The goal of the talk is to describe explicit generators and relations in the stable motivic homotopy groups. Methods include geometric constructions, Toda brackets, and the Adams spectral sequence.

In this talk, we will explore some properties of Chern classes. In the axiomatic way, these properties uniquely determine these Chern classes.

Examples will be given to convince you that I do not know when a closed subset of a Riemann surface is a set of uniform approximation by holomorphic or meromorphic functions.

A planar tiling is a repeating symmetric pattern in the plane. Because of their common everyday appearances such patterns are called "wallpaper groups." We follow Conway's orbifold notation, which describes the 17 wallpaper groups as certain topological spaces: quotients of the plane by some finite group. Completeness is given by computing the Euler characteristic of such spaces. No knowledge of groups, topology, orbifolds, or how to hang wallpaper required.

Let X be a smooth projective curve over the complex numbers. The topological space of complex points of X is fairly simple: It is a one-point union of spheres, at least up to stable homotopy equivalence. If X is a smooth projective curve over an arbitrary field, one may consider it within the motivic homotopy theory of Morel and Voevodsky. Under the assumption that X has a rational point, it is possible to split off a top-dimensional sphere if and only if the tangent bundle of X admits a square root.

TBA

We review a formal model of molecular self-assembly known as the abstract Tile Assembly Model (aTAM). The aTAM which models the interaction of artificial biochemical macromoleclues known as "DNA tiles", which are capable of binding to each other in specific and surprising ways. The goal of this and other models of self-assembly is to study the feasibility of engineering nanoscale structures through a bottom-up approach, through the "programming" of molecules to automatically assemble themselves, in contrast to top-down approaches such as lithography.

After presenting the aTAM and a few basic results that illustrate its power and its limitations, we survey some theoretical conjectures. These conjectures share the properties of being easy to state, easy to understand, "obviously true", and unresolved. A primary goal is to frustrate the audience with the simplicity of these problems, in the hopes that one of them will step in and solve them.