From the topological fact that the circle is the representing space for the functor $X \to H1(X,\mathbb Z)$ follows that by computing degree 1 cohomology and picking cocycle representatives corresponds to computing equivalence classes of continuous maps $X\to S1$. In a research project with Vin de Silva and Dmitriy Morozov, we use this in a data analysis context to produce intrinsic coordinate functions with values on the circle.

We shall discuss the derivation of circle-valued coordinates for point clouds using persistent cohomology to distinguish useful coordinate functions from functions appearing from noise in the data set, and discuss applications to the analysis of periodic signals and periodic dynamical systems.

After a short introduction on the symplectic packing problem, we will explain how recent results of B‐H. Li, T.‐J. Li, A. K. Liu, and Gao on symplectic cones lead to a concrete understanding of symplectic packings for rational ruled surfaces. If time permits, we will also explain how this relates to recent work of M. Hutchings on embedded contact homology.

This is joint work with O. Buse.

The horofunction compactification of a proper metric space (X,d), also known as the Busemann compactification, is obtained by using the distance function d to embed X into the space of continuous real valued functions on X and taking the closure. The horofunction boundary of X is the complement of the image of X in its horofunction compactification.

We explicitly find the horofunction boundary of the (2n+1)-dimensional Heisenberg group with the Carnot-Caratheodory metric and show that it is homeomorphic to a 2n-dimensional disk. We also show that the Busemann points correspond to the (2n-1)-sphere boundary of this disk and that the compactified Heisenberg group is homeomorphic to a (2n+1)-dimensional sphere. As an application, we find all isometries of the Carnot-Caratheodory metric. This is joint work with Tom Klein.

We will give an introductory talk to the theory of operads, giving basic definitions and examples. We will focus on examples in topology, category theory, and computer science.