Applied Algebraic Topology

Course Info

Instructors Crichton Ogle, o g l e @ m a t h . o h i o - s t a t e . e d u
Facundo Mémoli, m e m o l i @ m a t h . o s u . e d u
Course code Math 8800 -- Fall 2016
Times: MWF 3-4pm.
Location: EC 354.
Description: In recent years Topology has contributed key ideas to a new discipline sitting at the crossroads of mathematics, computer science, and statistics. These fields interact to create new methods that can be applied to interpret data coming from the life sciences, chemistry, engineering, etc.

This course will teach the main concepts in Applied Algebraic Topology. In particular we will study persistent homology, the most prominent tool for data analysis via topological methods. Persistent homology is a construction that attaches to a finite dataset (i.e. a finite metric space) a one parameter family of vector spaces called a persistence vector space.

It turns out that persistence vector spaces can be represented up to isomorphism by certain invariants called barcodes, which are easily interpretable and can be regarded as a generalization of the so called Betti numbers. In this course, we will also study the stabiltiy of persistent homology to perturbations, and how these ideas are applied to data analysis and shape recognition.

Several possible research directions will be discussed.

Prerequisites: The course has minimal requisites: Linear Algebra and exposure to Point Set Topology. The taregt audience are both graduate and undegraduate students.
Tentative Syllabus: (1) Introduction to Persistence. (2) Brief linear algebra review. (3) Categories: vector spaces, finite metric spaces, sets. Functors: clustering, simplicial homology with field coefficients. (4) Persistent homology: Persistent vector spaces, interval persistent vector spaces, Gabriel's theorem, barcodes (5) Stability of persistence: Metric geometry review, interleavings, barcode distance.


Meeting 1 (TBA/08). First meeting. Introduction to the main ideas. Readings: TBA.


Synergistic activities