Thursday 2-3 PM, Online (Contact organizer for link) 

Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of Topological Data Analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike.

Shape is foundational to biology. Observing and documenting shape has fueled biological understanding as the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. The vision of Topological Data Analysis (TDA), that data is shape and shape is data, will be relevant as biology transitions into a data-driven era where meaningful interpretation of large data sets is a limiting factor.

Signed persistence diagrams provide a concise representation of the rank invariant (persistent Betti numbers) or birth-death invariant for filtrations of topological spaces. When these filtrations are indexed by a linear poset, the signed persistence diagrams consist of nonnegative values, allowing for straightforward interpretation as the multiplicity of the bars in the barcode of the filtration. However, for filtrations indexed by non-linear posets, signed persistence diagrams can include negative values, which complicates their interpretability.

Title: Gauge Theory on Hyperkahler Manifolds and 3-Sasakian Links

One-dimensional persistent homology is arguably the most important and heavily used tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas. In this talk, we introduce a certain 2-dimensional persistence module structure associated with the persistent cohomology ring, where one parameter is the cup-length and the other is the filtration parameter.

Although Vietoris–Rips (VR) complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes show an interesting progression of homotopy types as the scale increases, moving from the circle S^1 to S^3 to S^5, and so on. However, much less is known about VR complexes of higher-dimensional spheres.