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Planar Networks
A circular planar graph is a planar (undirected) graph embedded in a disc, with vertices embedded in the boundary circle called boundary vertices, numbered $latex 1,\dots,n$ in circular order. A planar network is a circular planar graph with weighted edges; usually the weights are positive real numbers or formal indeterminates in a polynomial or power series ring.
- Imagining each edge weight to be a length, we can calculate the total lengths of the shortest paths between each pair of boundary vertices and put them in a symmetric matrix $latex M$. These matrix entries satisfy the triangle inequality ($latex M_{ij} + M_{jk} \geq M_{ik}$ for all $latex i,j,k$) as well as a “quadrangle inequality” ($latex M_{ik} + M_{j\ell} \geq M_{i\ell} + M_{jk}$ for all $latex i,j,k,\ell$ in circular order). Does every symmetric matrix with zero diagonal satisfying the triangle and quadrangle inequalities arise in this way?
- For which circular planar graphs is the edge weighting determined by the distance matrix $latex M$?
If we imagine each edge weight to be the resistance of an idealized resistor, and build the matrix of “effective resistances” between each pair of boundary vertices, then it is known which matrices arise in this way, and the matrix determines the edge weights iff the circular planar graph’s medial graph is lensless.
- If we instead build a matrix whose $latex ij$th entry is the sum, over all walks from $latex i$ to $latex j$, of the product of the weights of edges in the walk, then we again obtain a symmetric matrix. Which matrices arise in this way, and when does the matrix determine the original edge weights? (The “distance matrix” is a tropicalized version of this matrix.)
Polynomial laws
Given a ring $latex R$ (all rings and algebras on this page are commutative and unital unless otherwise noted), with modules $latex M$ and $latex N$, a polynomial law $latex p: M\to N$ is a family of functions $latex p_A: A\otimes_R M \to A\otimes_R N$ that commute with base change. A polynomial law is called homogeneous of degree d if each function $latex p_A$ satisfies $latex p_A(am) = a^d p_A(m)$ for all $latex a\in A, m\in A\otimes_R M$. (Degree-$latex 0$ polynomial laws are constant; degree-$latex 1$ polynomial laws are module homomorphisms; degree-$latex 2$ polynomial laws are quadratic forms.) If $latex M$ and $latex N$ are $latex R$-algebras, then $latex p$ is multiplicative if each function $latex p_A$ is.
- Suppose $latex p: M\to N$ is a multiplicative degree-$latex d$ polynomial law with $latex d \geq 1$. Is it always possible to base change along an injective ring homomorphism $latex R\to R’$ to make the polynomial law $latex p$ factor as an $latex R$-algebra homomorphism times a multiplicative polynomial law of degree $latex d-1$?
If $latex M$ is flat, then multiplicative homogeneous degree-d polynomial laws $latex M \to N$ correspond to $latex R$-algebra homomorphisms $latex (M^{\otimes d})^{S_d} \to N$. Given an $latex R$-algebra $latex A$ of rank $latex n$, the norm function $latex A\to R$ is a multiplicative degree-$latex n$ polynomial law, and therefore corresponds to a canonical $latex R$-algebra homomorphism $latex (A^{\otimes n})^{S_n}\to R$, called the Ferrand homomorphism.
G-closures
Given an $latex R$-algebra $latex A$ of rank $latex n$, and any subgroup $latex G \subseteq S_n$, a homomorphism $latex \varphi: (A^{\otimes n})^{G}\to R$ extending the Ferrand homomorphism is called a $latex G$-closure datum for $latex A$, and the $latex R$-algebra $latex A^{\otimes n} \otimes_{(A^{\otimes n})^G} R$, defined on the left by inclusion and on the right by $latex \varphi$, is called the $latex G$-closure algebra corresponding to $latex \varphi$.
- For any quadratic $latex R$-algebra $latex A$, there is a canonical $latex C_2$-closure datum for $latex A$, and the corresponding $latex C_2$-closure algebra is isomorphic to $latex A$. Given a cubic $latex R$-algebra $latex B$, under what conditions on a $latex C_3$-closure datum for $latex B$ is the corresponding $latex C_3$-closure algebra isomorphic to $latex B$?
- There is a canonical multiplication $latex \ast$ on quadratic algebras that reduces to addition of $latex C_2$-torsors in the étale case. Is there a similar operation on cubic algebras equipped with a $latex C_3$-closure datum?
Discriminant Algebras
- Given an $latex R$-algebra $latex A$ of rank $latex n$, define its discriminant algebra $latex \Delta_{A/R}$ to equal $latex (A^{\otimes n})^{A_n} \bigotimes_{(A^{\otimes n})^{S_n}} R$, where the left-hand map is inclusion and the right-hand map is the Ferrand homomorphism. Is the assignment $latex (R, A)\mapsto \Delta_{A/R}$ the only functor, up to natural isomorphism, that preserves base change and comes with an isomorphism $latex \bigwedge^2 \Delta_{A/R} \cong \bigwedge^n A$ that commutes with the discriminant bilinear forms?
- If $latex A$ and $latex B$ are $latex R$-algebras of ranks $latex m$ and $latex n$, then it is known that $latex \Delta_{A\times B/R} \cong \Delta_{A/R} \ast \Delta_{B/R}$. Is it also the case that $latex \Delta_{A\otimes B / R} \cong \Delta_{A/R}^{\ast n} \ast \Delta_{B / R}^{\ast m}$?
- If $latex B$ is an $latex A$-algebra of rank $latex m$, and $latex C$ is a $latex B$-algebra of rank $latex n$, then is $latex \Delta_{C/A} \cong \Delta_{\Delta_{C/B}/A} \ast \Delta_{B/A}^{\ast n-2}$?
Interval-valued probability measures
Half of all natural numbers are even. What this means, since there’s no uniform probability measure on the natural numbers, is that as $latex n \to \infty$, the fraction of natural numbers up to $latex n$ that are even converges to $latex 1/2$. This limiting fraction is called the density of the set of even numbers. However, not every subset $latex S$ of the natural numbers has a well-defined density; the limit $latex \lim_{n\to\infty} \dfrac{\#(S\cap\{1,\dots,n\})}{n}$ might not exist. (Example: the set of natural numbers whose base-ten expansion starts with $latex 1$.) However, we can always consider the set of all limits of convergent subsequences of $latex \left(\dfrac{\#(S\cap\{1,\dots,n\})}{n}\right)_{n=1}^\infty$; this is always a closed interval $latex \delta(S) \subseteq [0,1]$. This assignment $latex \delta$ from subsets of $latex \mathbb{N}$ to closed intervals in $latex [0,1]$ is an interval-valued probability measure in the sense that
- For all subsets $latex S$, we have $latex \delta(S^c) = 1-\delta(S)$.
- For all disjoint subsets $latex S,T$, we have $latex \delta(S\cup T) \subseteq \delta(S) + \delta(T)$.
- $latex \delta(\varnothing) = [0,0]$.
(The fact that #2 doesn’t hold for countable disjoint unions makes $latex \delta$ only a finitely-additive interval-valued probability measure.) There’s a similar finitely-additive interval-valued probability measure for Lebesgue measurable subsets of the real line, and for any inductive limit of probability spaces.
- Given a nonempty set $latex X$, a collection of “measurable” subsets $latex \Omega \subseteq P(X)$ closed under finite unions and complements, and a finitely-additive interval-valued probability measure $latex \delta$ defined for each subset in $latex \Omega$, must $latex \delta$ be consistent? In other words, is there always an ordinary finitely-additive probability measure $latex d: \Omega \to [0,1]$ such that for all $latex S\in \Omega$, we have $latex d(S) \in \delta(S)$?
If $latex X_1,\dots,X_n$ are a sequence of random variables, then define the information proportion of a subset $latex S\subseteq\{1,\ldots,n\}$ to be $latex h(S) = H(X_i : i\in S) / H(X_1,\dots,X_n)$. Then the assignment $latex S \mapsto [1-h(S^c), h(S)]$ is an interval-valued probability measure on $latex \{1,\ldots, n\}$.