Topology is the branch of mathematics that deals with questions of shape and space without measuring. It studies properties of shapes that are preserved under continuous deformations, such as stretching, twisting, or bending. Modern topology takes this to the extreme and studies shapes that are often infinite-dimensional and very hard to visualize!
Nevertheless, topologists studying the sub-field called equivariant stable homotopy theory have distilled some questions into the study of transfer systems. Transfer systems are concrete combinatorial objects that are easy to draw! Things that we learn about transfer systems often have important consequences for those infinite-dimensional shapes in equivariant stable homotopy theory.
Our math lab group has been studying duality for a certain class of transfer systems called flat transfer systems. Just knowing how many transfer systems there are can be a tough question. But if we can understand how transfer systems pair up with this duality, we can reduce the number of transfer systems that we need to study by half in certain favorable scenarios. This can be a huge win for making our computer programs faster!
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