Scientific and Engineering Applications of Algebraic Topology

Topology was invented as a tool for achieving, inter alia:

1. Qualitative analysis: Set-theoretic topology identifies properties (e.g., compactness, connectedness) useful in extensions of analysis beyond finite-dimensional Euclidean space (e.g., manifolds, functional analysis, calculus of variations). Such properties are said to be topological if they are robust to continuous deformation. The power of these extensions has been demonstrated repeatedly over the last 150 years.
2. Geometric pattern recognition: Poincaré found it useful to formalize the notion of loops and holes in a space and their higher dimensional analogues as a way to codify the qualitative properties of spaces. Algebraic topology was subsequently constructed as a rigorous formalization. What was arrived at is a collection of generalizations of the notion of connectivity to higher connectivity information, which are encoded by algebraic objects.

The demands of modern science and engineering have placed us in a position where it is vital to develop methods for qualitative analysis and recognition problems in contemporary contexts, including data (finite metric spaces as samples from experiments, surveys, or sensors), networks (internet traffic, gene regulation, coordinated robotics, communications), and dynamics (systems equipped with only finite resolution or which are stochastic). Examples include:

1. Data of various kinds is being collected at an enormous rate, and in many different forms. Often the data is equipped with a notion of distance that reflects certain notions of similarity, but which may be far from Euclidean (think genomic sequence analysis). It is also frequently the case that the metrics are not defined by any precise theory, but are chosen in a relatively ad hoc way to reflect the investigator's intuitive notions of similarity. For this reason, it is important to make computations that are reasonably robust to changes in the metric, since one expects that the interesting scientific properties should not change if the metric is changed via deformations.
2. In the area of sensor networks, one studies families of sensors with relatively weak computational ability and wishes to study coverage questions. The sensors will likely not even have their own positions available, but rather only information about what the neighboring sensors are. It is very desirable to solve the coverage question, and therefore develop a methodology that solves them given only the adjacency information referred to above. The adjacency information produces an undirected graph, and it is from this information one must develop methods for resolving the coverage question.
3. Many problems in biology, from protein folding to gene regulatory networks, can be usefully formulated as qualitative questions about dynamical systems. The large systems in question are given in terms of various different kinds of metrics and are often best formulated with a stochastic component. Methods of understanding the qualitative features using such “fuzzy” inputs are vital to properly interfacing with biology.

Many efforts to address these problems have been under development over the last decade. There has been a great deal of work in various kinds of persistent homology (a methodology for inferring topological invariants of a geometric object from finite samples with error from the object), the homological properties of sensor networks and their implications for coverage and other questions, and the extension of algebraic topological tools for qualitative analysis of dynamical systems (Conley indices, for example) to tools in the finite approximation and stochastic settings. We believe that the importance of the problems addressed by these methods are of such fundamental importance that a program that will bring together the various groups (topologists, computational geometers, networks experts, statisticians, biologists, and other application domain specialists) who are critical to the further development and implementation of the methods is warranted. We believe that the subject is now at a point where such a gathering would allow decisive progress in a number of different directions.