Schedule & Abstracts
Coronavirus Update (12 March):
It is with great regret we have been forced to cancel the Oxford Applied Topology School.
We have resisted cancellation, but unfortunately our participant accommodation has been cancelled and two of our three lecturers have been prevented from travelling.
We are hoping to reschedule the School to take place at a later date.
Apologies for the disruption caused by this exceptional set of circumstances!
Geometric and Topological Inference
Frédéric Chazal
The estimation of topological and geometric properties of data, usually represented as point clouds sampled in Euclidean spaces or more generally metric spaces, is attracting increasing interest with the emergence of the fast growing field of Topological Data Analysis (TDA). The mathematical and statistical approaches to geometric inference play a fundamental role for the design of mathematically well-founded tools for TDA in practice. The goal of this mini-course is to provide a short introduction, through a few selected topics, to the mathematical and statistical foundations of topological and geometric inference.
Topological Complexity
Petar Pavešić
Topological complexity is an invariant associated to a topological space describing the minimal size of an open cover over which local sections of the path space projection map exist. Using the local sections of the projection map we may continuously prescribe a unique path between two points of the topological space within each open set. As such, topological complexity is an important invariant in the motion planning problem in robotics, where we seek to find a path from an initial state to a final state for a robot. In this course we will explore topological complexity and applications to robotics. We will see configuration spaces, kinematic maps, motion planning constructions and obstructions as well as techniques to bound the topological complexity.
Sheaf Theory, Homological Algebra & Data Analysis
Michael Robinson
In this mini-course, I will motivate the algebraic and geometric foundations of computational topology and extend them to treat predictive and descriptive models. We will traverse a pipeline starting with models written as systems of equations, encoding them as sheaves, exposing structural features of these sheaves as homological algebraic invariants, and using geometric tools to measure data-model agreement. I will show how to construct models diagrammatically, using category theory as a starting point. This diagrammatic theory reinterprets models as sheaves of pseudometric spaces on finite spaces. The benefit of this encoding is twofold: we obtain homological algebraic invariants for understanding the model structure, and we obtain geometric invariants for understanding data-model consistency. Since sheaves preserve the structure of the original models, we can also interpret the algebraic invariants in terms of model behavior. Using the consistency radius of a sheaf assignment, we will quantify the fit between data and models, and will tune this quantification to improve data or model quality. Finally, we will discuss the structure of the consistency filtration, a tool that applies traditional topological data analysis tools to sheaf geometry. The consistency filtration provides a deep diagnosis of data and model defects and can be used to extract both qualitative and quantitative features that aid in classification tasks.