Weekly Colloquium

Surfaces of Revolution with Constant Mean Curvature in Hyperbolic 3-Space

I discuss how to construct surfaces of revolution with constant mean curvature H=c in hyperbolic 3-space H³(-c²) of constant sectional curvature -c². It is intriguing to see that while the hyperbolic 3-space flattens to Euclidean 3-space E³ as c → 0, those surfaces approach catenoid, the minimal surface of revolution in E³. I also discuss how to construct minimal surface of revolution in H³(-c²). This work was done with Kinsey Zarske as her undergraduate research project.

Analytic Logical Functions that Provide Weak Interpretations of Satisfyability

We introduce a parallel Analytic Logic. We utilize a hyperspace over Boolean disjunctive clauses with a new semantics. The goal of this work is the development of an analytical logical calculus that provides weak solutions to satisfyablility problems in polynomial time. The approach utilizes three novel techniques: a parallel tri-state hyperspace over 2-state Boolean Algebra, a parallel tri-state encoding of disjunctive clauses that induces a metric, and a parallel tri-state concurrent intersection operator that does not require enumeration of all intersectors.

The mean ergodic theorem of matrices and its application to solving the Yang-Baxter matrix equation

We construct some solutions to the Yang-Baxter matrix equation with the help of the mean ergodic theorem for matrices.

Incidence Algebras

Gian-Carlo Rota defined an incidence algebra as a tool for solving combinatorial problems. An incidence algebra is a specific ring of functions defined on the ordered pairs of a partially ordered set to a given ring, and equipped with a module action by this ring. A notable element of an incidence algebra is its Möbius function, which coincides with the well-known Möbius function of number theory. Subsequent work of Rota and other authors investigate incidence algebras as a tool in algebraic topology. After a review of these ideas, I will summarize what I study in the algebraic context of incidence algebras.

Nim^∞

While trying to explain my research to students, I invented a game that turned out to be a generalization of the game of Nim, whose mathematical implications are well known. We review some of these implications, and explore how “Nimfinity” distills fundamental ideas of commutative algebra into a challenging game.