Papers

April 18th, 2024 - A Universal Property of the Spectrum of a Ring and the Semiring of its Ideals
by William Bernardoni
Preprint
Arithmetic valuations are intimately connected with the structure of the ideals of a commutative ring. We show how the generalized idempotent semiring valuations of Jeffrey and Noah Giansiracusa can be used to make this connection explicit. Through this generalized valuation theory sufficiently complete positive valuations give rise to Galois correspondences with the lattice of ideals of a commutative ring. Up to isomorphism the semiring of ideals of a commutative ring can be defined as a universal factoring semiring for positive valuations. We then further show that this valuation theory can formally connect Joyal's notion de zéros and universal property of the spectrum of a ring to arithmetic valuation theory. We show that up to isomorphism as a coframe, the closed Zariski topology on a spectrum of a commutative ring can be defined as the universal factoring semiring of additively and multiplicatively idempotent valuations.

March 7th, 2023 - Network Storage Analysis via Semiring Geometry
by William Bernardoni, Robert Kassouf-Short, Robert Cardona, Brian Heller, Justin Curry, David Spivak, Juan A. Fraire
Accepted at the IEEE Aerospace Conference, 2024.
We introduce a novel semiring model for contact-based routing protocols that includes a means of determining storage needs. Through proper analysis of the semiring structure, we show how to determine optimal storage structures in satellite networks. In addition, we run our analysis on simulated satellite networks to demonstrate the potential for working with these semiring models in a computational framework. We conclude by indicating future directions for semiring analysis is space communications.

May 15th, 2023; published June 29th 2024 - Applications of Generalized Universal Valuations
by William Bernardoni
Published in Communications in Algebra
Preprint available here.
We introduce a generalization of the universal valuation semiring defined by Jeffrey and Noah Giansiracusa. We then explicitly characterize the additive structure of this semiring and show that, when applied to \(\mathbb Q\), this characterization gives the Non-Archimedean case of Ostrowski's theorem. We conclude with examples of non-commutative valuations and their applications, such as the detection of the existence of representations of rings in ultrametric vector spaces.

April 3rd, 2023 - Algebraic and Geometric Models for Space Networking
by William Bernardoni, Robert Cardona, Jacob Cleveland, Justin Curry, Robert Green, Brian Heller, Alan Hylton, Tung Lam, Robert Kassouf-Short
Preprint, Submitted to SIAGA
In this paper we introduce some new algebraic and geometric perspectives on networked space communications. Our main contribution is a novel definition of a time-varying graph (TVG), defined in terms of a matrix with values in subsets of the real line \(P(\mathbb R)\). We leverage semi-ring properties of \(P(\mathbb R)\) to model multi-hop communication in a TVG using matrix multiplication and a truncated Kleene star. This leads to novel statistics on the communication capacity of TVGs called lifetime curves, which we generate for large samples of randomly chosen STARLINK satellites, whose connectivity is modeled over day-long simulations. Determining when a large subsample of STARLINK is temporally strongly connected is further analyzed using novel metrics introduced here that are inspired by topological data analysis (TDA). To better model networking scenarios between the Earth and Mars, we introduce various semi-rings capable of modeling propagation delay as well as protocols common to Delay Tolerant Networking (DTN), such as store-and-forward. Finally, we illustrate the applicability of zigzag persistence for featurizing different space networks and demonstrate the efficacy of K-Nearest Neighbors (KNN) classification for distinguishing Earth-Mars and Earth-Moon satellite systems using time-varying topology alone.