We take the unit square and have it divided by $n$ lines which are chosen randomly. We choose the lines as follows, choose one of the four sides of the square at random and then choose a random point on the square. Then choose a random point from one of the other three sides and connect the dots to create a line. Do this $n$ times.

What is the expected largest piece of area after $n$ trials?

Let $X=\proj_nX_n$ be a Fr\'echet space. What is the relation between $\proj_nX_n''$ and $X''$? This should be known but I cannot find the reference.

Define a tree T that has the same nodes as the ordinary complete infinite Binary Tree but instead of paths running through the nodes let three paths begin at every node of T without crossing any further nodes. Like lametta, shown red. For every $n \in \mathbb{N}$ the set of lametta paths crossing level $n$ is larger than the set of distinct paths crossing level $n$ in the ordinary Binary Tree. The set of paths of T has cardinal number $3\aleph_0 = \aleph_0$.

In classical mathematics this means that the lametta paths are never less than the paths in the ordinary Binary Tree - even in the limit. How can one argue that nevertheless the paths in the ordinary Binary Tree are more numerous in the limit?

The only argument that I can imagine is this: The number of distinct paths as a function of levelnumber $n$ is not continuous. The direct comparison theorem fails. But how can it then be argued that the number of nodes is a continuous function of levelnumber $n$ and does not leap to uncountable cardinality in the limit?

Let $G$ be a locally compact group and $I$ be an arbitrary closed two sided ideal in $L^1(G)$. Is there any locally compact group $N$ such that $\frac{L^1(G)}{I}\cong L^1(N)$?

Or can we add some conditions to $G$ or $N$ or $I$ and conclude the above question?If yes, what is the relation between $G$ and $N$?

In the spatial case, let $H$ be a closed normal subgroup of $G$, $\frac{L^1(G)}{I}\cong L^1(\frac{G}{H})$. Since $T_H:L^1(G)\to L^1(\frac{G}{H})$ is a continuous onto homomorphism and $I=\ker(T_H)$.

In algebraic geometry, the intersection number of two plane curves is ubiquitous. I am looking for interesting characterizations and alternative definitions of the intersection number.

I know the intersection number can be defined axiomatically as satisfying some interesting properties. I also know it can be defined in terms of the resultant. But none of these characterizations strike me as very geometrical. So I am looking for some more geometrical definitions, although other non-geometrical characterizations are always welcome.

In particular, it is likely the case that the intersection number of two curves can be defined as the actual number of intersection points when the curves are perturbed slightly. Does anybody happen to know of a formal proof of this fact?

Also, and related, I am interested in an interpretation of the intersection formula in terms of infinitesimals. For example, the intersection of the parabola $Y-X^2 =0$ and the $X$-axis $Y=0$ is in the point $(0,0)$. However, given any infinitesimal $e^2=0$, we also have that $(e,0)$ is a root. Thus the intersection number is $2$ since the 'order' of the infinitesimal is $2$, i.e. $e^2 = 0$. But it is not at all obvious how to generalize this to more complicated curves, and whether it is related to the usual resultant at all. Does anybody know of a good reference?

My question is about the computation the fundamental group of graph of groups:

First let me give a reference for my question:

In the page 14 of Groups acting on graphs by Dunwoody and Dicks.

It says that

The fundamental group of graphs of groups can be obtained by successively performing one free product with amalgamation for each edge in the maximal subtree and then one HNN extension for each edge not in the maximal subtree.

I cannot see this procedure in the following example.

Suppose that our graph $Y$ is the complete graph with three vertices $K_3$ and the vertex groups are $A,B,C$ and the edge groups are $E,F,D$.

In elementary calculus, the derivative is often motivated as the slope of the tangent line. This is often illustrated by a picture of a line through two points somehow converging to a tangent line.

My question is how to make this convergence rigorous. Is there some kind of topology on the subsets of $\mathbb{R}^n$ which describes the convergence of lines to the tangent line, and possible other objects like Taylor polynomials, oscullating circles or tangent planes.

Of course it seems clear that we somehow want that if $f_n:\mathbb{R}^n\rightarrow \mathbb{R}$ converges to some $f$ (for some convenient notion of convergence) then $f_n^{-1}(C)$ converges to $f^{-1}(C)$ for all well-behaved $C$, and we probably want some kind of converse too.

Sorry that I cannot be more formal, but my question is whether there is some kind of theory that deals exactly with this. Is there some kind of book that covers this subject?

I already know about the Hausdorff metric, but it doesnt give us the right idea of convergence in our case, although I think it would give the right idea of convergence for compact sets such as circles converging to the oscullating circle. But even if this is the case, it seems nontrivial to show that circles converge in the Hausdorff metric iff their equations converge.

Let $\mathcal{C}$ be a braided $\mathbb{k}$-linear fusion category ($\mathbb{k}$ algebraically closed; if necessary to answer my question you can also assume $\mathcal{C}$ to be pivotal or even modular). Two algebras $A,B$ in $\mathcal{C}$ are called *Morita equivalent* if the categories $A\hbox{-}\mathsf{Mod}(\mathcal{C})$ and $B\hbox{-}\mathsf{Mod}(\mathcal{C})$ of (left or right) modules internal to $\mathcal{C}$ are equivalent.

**Questions:**

(1) Is there any alternative, equivalent characterization of Morita equivalence of two algebras which circumvents showing that their module categories are equivalent?

(2) In particular, is there a way to "calculate" the Morita equivalence class of a given algebra?

(3) Is there anything special one can say about the Morita equivalence class of an algebra, e.g. some classification result, maybe by imposing additional properties on the algebra?

Please also refer to literature, thanks!

Need this for probabilistic factoring algorithm.

Let $p$ be sufficiently large prime and $E$ the elliptic curve $E /\mathbb{F}_p: y^2=x^3+ax+b$. Let $o=\#E(\mathbb{F}_p)$. $\psi_n$ denote the $n$-the division polynomial of $E$.

If $X$ is the $x$ coordinate on $E$ we have $\psi_o(X)=0$.

What hypothesis on $E$ do we need to make it only if?

$$\psi_o(X)=0 \iff X^3+aX+b=\square$$

I suspect it is enough $E$ to not be supersingular.

Would prefer reference rather than proof.

I am looking for a good overview on quasitoric manifolds. I have read *Toric Topology* by Buchstaber and Panov which was good but I was wondering if there is something that has more. Something like a (almost) complete history of the subject. Does someone know of such a source?

Thank you.

Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$

It is easy to see that $\leq_{RK}$ is reflexive and transitive, but not anti-symmetric. Set ${\cal U}\simeq_{RK} {\cal V}$ if ${\cal U}\leq_{RK}{\cal V}$ and ${\cal V}\leq_{RK}{\cal U}$. So $\text{NPU}(\omega)/\simeq_{RK}$ is a poset with the Rudin-Keisler order applied to equivalence classes.

**Question.** If ${\cal C},{\cal D}$ are totally ordered subsets of $\text{NPU}(\omega)/\simeq_{RK}$ that are maximal with respect to $\subseteq$, do they necessarily have the same cardinality?

Can anyone tell me where to find a good book or lecture note on Riesz-Dunford -Schwartz functional calculus for commuting tuple of operators?

Given that the following identity $\sum_{n=1}^{\infty}e^{-\pi n^2 a}=(\frac{1}{2\sqrt{a}}-\frac{1}{2})+\frac{1}{\sqrt{a}}\sum_{n=1}^{\infty}e^{-\pi n^2 /a}, a>0$ which can be proved using Poission summation formula, is it possible to derive a similar identity for the following sum $\sum_{n=\frac{1}{2}}^{\infty}e^{-\pi n^2 a}$ ? Here $n$ increases in the step of 1. I have tried the following equivalent form $\sum_{n=0}^{\infty}e^{-\pi (n+\frac{1}{2})^2 a}$. Is there any way to solve it?

To be precise, I am curious about the equation $$Ax+by=y^2$$

where all variables are positive integers, and $A$ is given as some fixed constant. I know you can do it with a Pell equation/continued fraction; to be clear, I'm wondering if there are any other ways to solve this other than that, or brute force---I suspect there aren't, but would love to be corrected, as this is not really my area.

I've googled pretty extensively and was surprised that I couldn't find this already addressed somewhere. Thanks for any help!

Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.

Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract embeddings (ie, as $\mathbb{Q}$-algebras). Let $X_\mathbb{C},X_{\mathbb{C}}'$ be the base changes of $X_{\overline{K}}$ to $\mathbb{C}$ via $i$ and $i'$. Then, $X_\mathbb{C}(\mathbb{C})$ has the structure of a Riemann surface. Let $x\in X_\mathbb{C}(\mathbb{C})$ come from a $\overline{K}$-rational point. We may consider its topological fundamental group $\pi_1^{top}(X_\mathbb{C}(\mathbb{C}),x)$. Since every finite cover of the Riemann surface $X_\mathbb{C}(\mathbb{C})$ is algebraic, for every loop in $\pi_1^{top}(X_\mathbb{C}(\mathbb{C}),x)$, its monodromy action on the fibers at $x$ of its finite covers determines an automorphism of the fiber functor at $x$, and hence we obtain a homomorphism $$\pi_1^{top}(X_\mathbb{C}(\mathbb{C}),x)\rightarrow \pi_1^{et}(X_\mathbb{C},x)$$ which is known to be the embedding of the first group into its profinite completion. The map $X_\mathbb{C}\rightarrow X_{\overline{K}}$ given by base change induces an isomorphism on etale fundamental groups, and composing these maps we get $$\pi_1^{top}(X_\mathbb{C}(\mathbb{C}))\longrightarrow \pi_1^{et}(X_\mathbb{C})\stackrel{\sim}{\longrightarrow}\pi_1^{et}(X_\overline{K})$$ where I've omitted the base points because I only care about these maps up to conjugacy (say, inside $\pi_1^{et}(X_{\overline{K}})$). Similarly, with $X_\mathbb{C}'$, we get a map

$$\pi_1^{top}(X'_\mathbb{C}(\mathbb{C}))\longrightarrow \pi_1^{et}(X'_\mathbb{C})\stackrel{\sim}{\longrightarrow}\pi_1^{et}(X_\overline{K})$$

Both of these maps give embeddings of the topological fundamental groups inside $\pi_1^{et}(X_{\overline{K}})$, canonical up to conjugation.

My question is:

**When are the images the same (up to conjugation)?**

Are there examples when the images are not the same?

I'm particularly interested in the case when $X_\mathbb{C}(\mathbb{C})$ is hyperbolic.

References would also be appreciated.

Let $P$ be a finite polyhedron and $N$ be a normal subgroup of $G=\pi_1 (P)$. It is known that there exists a covering space $(\tilde{P},p)$ so that $p_* \pi_1 (\tilde{P})=N$. It follows that for the finite polyhedron $\tilde{P}$ (which is related to $P$), we have $\pi_1 (\tilde{P})\cong N$.

My question is that:

Is there any finite polyhedron $Q$ (connected to $P$) so that $\pi_1 (Q)\cong \frac{G}{N}$?

Thanks in advance.

Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$. I have seen quoted in the literature that \begin{align} \mathbb{P}[|\left\| P_V(x)\right\|_2 - \sqrt{k/n} | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2). \, \, \, \, \, \, \, (1) \end{align} However, i can still not find a concrete proof. What i do understand is that for a $1$-Lipschitz function $f:\mathbb{S}^{n-1} \rightarrow \mathbb{R}$ such as $x \mapsto |\left\| P_V(x)\right\|_2$, we have that \begin{align} \mathbb{P}[|f - M_f | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2), \, \, \, \, \, \, \, (2) \end{align} where $M_f$ is the median of $f$. (2) mostly follows from the isoperimetric inequality on the sphere. The issue though with (1) is that $\sqrt{k/n}$ does not seem to be the median of $x \mapsto |\left\| P_V(x)\right\|_2$. Is anyone able to provide a clean argument for (1) or a self-contained reference in the literature? Many thanks.

Does there exist a model $V$ of $ZFC$ with the following property?

Suppose that $X$ is a set and $\mathcal{A}\subseteq P(X)$ is a collection of subsets such that $X\in\mathcal{A}$ and where $\mathcal{A}$ is a join-semilattice with respect to $\subseteq$ and where $\mathcal{A}$ has a smallest subset with respect to $\subseteq$. Then there is some $Y$ and bijection $f:X\rightarrow Y$ such that $A\in\mathcal{A}$ if and only if there is some ground model $W\subseteq V$ where $Y\cap W=f[A]$.

If not, then what extra conditions on $\mathcal{A}$ should one impose?

Let $F$ be a number field and $\chi$ a one dimensional Artin character. That is, it is a map $\chi: Gal(\overline F/F) \to \mathbb C^\times$ and let $L(s,\chi)$ be it's L-series.

What is known about the values of $L(s,\chi)$ at negative integers? Are they always algebraic integers contained in $F(\chi)$? When are they zero? Is there a (conjectured) explicit representation?

What about the case when $\chi$ is trivial? What about higher dimensional characters? Are there any conjectures dealing with these values?

I only know the case $F = \mathbb Q$ and $\chi$ a Dirichlet character where you can find an explicit representation in terms of (generalized) Bernoulli numbers.

The proof that the Seiberg-Witten invariants of a 4-manifold $X$ with fixed Spin$^c$ structure really are invariant wrt the metric used to define them goes roughly as follows (for simplicity let $b_2^+(X) > 1$ and assume the expected dimension of the SW moduli space is 0):

If $g_0$ and $g_1$ are metrics on $X$ then they may be joined by a path $(g_t)_{0 \le t\le 1}$ of metrics where each $g_t$ is such that the corresponding SW moduli space avoids reducible solutions. Now the collection of moduli spaces $(M_t)_{0\le t\le1}$ of the SW equations for each metric $g_t$ can be viewed as a cobordism between the 0-dimensional oriented manifolds $M_0$ and $M_1$. Hence the number of solutions with orientation accounted for is the same in each.

This allows for an "annihilation" phenomenon to occur among solutions: for example, {+,+,-} is cobordant to {-} via a cobordism consisting of one line from + to - and a U-shaped line along which + and - annihilate.

My question is whether anyone can illustrate an explicit example of two solutions to the SW equations wrt a given metric $g_0$ that have opposite orientation and "annihilate" along a path $(g_t)_{0 \le t \le 1}$ to some final metric $g_1$. I would also be interested in any qualitative discussion of this phenomenon.