Math 423 INTRODUCTION TO COMPUTATIONAL TOPOLOGY

Course Number and Title: 2360423, Introduction to Computational Topology

METU Credit & ECTS Credit: (3-0)3 and 6.0
Prerequisite Courses: Math 251 or Math 120, knowledge of any programming is not required!

Catalogue Description: Basic topology, Surfaces and their triangulations, Complexes, Homology, Persistence homology, Morse functions, Discrete Morse Functions, Applications.

Aim of  the Course: Topology studies how a shape or object is connected. In the past few years, there has been an increased interest in the development and use of topological methods for solving various problems in science and engineering. This new line of study is called Computational Topology or Applied Algebraic Topology. Computational topology combines topological results with efficient algorithms to analyze data and solve problems in many fields, including computer graphics and image analysis, neural networks and sensor networks, clustering, robotics, genetics, protein biochemistry, geography, and others. The aim of this course is to give a crash course on topology and geometry with an emphasis on the algorithmic computations for students who are interested in the application side of mathematics.

Textbooks: Computational Topology, An Introduction by H. Edelsbrunner and J.Harer, available online.

Course Objectives: By the end of semester a successful student will be able to use efficient algorithms to compute homology of a given complex,
and  compute persistence homology, relate discrete Morse functions and persistence, sketch persistence diagrams.

Course Outline (14 weeks):
1.Basic topology, surfaces and their triangulations
2.Complexes; simplicial, Delaunay, alpha and Cech complexes
3.Homology groups
4.Relative homology
5.Persistent homology; filtrations, birth and death
6.Barcodes and Persistence diagrams
7.Matrix reduction
8.Smooth Morse theory
9.Discrete Morse theory
10.Reeb Graphs
11.Sublevel sets and bottleneck distance
12.Stability
13.Applications
14. Applications

Important Notice! The classes are planned to be live in classroom and at the same time the lecture will be live on zoom.  The students satisfying  Metu's regulations during pandemic may attend to the classes. (In order to attend the face-to-face classes and enter common places such as studying areas, all students and personnel must complete their vaccination processes as specified on e-Nabız. It is compulsory to wear face masks in the classrooms and all the other indoor places. Those who do not want to fulfill this requirement will not be admitted to the classes and other related places.)
Schedule:
 Tuesdays   10:40-12:30 M-102
 Thursdays  11:40-12:30 M-102

Grading: There will be a midterm exam, a final exam and a term project. Details will be explained later.