Topological Transforms for Statistical Shape Analysis
Kate Turner
Abstract: Persistent homology and other topological summaries like Euler curves are a classic way to characterise geometric shape. One common filtration is a height function which will capture information about geometry of a shape with respect to a specific direction. The Persistent Homology Transform (PHT) basically expands on this idea by considering the height functions in all directions. This has nice theoretical properties, in particular that it completely describes a compact nice subsets of Euclidean space, and also can provide new metrics for quantitatively measuring the difference between geometric objects. We can also make different variants of the PHT by using different topological summaries, most notably the Euler Characteristic Transform which constructs the Euler Curves in each direction. This lecture series will cover the mathematical theory behind these topological transforms, outline some statistical methodology and also survey a variety of nice applications from disease prognosis from the shapes of brain tumours to the identification of plants from their leaf contours.
Six functor formalisms and motivic homotopy theory
Marc Hoyois
Abstract: I will introduce motivic homotopy theory with a view towards the formalism of six functors. The latter is a general categorical framework for homology and cohomology devised by Grothendieck, which explains and generalizes many non-trivial facts like Poincaré duality and the Lefschetz trace formula. I will start with a description of this formalism in the classical setting of topological spaces. I will then define the stable motivic homotopy category and give an overview of its features, which in particular include a formalism of six functors for schemes.