Georgia Institute of Technology College of Sciences

When visualising topological objects via 3D printing, we need athree-dimensional geometric representation of the object. There areapproximately three broad strategies for doing this: "Manual" - usingwhatever design software is available to build the object by hand;"Parametric/Implicit" - generating the desired geometry using aparametrisation or implicit description of the object; and "Iterative" -numerically solving an optimisation problem.The manual strategy is unlikely to produce good results unless the subjectis very simple. In general, if there is a reasonably canonical geometricstructure on the topological object, then we hope to be able to produce aparametrisation of it. However, in many cases this seems to be impossibleand some form of iterative method is the best we can do. Within theparametric setting, there are still better and worse ways to proceed. Forexample, a geometric representation should demonstrate as many of thesymmetries of the object as possible. There are similar issues in makingthree-dimensional representations of higher dimensional objects. I willdiscuss these matters with many examples, including visualisation offour-dimensional polytopes (using orthogonal versus stereographicprojection) and Seifert surfaces (comparing my work with Saul Schleimerwith 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 withMarco Mahler), "Triple gear" and a visualisation of the Klein Quartic(joint work with Saul Schleimer), and hinged surfaces with negativecurvature (joint work with Geoffrey Irving).

I will give an elementary introduction to Majid's theory of braided groups and how this may lead to a more geometric, less quantum, interpretation of knot invariants such as the Jones polynomial. The basic idea is set up a geometry where the coordinate functions commute according to a chosen representation of the braid group. The corresponding knot invariants now come out naturally if one attempts to impose such geometry on the knot complement.

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links. I'll show that every such surface admits a decomposition into three standard pieces called a bridge trisection. I'll also describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface. This is joint work with Alexander Zupan.​

I will describe the results of a joint project with Mike Mandell on the algebraic K-theory of the sphere spectrum, focusing on recent work that describes the fiber of the cyclotomic trace using a spectral lift of Tate-Poitou duality.