Jan

25

2022

Nakada’s α-expansions move from the regular continued fractions (α = 1), Hurwitz singular continued fractions (obtained at α =(-1+\sqrt{5})/2 ), and nearest integer continued fractions (α=1/2), to more unusual cases for α less than sqrt{2}-1. This talk will look at similar continued fraction expansions with odd partial quotients. I will describe how restricting the parity of the partial quotients changes the Gauss map and natural extension domain. This is joint work with Florin Boca as well as Yusef Hartono, Cor Kraaikamp, and Niels Langeveld.

Jan

26

2022

Topological data analysis (TDA) studies the shape patterns of data. Persistent homology (PH) is a widely used method in TDA that summarizes homological features of data at multiple scales and stores them in persistence diagrams (PDs). However, a sufficiently large amount of PDs that allow performing statistical analyses is typically unavailable or requires inordinate computational resources.

Jan

27

2022

Data sets can be considered as geometric objects. For example, a point cloud can be endowed with a metric structure through Euclidean or Lp distances, digital images can be considered as cubical complexes. Furthermore, metric data can be enriched with the normalized counting measure, and graphs and simplicial complexes can be built out of it through Vietoris-Rips, Cech or similar constructions. Thus, one can apply methods coming from fields like Algebraic Topology, Metric Geometry and Optimal Transport to extract geometric information from data.

Jan

28

2022

Jan

31

2022

Given the classical Apollonian gasket, what are all the circles that intersect it in countably many points, and how are they distributed? Equivalently, what are all the elementary geodesic planes in the Apollonian orbifold, and how do they behave topologically?