Melody Chan

Stably computable properties of network graphs

We consider a scenario in which anonymous, finite-state sensing devices are deployed in an ad-hoc communication network of arbitrary size and unknown topology, and explore what properties of the network graph can be stably computed by the devices. We show that they can detect whether the network has degree bounded by a constant d, and, if so, organize a computation that achieves asymptotically optimal linear memory use. We define a model of stabilizing inputs to such devices and show that a large class of predicates of the multiset of final input values are stably computable in any weakly-connected network. We also show that nondeterminism in the transition function does not increase the class of stably computable predicates.

Note: Improved pebbling bounds

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f(G), is the minimal number of pebbles such that every configuration of f(G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.

Theta Characteristics of Tropical K 4 -Curves

A K4-curve is a smooth proper curve X of genus 3 over a field with valuation whose Berkovich skeleton Γ is a complete graph on four vertices. The curve X has 28 effective theta characteristics—the 28 bitangents to a canonical embedding—while Γ has exactly seven effective tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a K4-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.

Genera of Brill-Noether curves and staircase paths in Young tableaux

In this paper, we compute the genus of the variety of linear series of rank

Three notions of tropical rank for symmetric matrices

r

Sandpiles, Spanning Trees, and Plane Duality

and degree

Fano schemes of determinants and permanents

d

Lectures on Tropical Curves and Their Moduli Spaces

on a general curve of genus

Tropical hyperelliptic curves

g

Note: The distinguishing number of the augmented cube and hypercube powers

, with ramification at least