Events of the Week

We are in a position to establish stability for Morse functions in terms of the interleaving distance.

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This learning seminar introduces the foundations of persistent homology, a key tool in topological data analysis for capturing multiscale topological features of general metric spaces. We will study persistence modules, barcodes, and distances such as the bottleneck and interleaving metrics, with a focus on their geometric meaning.

A central goal of the seminar is to understand and prove stability theorems for persistent homology, including bounds relating bottleneck distance to the Gromov–Hausdorff distance. Core examples will come from Morse theory and Vietoris–Rips filtrations of metric spaces.

The seminar is structured as a guided, collaborative reading course and is aimed at graduate students and faculty generally interested in this aspect of topology and geometry.

By cutting a 3D convex polyhedron by a plane, we obtain a convex cap. By cutting on a boundary rooted spanning forest of the edge graph, we can unfold the cap into the plane. Nearly flat caps have unfoldings which are very close to their orthogonal projections. We take advantage of this to construct a necessary and sufficient condition for detecting overlap in the unfolding based on the orthogonal projection of the cap's edge graph. This is a recent result which is a joint work with Tyson Trauger. We also discuss two possible applications of this condition: a positive resolution to a special case of Durer's problem, and a necessary and sufficient condition for detecting overlap in infinitesimal edge unfoldings of arbitrary convex caps.

Title and abstract to be announced

Given a local ring R we can ask when saturation of ideals in R commutes with other operations on ideals (such as extension to a ring containing R). We show that the condition that extension of ideals along a ring map R \to S commutes with saturation controls inherent properties of the rings R & S, such as Cohen-Macaulayness and unramifiedness.

Title and abstract to be announced

Title and abstract to be announced

We will share a story that brings together geometry, dynamics, and number theory in interesting and novel ways, and also creates some very compelling imagery- the talk will have lots of pictures, and all relevant background notions will be introduced and explained. Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot-Vijayraghavan expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

Title and abstract to be announced

Title and abstract to be announced