Topics in Geometry - Computational Topology

Lecture schedule

The section numbers correspond either to Munkres - Topology, or Edelsbrunner, Harer - Computational Topology.

Date Topic
1/18 Intro, Topological spaces (M. 12)
1/20 Bases, subspace topology (M. 13, 16)
1/23 Closed sets, limit points (M. 17)
1/25 Continuous functions (M. 18)
1/27 The metric topology (M. 20)
1/30 The quotient topology (M. 22)
2/1 Connected spaces (M. 23), Graphs (EH I.1)
2/3 Surfaces (EH II.1)
2/6 Simplicial complexes (EH III.1)
2/8 Convex set systems (EH III.2)
2/10 Homology (EH IV.1)
2/13 Homology (EH IV.1)
2/15 Induced maps on homology (EH IV.1)
2/17 Euler-Poincare formula (EH IV.2)
Giusti et. al. - Clique topology reveals intrinsic geometric structure in neural correlations
Masulli, Villa - The topology of the directed clique complex as a network invariant
2/20 Relative homology (EH IV.3)
2/22 Exact sequences (EH IV.4)
2/24 Exact sequence of a pair (EH IV.3)
Homology of a genus g surface
2/27 The mapper algorithm
Python Mapper - software implementation of mapper
Singh, Mémoli, Carlsson - Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition - original paper
Kraft - Illustrations of Data Analysis Using the Mapper Algorithm and Persistent Homology - more detailed explanation
3/1 Persistent Homology (EH VII.1)
3/3 Persistent Homology (EH VII.1)
3/6 Stability Theorems (EH VIII.2)
3/8 Midterm - Covers sections III.1, IV.1, IV.2 (Euler-Poincaré formula), IV.3 in Edelsbrunner-Harer's book
Practice problems
3/10 Applications
Persistent homology transform for modeling shapes and surfaces
Sliding windows and persistence: an application of topological methods to signal analysis
3/13-3/17 Spring Break
3/20 Morse theory (EH VI.1)
3/22 Morse theory (EH VI.1)
3/24 Transversality (EH VI.2)
3/27 The Morse-Smale complex (EH VI.2)
3/29 CW complexes
REU paper on cellular homology
3/31 Cellular homology
4/3 Cellular homology
4/5 Discrete Morse theory
Forman - A user's guide to discrete Morse theory
Forman - Morse theory for cell complexes
4/7 Discrete Morse theory
4/10 Discrete Morse theory
4/12 Discrete Morse theory
4/14-4/17
4/19
4/21
4/24
4/26
4/28 Presentations
9:25 -- 9:35 Christian Gorski
9:40 -- 9:50 Shijie Jin
9:55 -- 10:05 Zhixing Zhang
5/1 Presentations
9:18 -- 9:28 Ariel Navotas
9:30 -- 9:40 Alex Sitaras
9:42 -- 9:50 Luke Vandertie
9:52 -- 10:02 Marissa Koscielski
10:04 -- 10:14 Erin Denison
5/3 Presentations
9:25 -- 9:35 Christian Hokaj / Matthew Drnevich
9:38 -- 9:48 Robert Black / Jonathan Baker
9:51 -- 10:01 Sean Kent
10:04 -- 10:14 Samantha Kerper