Up
 
Computational Topology
in Science & Engineering
Reading Group, Winter 2008
Organizer: Abubakr Muhammad
Time: Wednesdays at 4:15 pm
Venue: McConnell Bldg. Room 321 (SOCS Lounge)
What is Computational Topology?
In recent years, there has
been an enormous interest among researchers in various disciplines to develop and use
topological methods for solving various problems in science and engineering. These
algorithmic methods provide robust measures for global qualitative features of
geometric and combinatorial objects that are relatively insensitive to local details. This
makes topological abstractions into useful models for understanding qualitative geometric and combinatorial questions
in several settings. The abstract machinery of algebraic topology has been used in
various contexts related to data analysis, object recognition, discrete &
computational geometry and distributed computing.
Summary of Objectives & Activities
The aim of this reading group is to communicate some of
these recent developments to the participants with a minimal background in algebraic topology. Our
focus will be on applications, although the proper appreciation of this research will
require the understanding of some sophisticated mathematical methods.
Since it is expected that the attendees will come
from diverse backgrounds in science, mathematics and engineering; the organizer will
provide tutorials on the required background in topology. Moreover, the organizer will
demonstrate how to use various computational topology software tools. The majority of
meetings will be dedicated to discussing various research articles written by the leading
experts in the field. Hopefully, these activities will enable the participants to generate
new mathematics as well as new applications.
Topics
Mathematics: Homology, homotopy,
Morse theory, Conley index theory,
configuration spaces
Computational Methods:
Cech, Rips, witness
and alphacomplexes, persistent
homology of filtrations, harmonic methods for
computing homology, software tools
Applications:
Coordination,
navigation and reconfiguration in robotics, coverage and routing in sensor networks, visualization and qualitative analysis of highdimensional
data sets, analysis of
nonlinear dynamical systems, structural biology, image classification,
distributed algorithms
Who should attend?
 Mathematicians
with interest in topology, geometry and dynamical systems 
 Computer
scientists investigating computational geometry, machine learning, visualization &
data analysis 
 Engineers
interested in algorithmic aspects of robotics, networked sensing and control theory 
 Life
scientists dealing with large data sets in molecular biology, neuroscience, systems
biology 
Tentative Schedule
DATE 
TOPICS 
FRONTIERS 
SUGGESTED READING 
Background

Jan 16 
Overview of computational
topology 
. 
Barcodes: The persistent
topology of data by Robert Ghrist. 
Jan 23 
Simplicial & cubical
complexes; homotopy 
Math 
Notes on homology
theory by Abubakr Muhammad. Also check Afra Zomorodian's course
notes. 
Feb06 
Homotopy; simplicial homology 
Math, CS 
Same as last week. 
Feb 13 
Filtrations & persistent
homology 
Math, CS 
Computing
Persistent Homology by Zomorodian and Carlsson. 
Feb 20 
Handson training: Plex
software package 
CS 

Data Analysis, Learning & Visualization 
March 05 
Manifold Learning from point
cloud data sets (PCD) 
Math, CS, Bio 
Finding
the homology of submanifolds with high confidence from random samples by Niyogi, Smale and Weinberger 
March 12 
Persistence and its Stability
in PCDs 
Math, CS, Bio 
Persistent
Homology  a Survey by Herbert Edelsbrunner and John Harer. 
April 02 
Natural Image Classification 
EE, CS, Bio 
A Topological
Analysis of the Space of Natural Images Gunnar Carlsson and Tigran
Ishkhanov. 
April 09 
Homology computation using
harmonic analysis 
CS, Math 
Computing
Betti numbers via Combinatorial Laplacians by Joel Friedman 
Networks and Sensing 

Coverage problems in sensor
networks –I 
EE, CS 
Homological
sensor networks by deSilva and Ghrist (survey). Blind swarms for coverage in
2D by Ghrist, deSilva, Muhammad 

Coverage problems in sensor
networks –II 
EE, CS 
Coordinatefree coverage in sensor
networks with controlled boundaries via homology by deSilva and Ghrist. 

Landmarks, routing and homology
feature size in sensor networks 
EE, CS 
Geodesic
Delaunay Triangulation and Witness Complex in the Plane by Gao, Guibas, Oudot, and
Wang. 
Robotics and Coordination 

Morse theory – continuous,
discrete & combinatorial 
Math 


Navigation in robotics 
EE, ME, CS 


Configuration spaces – I:
Distributed coordination 
Math, ME, EE 


Configuration spaces – II:
Reconfigurable systems 
Math, ME, EE 

Dynamical Systems 

Conley index theory 
Math 


Computer assisted proofs in
dynamical systems 
Math, Phys, Bio 


Handson training: CHomP
software package 
Math, CS 

Miscellaneous Topics 

Topology of random data and
random fields 
Math, CS 


Protein docking and structural
biology 
Bio, CS 

Resources in Computational Algebriac
Topology
(If you find someone missing in
these lists, please email me!)
Workshops/Programs
Courses
Software
Books
 Tomasz Kaczynski, Konstantin Mischaikow,
Marian Mrozek (2004), Computational Homology, Springer, ISBN 0387408533. 
 Afra J. Zomorodian (2005). Topology
for Computing, Cambridge, ISBN 0521836662. 
 William
Brasener (2006), Topology and
its applications, John Wiley, ISBN: 9780471687559. 
 Allen
Hatcher, Algebraic
Topology. 
