Annual Report 2005 - Fields Institute - University of Toronto

GENERAL SCIENTIFIC ACTIVITIES
Workshops
Algebraic Topological Methods in Computer Science, II
July 16–20, 2004
Held at the University of Western Ontario
Organizers: Gunnar Carlsson (Stanford) and Rick Jardine
(UWO)
The purpose of the meeting was to study and develop the
applications of techniques of algebraic topology in various
areas of theoretical computer science, including concurrency
theory, computational geometry and problems
related to combinatorics. There has been much progress in
these areas since the time of the first meeting in Stanford.
It is now clear that the higher dimensional automata which
model parallel processing behaviour and their associated
“fundamental categories” can be encoded in a purely
combinatorial homotopy theory of cubical sets. Other
new and interesting approaches to studying concurrent
behaviour involve, respectively, a Quillen model structure
on the category of 2-categories, and the homotopy theory
of simplicial presheaves on a category of spaces admitting
local partial orders.
Several talks were presented on the detection of shapes
from point cloud data. The most effective and well understood
method of carrying out such an analysis involves
persistent homology, which amounts to keeping track of
Betti numbers of simplicial complexes built from the data
set as more data points are taken into account. The basic
idea, which has proven to be quite robust experimentally,
is that good Betti numbers (i.e. those which represent real
features) should appear and persist once the resolution of
the data is sufficiently fine.
There was an interesting discussion of a program initiated
by Lovasz, concerning his cell complex Hom(G,H), which
is defined for graphs G and H. The idea is that this complex
contains obstructions to the existence of maps from G to H.
The question of the existence of such maps in the case when
H is a complete graph amounts to the graph colouring
problem. Kozlov has recently proved a conjecture of Lovasz
which asserts that if G is a cycle of odd length and the cell
complex Hom(G,H) is k-connected, then the chromatic
number of H is greater than or equal to k + 4. The method
of proof involves spectral sequence calculations and manip-
G e n e r a l S c i e n t i f i c A c t i v i t i e s
ulations of Stiefel-Whitney classes. The homotopy theory of
graphs which should underly this discussion remains to be
explored.
Support for the meeting was received from NSF and the
Fields Institute.
Speakers:
Saugata Basu (Georgia Tech.)
Efficient algorithms for computing the Betti numbers of semialgebraic
Sets
Peter Bubenik (EPF, Lausanne)
Towards a model category for local po-spaces
Gunnar Carlsson (Stanford)
Topology of point cloud data
Anne Collins (Stanford)
A barcode shape descriptor for curve point cloud data
Peter Csorba (ETH, Zürich)
Homotopy types of box complexes
Vin de Silva (Stanford)
Harmonic methods in computational topology
Herbert Edelsbrunner (Duke)
Elevation on a 2-manifold
Robin Forman (Rice)
A topological approach to the game of “20 Questions”
Robert Ghrist (Illinois)
Reconfiguration and the geometry of cube complexes
Eric Goubault (CEA Saclay)
Algorithms for computing fundamental categories, and applications
to the static analysis of concurrent programs
Kathryn Hess (EPF, Lausanne)
Quillen model categories applied to concurrency theory
Rick Jardine (UWO)
Higher order automata, cubical sets, and some conjectures of
Grothendieck
Fields Institute 2005 ANNUAL REPORT 54