Henry Segerman (Oklahoma State University)

Title: Design of 3D printed mathematical art

Abstract: When visualising topological objects via 3D printing, we need a three-dimensional geometric representation of the object. There are approximately three broad strategies for doing this: "Manual" - using whatever design software is available to build the object by hand; "Parametric/Implicit" - generating the desired geometry using a parametrisation or implicit description of the object; and "Iterative" - numerically solving an optimisation problem.

The manual strategy is unlikely to produce good results unless the subject is very simple. In general, if there is a reasonably canonical geometric structure on the topological object, then we hope to be able to produce a parametrisation of it. However, in many cases this seems to be impossible and some form of iterative method is the best we can do. Within the parametric setting, there are still better and worse ways to proceed. For example, a geometric representation should demonstrate as many of the symmetries of the object as possible. There are similar issues in making three-dimensional representations of higher dimensional objects. I will discuss these matters with many examples, including visualisation of four-dimensional polytopes (using orthogonal versus stereographic projection) and Seifert surfaces (comparing my work with Saul Schleimer with Jack van Wijk's iterative techniques).

I will also describe some computational problems that have come up in my 3D printed work, including the design of 3D printed mobiles (joint work with Marco Mahler), "Triple gear" and a visualisation of the Klein Quartic (joint work with Saul Schleimer), and hinged surfaces with negative curvature (joint work with Geoffrey Irving).

Ileana Streinu (Smith College)

Title: Maxwell's Problem, 150 years later: from bridges to nano-mechanics

Abstract: Finding a combinatorial characterization for (minimally) rigid bar-and-joint frameworks in dimensions higher than 3 is an easy-to-state, yet elusive, long standing open problem in rigidity theory, originating in two geometry papers from the 19th century of the renowned physicist James Clerk Maxwell. I will summarize our current state of knowledge on Maxwell's problem, and present recent developments leading to a surprising range of applications, from folding robot arms and origami to anayzing the flexibility of proteins and crystalline matter. No advanced prerequisites are necessary. To help build the geometric and kinematic intuitions, the relevant mathematical concepts and techniques (from discrete and algebraic geometry, rigidity theory, kinematics, graph theory, linear programming, matroid theory) will be introduced primarily through physical models and animated graphics.

Melanie Wood (University of Wisconsin, Madison)

Title: The Chemistry of Primes

Abstract: We are familiar with the prime numbers as those integers which cannot be factored into smaller integers, but if we consider systems of numbers larger than the integers, the primes may indeed factor in those larger systems. We discuss various questions mathematicians ask about how primes may factor in larger systems, talk about both classical results and current research on the topic, and give a sense of the kind of tools needed to tackle these questions.