f : \{0,1\}^{\mathbf{N}} \to \mathbf{N}$, and we want to maximize the winning probability over pairs of strategies $(f_A,f_B)$.

Constant strategies win with probability $1/4$, and it is perhaps counterintuitive that you can do better. Choosing $f$ to be the index of the first $1$ wins with probability $1/3$. This is not optimal though, by running a little program trying randomly modified strategies on a finite window I could find that $p_{opt} \geq 358/1023 \approx 0.3499$, with some pair (with $f_A=f_B$) lacking any apparent pattern.

But a more interesting question is: can you prove any upper bound on $p_{opt}$, besides the trivial $p_{opt} \leq 1/2$?

**Edit**. As has been pointed out by Édouard Maurel-Segala, the problem has been studied in this paper, where it is proved (as is also proved in the present thread) that $0.35 \leq p_{opt} \leq 0.375$, stated without proof that $p_{opt} \leq \frac{81}{224} \approx 0.3616$, and conjectured that $p_{opt} = 0.35$.

**Edit** (clarifying what I said in the comments). You can ask the same question for the finite version of the game, with strings $(A_1,\dots,A_N)$ and $(B_1,\dots,B_N)$, giving optimal winning probability $p_N$. It can be checked than $(p_N)$ is non-decreasing with limit $p_{opt}$. Moreover the inequality $p_{opt} \geq \frac{4^N}{4^N-1} p_N$ holds, because in the infinite game, when a player sees a string of $N$ $0$s, he may discard them and apply the strategy to the next $N$ bits. We have $p_1=1/4$, $p_2=5/16$, $p_3=22/64 > p_2$. It seems that $p_4=89/256$ (therefore
$p_4 > p_3$, but $\frac{256}{255} p_4 < \frac{64}{63} p_3$, so $4$-bit strategies are worse than $3$-bit for the infinite game), and I know that $p_5 \geq 358/1024$ and $p_6 \geq 1433/4096$. For $p_3$ and $p_4$ one strategy achieving the value is: when the observed string contains a single block of $1$s, Alice (resp. Bob) picks the index of the $0$ immediately after (resp. before) that block; what they do in the remaining cases is irrelevant.

For each *n*, let $a_n$ be the least integer, greater than *n*, such that the numbers $a_n$, $a_n$+ 1, $a_n$+ 2, ..., $a_n$+ (*n* – 1) are divisible, in some order, by 1, 2, 3, ..., *n*. For example $a_{12}$ = 110.

What are the best estimates known for $a_n$?

I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.

The Prouhet-Thue-Morse constant, defined as

$$ \tau =\sum _{{i=0}}^{{\infty }}{\frac {t_{i}}{2^{{i+1}}}}=0.412454033640\ldots $$

where the $t_i$ are elements of the Thue-Morse sequence, is transcendental. But is

$$ \tau_b =\sum _{{i=0}}^{{\infty }}{\frac {t_{i}}{b^{{i+1}}}} $$

also transcendental, for $b>2$?

Let $\scr{F}$ be free filter ($\cap\scr{F}=\emptyset$) on a countable set $X$ and $B\in\scr{F}$. We define the trace of $\scr{F}$ on $B$ as follows $\mathscr{F}_B=\{Y\cap B:~Y\in\scr{F}\}$. $\scr{F}$ and $\mathscr{F}_B$ are isomorphic in the case of Frechet filter or in the case of any ultrafilter. Are $\scr{F}$ and $\mathscr{F}_B$ isomorphic in all cases ?

Let $K\subset \mathbb{R}^n$ be compact, and consider a sequence of probability measures $\mu_n$ on $K$. Being weakly bounded, we know there exists a weakly convergent subequence of the $\mu_n$. By weak convergence, I mean convergence given by action against $C(K)$. My question is what sort of conditions can be imposed on the sequence $\mu_n$ in order to guarantee that the original sequence converges as opposed to a subsequence?

Relatedly, let me modify the question by now requiring that the $\mu_n$ be discrete i.e. $\mu_n = \frac{1}{n}\sum_{i=1}^n\delta_{x_i}$ for some set (possibly depending on n) of points $x_1,...,x_n \in K$.

Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?

I am wondering if there is some example of a mathematician or physicist who published other papers at the same time as their PhD work and independently of it which actually eclipsed the content of the PhD thesis.

The only semi-example I can think of immediately is Einstein, whose other publications in 1905 (especially on special relativity and the photoelectric effect) eclipsed his PhD thesis which was published in the same year. Although it contained important insights, it was somewhat forgotten to the point where he felt that he had to remind people about it.

Although this is a soft question, I didn't ask in Academia as I didn't want examples outside of mathematics and physics.

*Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow.*

Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability density supported on the (compact) interval $[a,b]$. Let $\Vert\cdot\Vert$ be some norm (e.g. $L^2$, total variation). What is the best approximation to $f$ by a single, univariate Gaussian? In other words, solve $$ \text{argmin}_{m\in\mathbb{R},v^2\ge 0}\Vert f-g_{m,v^2}\Vert $$

where $g_{m,v^2}$ is the Gaussian PDF with mean $m$ and variance $v^2$. As I mention above, this question is related to a previous question; I realized while trying to solve that problem that even the simple case of a single Gaussian seems nontrivial.

**Hint:** I have verified numerically, for both $L^2$ and TV norm, that **the answer is not $m=\mathbb{E}_{X\sim f} X$ or $v^2=\text{var}_{X\sim f} (X)$**.

**Note:** The details of which norm (or even metric) or regularity assumptions are made are not important here. I am mostly curious to see if this computation can be carried out for any "reasonable" distance and regularity. In fact, even the case of minimizing over $m$ alone (holding $v^2$ fixed) appears difficult. Note that for the KL-divergence (which is neither a norm nor a metric), this can be solved in closed form quite easily.

**I simplified the question:**

Take the function $f(x)=e^{-x^2}$. This function is **log concave**.

Now consider another function $e^{g}$ where $g \in C^{\infty}(\mathbb R)$ is strictly concave and negative outside some fixed interval $I=[a,b]$ at which ends the function $g$ vanishes $g(a)=g(b)=0.$ (the strictly concave also only holds outside of $I$.)

The behaviour of $g$ inside $I$ shall not be further specified, but we assume $g$ to be sufficiently regular and sufficiently fast decaying to $-\infty$ so that everything is well-defined.

We then define a function $h:= g\vert_I$ which coincides with $g$ on $I$ and is continued as $h(x)=0$ for $x \notin I.$

Now, we define two functions as convolutions $z_1:=f*e^{h}$ and $z_2:=f*e^{g}.$

I ask: Can one find an explicit constant $C(g)>0$ that one can estimate from the function $g$, only such that $$\sup_x \frac{d^2}{dx^2}\log(z_1(x)) \ge C(g) \sup_x \frac{d^2}{dx^2}\log(z_2(x))?$$

I suspect this to be true, as $g-h$ is concave, so I'd expect $z_1$ to be more "log convex" than $z_2.$

The point behind these operations is that log-concavity is preserved by convolutions see wikipedia for details.

**EDIT:** It would also be interesting for me to know if there is anything like what I wrote down that holds "in spirit." So imagine you have a function $g$ that is very concave outside an interval. Can you find a compactly supported function (or at least bounded) $h$ such that $e^{h}$ gives the above inequality. Please feel free to point out other natural analogues of my question to me. I am curious to hear about them.**

Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.

But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?

Let $C$ be a hyperelliptic curve $y^2 = f(x) $ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian. I know Zarhin's result [Hyperelliptic Jacobians without complex multiplication], which states that if the Galois group $G= Gal(f)$ of $f$ is either the Symmetric group $S_n$ or the alternating group $A_n$, then the endomorphism ring of $J$ is $\mathbb{Z}$ (that is, $J$ has no complex multiplication).

My question is that whether there is a condition on $G$ under which $J$ has CM.

The reason why I ask this is the follwing; In [Wamelen, Examples of genus two CM curves defined over the rationals], Wamelen found 19 curve (of gunus 2) whose Jacobian has CM. In all of these examples, the Galois group of $f$ is either the cyclic group $C_4$ of order $4$ or the Frobenius group $F_5$ of order $20$. So my question is that;

Is there any example of $C$ without CM and with $G = C_4$ ?

Note: This is an edit of the previous question.

By $T_2$ conservatively extends $T_1$ if and only if:

There exists a function $F$ such that $T_2$ extends $T_1$ through $F$; and for every function $G$ such that $T_2$ extends $T_1$ through $G$, we have $T_2$ conservatively extends $T_1$ through $G$.

For definitions of the above terminology see "About conservative extensions of First Order theories?"

Now lets assume $T_1$ and $T_2$ to have the same language, like for example with the case of $ZFC$ and $Predicative \ MK$ set theories, where the later is a weakening of Morse-Kelley set theory by restricting class comprehension scheme to formulas in which all quantifiers are bounded in $V$. I think that $Predicative \ MK$ would conservatively extends $ZFC$ according to the above definition. The former theory is finitely axiomatizable.

Now my question is can we apply methods present in the answer to this question as to get also finitely axiomatizable conservative extensions [in the above sense] for all first order theories [meeting qualifications in that posting] even if they were written in the same language?

Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the internal categorical product of quotient maps is a quotient map." I have a situation where an argument works regardless of the particular "convenient" subcategory that I pick. Hence, as long as a space lies in *some* coreflective Cartesian closed subcategory, I may apply my argument.

What is the class of topological spaces, that lie in *some* coreflective Cartesian closed subcategory of $\mathbf{Top}$?

**Update:** It is well-known that every class of spaces generates a Cartesian closed subcategory of $\mathbf{Top}$ (See Booth-Tillotson) but it may not be coreflective. On the other hand, many coreflective categories like the category of locally path-connected spaces are not Cartesian closed. As pointed out by David White below, it is a result of Juraj Cincura that there is no largest coreflective Cartesian closed subcategory of $\mathbf{Top}$.

I'd be content to just to know that there is a space that does not lie in any coreflective Cartesian closed subcategory of $\mathbf{Top}$.

Philosophy of Logic – Reexamining the Formalized Notion of Truth https://philpapers.org/archive/OLCPOL.pdf

Given an abstract simplicial complex $K$, one can make a simplicial set $X(K)$ with $n$-simplices given by sequences $(x_0, \ldots, x_n)$ such that $\{x_0, x_1, \ldots, x_n\}$ is a simplex of $K$. The face maps delete entries and the degeneracy maps repeat entries. I'd like a reference for the fact that the geometric realization of $X(K)$ is homotopy equivalent to the geometric realization of $K$ itself. (Note that $|X(K)|$ is typically very big: for $K$ a single edge, $|X(K)|$ is the infinite-dimensional sphere $S^\infty$.)

I've sketched a proof of this fact here, but hope there is a reference I can just cite since, as I expected, every algebraic topologist I've asked in person already knew the fact. :)

Also, does this $X(K)$ have a standard name or notation? Or if not, can someone think of a catchy name or nice notation?

Let $F_n$ be the free group generated by $x_1,\ldots,x_n$ and let $S_n$ be the symmetric group on $\{1,\cdots,n\}$. Let $w=x_{i_1}^{\pm1}\cdots x_{i_s}^{\pm1}$ be a word and for each $\sigma \in S_n$, define $\sigma(w)=x_{\sigma(i_1)}^{\pm1}\cdots x_{\sigma(i_s)}^{\pm1}$. We consider groups of the form

$$G_n(w)=\langle x_1,\ldots,x_n\mid\sigma(w), \sigma\in S_n\rangle,$$ where $w$ is a given word in $F_n$. Such groups are called symmetrically presented. For example, it can be proven that $$G_4(x_1x_2^2x_3x_4^{-1})=\langle x_1,x_2,x_3,x_4\mid\sigma(x_1x_2^2x_3x_4^{-1}), \sigma\in S_n\rangle$$is a non-Abelian group of order $96$.

My question is, given $n$, what is the smallest non-Abelian symmetrically presented group? Any list of examples of non-Abelian symmetrically presented groups will also be much appreciated.

For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows: $$V = \{s\in\Sigma^\star\colon |s|\le n\}$$ $$E = \{(s_1,s_2)\colon s_1=\text{insert}(s_2,p,c), c\in\Sigma, 0\le p\le |s_1|\}$$ in which $insert(s,p,c)$ takes string $s$ and inserts character $c$ in position $p$ of string $s$, and gives the result as output.

Finally we define the undirected distance between two nodes as: $$d(s_1,s_2) = \text{length of shortest path between } s_1 \text{ and } s_2 \text{ discarding edge directions}$$

For a given $n$ and alphabet $\Sigma$ is it possible to embed $G$ in a Poincare disk with dimension $d$, such that the geodesic distance between each pair $u,v$ is close to $d(u,v)$? I know that there is a hyperbolic embedding due to Gromov for trees. But are there similar results for DAGs? If yes, what are the assumptions about DAGs that must hold?

Define the measure of algebraic independence of the numbers $a_1, \ldots, a_n \in \mathbb{C}$ as

$\Phi(a_1, \ldots, a_n; m, H) = \min |P(a_1, \ldots, a_n)|$,

where the minimum is taken over all polynomials of degree at most $m$, with integer coefficients not all of which are zero, and of magnitude at most $H$.

**Question:** 1) How large can be $\Phi(a_1, \ldots, a_n; m, H)$ for fixing $n, m, H$?7

2) What lower bounds of $\Phi(a_1, \ldots, a_n; m, H)$ are known for *explicit* $a_1, \ldots, a_n$?

( I want to find $a_1, \ldots a_n$ such that $\Phi$ is large.)

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\pi^{-1}(X_{\mathrm{sm}}) \cong X_{\mathrm{sm}}$, where $X_{\mathrm{sm}}$ is the regular locus of $X$ and the isomorphism is simply the restriction of $\pi$. Denote by $E$ the exception divisor associated to the morphism $\pi$ (set-theoretically isomorphic to $\widetilde{X} \backslash \pi^{-1}(X_{\mathrm{sm}})$).

Let $A$ be a local artinian ring, $f_A: C_A \to \mathrm{Spec}(A)$ a smooth family of irreducible, affine curves and an $A$-morphism $g_A: C_A \to X \times_{\mathbb{C}} \mathrm{Spec}(A)$. Denote by $C_o$ the special fiber of $C_A$ (under the morphism $f_A)$ and $g_o:C_o \to X$ the restriction of $g_A$ to the special fiber. Suppose that $C_o$ does not contract to a point on $X$. Since $C_o$ is a curve, the universal property of blow-up implies that the morphism $g_o$ lifts to $\widetilde{X}$ i.e., there exists a morphsim $h_o:C_o \to \widetilde{X}$ such that $g_o=\pi \circ h_o$.

Is it then true that the morphism $g_A$ also lifts to $\widetilde{X} \times \mbox{Spec}(A)$ i.e., there exists a morphism $h_A: C_A \to \widetilde{X} \times \mbox{Spec}(A)$ such that $g_A=(\pi \times \mathrm{id}) \circ h_A$? If not true in general, is there any known condition under which this holds true? Any hint/reference will be most useful.