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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 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 . These matrix entries satisfy the triangle inequality ( for all ) as well as a “quadrangle inequality” ( for all 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 ?

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 th entry is the sum, over all walks from to , 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 (all rings and algebras on this page are commutative and unital unless otherwise noted), with modules and , a *polynomial law* is a family of functions that commute with base change. A polynomial law is called *homogeneous of degree d* if each function satisfies for all . (Degree- polynomial laws are constant; degree- polynomial laws are module homomorphisms; degree- polynomial laws are quadratic forms.) If and are -algebras, then is *multiplicative* if each function is.

- Suppose is a multiplicative degree- polynomial law with . Is it always possible to base change along an injective ring homomorphism to make the polynomial law factor as an -algebra homomorphism times a multiplicative polynomial law of degree ?

If is flat, then multiplicative homogeneous degree-d polynomial laws correspond to -algebra homomorphisms . Given an -algebra of rank , the norm function is a multiplicative degree- polynomial law, and therefore corresponds to a canonical -algebra homomorphism , called the *Ferrand homomorphism*.

### G-closures

Given an -algebra of rank , and any subgroup , a homomorphism extending the Ferrand homomorphism is called a –*closure datum* for , and the -algebra , defined on the left by inclusion and on the right by , is called the –*closure algebra corresponding to* .

- For any quadratic -algebra , there is a canonical -closure datum for , and the corresponding -closure algebra is isomorphic to . Given a cubic -algebra , under what conditions on a -closure datum for is the corresponding -closure algebra isomorphic to ?
- There is a canonical multiplication on quadratic algebras that reduces to addition of -torsors in the étale case. Is there a similar operation on cubic algebras equipped with a -closure datum?

### Discriminant Algebras

- Given an -algebra of rank , define its discriminant algebra to equal , where the left-hand map is inclusion and the right-hand map is the Ferrand homomorphism. Is the assignment the only functor, up to natural isomorphism, that preserves base change and comes with an isomorphism that commutes with the discriminant bilinear forms?
- If and are -algebras of ranks and , then it is known that . Is it also the case that ?
- If is an -algebra of rank , and is a -algebra of rank , then is ?

### 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 , the fraction of natural numbers up to that are even converges to . This limiting fraction is called the *density* of the set of even numbers. However, not every subset of the natural numbers has a well-defined density; the limit might not exist. (Example: the set of natural numbers whose base-ten expansion starts with .) However, we can always consider the set of all limits of convergent subsequences of ; this is always a closed interval . This assignment from subsets of to closed intervals in is an *interval-valued probability measure* in the sense that

- For all subsets , we have .
- For all disjoint subsets , we have .
- .

(The fact that #2 doesn’t hold for countable disjoint unions makes 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 , a collection of “measurable” subsets closed under finite unions and complements, and a finitely-additive interval-valued probability measure defined for each subset in , must be
*consistent*? In other words, is there always an ordinary finitely-additive probability measure such that for all , we have ?

If are a sequence of random variables, then define the *information proportion* of a subset to be . Then the assignment is an interval-valued probability measure on .