Nothing as applicable as an abstract theory. Topology might seem abstract and foundational (like number theory) but has a wide range of applications in life. The combination of topology and data science, in particular, is called topological data analysis (TDA) and has recently been a hot new field. The reason why topology is so useful in general is because it deals with the local vs. global aspects of data. What are the global features of data preserved under deformations? In the context of big-data one asks what are the global characteristics discarding noise. Below you can find a short overview of how topology can help tackle diverse real-world problems.

Knots and quantum computing

One of the more promising implementations of quantum computing is based on knots and braids. The simple reason is that knots are (almost by definition) invariant under deformations and as such not influenced by noise. Since error correction and instability is a main issue in quantum computing the topological implementation comes as a magical solution. You can read the full story on this site, see Topological Quantum Computing notes.

Optimization and duality

There is a topological essence to all optimization problems. For example, Von Neumann’s original minimax theorem was proved using the Brouwer fixed point theorem; Nash’s subsequent improvement used an improved fixed point theorem. How much of modern optimization theory can be reduced to or extended via algebraic topology? Perhaps a great deal. The classical max-flow-min-cut theoremhas a sheaf-theoretic version that connects LP duality with Poincare duality, the implications of which are still being explored. The max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in the minimum cut, i.e. the smallest total weight of the edges which if removed would disconnect the source from the sink. Image segmentation can be stated as a max-flow problem.

Topological signal processing 

Much of the work in radar signal processing depends sensitively on geometry; what can you do with data that is too coarse or noisy to retain geometry well? What if the only available information is topological in nature? Tools for topological signal processing are being developed, including Euler integration (a topological alternative to Riemann integrals), nerve complexesfor signals, and modal Lyusternik-Schnirelman categories. All of these tools are relevant to understanding and reconstructing data from topological signals.

For an intro see for example Topological signal processing by Michael Robinson.

Geometric and topological robotics

Robotics is an ideal domain for a mathematician to work in: here, one has a genuine need for rigor. Imagine trying to verify that a control system for a robotic brain surgeon works. Would you prefer to have a successful computer simulation or a theorem guaranteeing performance? Using methods and ideas from topology and geometric group theory yields rigorous results about robot motion-planning and control. Examples include spaces of nonpositive curvature as applied to metamorphic and reconfigurable robots, and also to robot coordination problems. CAT(0) geometry answers questions about pursuit-evasion algorithms and optimal multi-agent planning. These ideas also have strong overlap with work in self-assembly.

Topological target tracking 

Sheaves and sheaf cohomology are extremely useful in solving local-to-global problems in many contexts. One of the more interesting uses is in target-tracking, where sheaves of semigroups over the time-axis can encode the directionality constraints in pursuit-evasion games. Better still is the use of sheaves over more general posets to fuse different types of sensor data, converting detection-based tracking into a coverage problem for co/homology.

Algebraic topology and sensor networks

As technology for sensors progresses, we will be able to replace large, expensive sensors with swarms of small, cheap, local sensors. One problem facing the sensors community is how to integrate local data into a global picture on an environment and how to manage the information overload. Imagine, for example, that you have thousands upon thousands of mobile video cameras and one of them catches something important. How should the system self-organize to trap the event? And, to make it interesting, let’s assume that you do not have GPS, range finders, orientation sensors, or a compass.

Fortunately, topologists solved a similar problem of going from local combinatorial data to a global picture. Homology & cohomology are surprisingly effective at answering questions about coverage and other problems in sensor networks. Recent advances in computational homology and persistent homology make these classical theories newly relevant to a wide variety of problems in security and communication. Sheaf theory is surprisingly useful in data aggregation problems over networks: a simple sheaf-theoretic integral using the Euler characteristic as a measure is very effective in problems of target enumeration over networks, and problems of information flow capacities reduce to sheaf cohomology.

Topological data analysis (TDA)

The need for efficient (in time and memory) computation of co/homology is ever-pressing, given the recent revolution in topological data analysis. There are particular challenges in computing persistent co/homology and sheaf cohomology, with the need to extract not only barcodes, but explicit generators. Work-in-progress uses a combination of discrete Morse theory, efficient numerical linear algebra, and matroid theory, with anticipated applications to neuroscience data sets.

See also What is persistent homology?

Contact topology and fluid dynamics

Rigorous results about fluid dynamics are rare for fully three-dimensional flows. I use global techniques from contact topology (an odd-dimensional variant of symplectic topology) to prove results about the most difficult classes of steady inviscid fluid flows. These techniques, e.g., contact homology, can be used answer questions about concrete physical phenomena such as hydrodynamic instability.