Path signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. The theory of path signatures can be lifted to the setting of Lie group valued time series while retaining their universal and characteristic properties. This talk will introduce these generalized path signatures on Lie groups and…

A trisection of a smooth, connected 4-manifold is a decomposition into three standard pieces. Like the case of Heegaard splittings in dimension three, a trisection is described by a trisection diagram: three sets of curves in a surface satisfying some properties. In general, it is not evident whether two trisection diagrams represent the same decomposition…

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only 2-torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported…

The Heegaard Floer homology introduced by Ozsvath and Szabo provides a lot of link invariants to study links in the three-sphere and its surgery manifolds. In this talk, we exact some invariants from the package to give bounds for the four-ball genus of links in the three-sphere. We also give a family of links where the…

This talk will focus on using the Euclidean Signature to determine whether two smooth planar curves are congruent under the Special Euclidean group. Work done by Emilio Musso and Lorenzo Nicolodi emphasizes that signatures must be used with caution by constructing 1-parameter families of non-congruent curves with degenerate vertices (curve segments of constant curvature) with identical signatures. We address the claim…