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Opinion: Mathematicians are increasingly turning to topology, the study of shapes and their holes, to mathematically prove whether defensive networks can completely block drone swarms or if hidden gaps still let intruders slip through, explains Adjunct Professor Sidney A Morris.
In the race to protect cities, convoys and military bases from drone swarms, hidden geometric gaps can mean the difference between safety and failure.
Mathematicians call the study of such gaps topology, the branch of mathematics that classifies shapes like coffee cups and donuts by their holes.
Surprisingly, these ideas can do more than describe shapes, they can prove whether a swarm of drones can slip through a defensive shield.
The problem
Most anti-drone systems rely on interceptors, sensors or electronic jammers arranged around a protected area.
Even when coverage appears solid on a map, small unseen “corridors” may remain. Computer simulations can miss these weak points, but topology can certify whether such paths exist – or are mathematically impossible.
The idea
Topology studies how spaces connect and whether they contain “holes”.
If defenders’ coverage zones overlap so that together they form a single connected surface with no holes, then no continuous path exists for an intruder to reach the target.
This is not a probabilistic estimate – it’s a mathematical guarantee! Evasion becomes impossible unless coverage breaks and a hole opens.
A simple example
Imagine covering a playground with umbrellas. If the umbrellas overlap enough, every patch of ground is shaded – no sunlight escapes.
Replace sunlight with a drone’s flight path, and you have the essence of a topological defence network.
Each overlap closes a potential gap in the protective shield.
Practical uses
Item base and convoy defence: topology can determine the minimal number of interceptors, sensors or radars needed for guaranteed protection.
Urban security: by modelling buildings as obstacles, the same mathematics reveals where drone swarms could hide or slip through.
Network resilience: the approach also applies to communication networks, detecting “holes” where signal coverage or data links fail.
The future
As autonomous defence systems grow more complex, commanders will need not only faster algorithms but also proofs – rigorous mathematical guarantees that no swarm can succeed under given conditions.
Topology provides exactly that: a way to certify coverage completeness before the first drone flies.
This could make defence planning more reliable and transparent, replacing guesswork with geometry.
Conclusion
Defending against drones is often seen as an arms race of speed and firepower. But sometimes, the strongest defence lies not in bigger weapons but in smarter mathematics.
By understanding the shapes and holes in the skies we guard, we can ensure that no threat finds a way through.
Sidney A Morris is adjunct professor at La Trobe University and emeritus professor of the Federation University, Australia.
