Using Math to Understand the Universe
If you take a tangled shoelace and glue the ends together, you get a knot—but you also get the basis for a complex mathematical puzzle.
A fundamental question in topology (the study of the properties that are preserved by deforming, twisting and stretching an object) is whether any two knots are the same. By untangling one—without cutting it—can you turn it into the second?
This seemingly simple question can only be solved using highly sophisticated math. As Canada Research Chair in Mathematics, Dr. Sabin Cautis is exploring this problem and others like it to better understand fundamental questions in geometry and representation theory (the study of abstract algebraic structures). This enhanced comprehension of mathematical theories could lead to a more profound understanding of our universe.
Techniques from representation theory can construct certain “invariants” of knots, which can distinguish some, but not all, knots. These invariants have been studied by theoretical physicists in connection with string theory—a subject that hopes to provide a working model for the universe by combining quantum mechanics and general relativity.
Cautis and his research team are approaching the matter from a more philosophical perspective, using a new technique known as “categorification” to understand the deeper structures behind various theories. In mathematical terms, this means replacing vector spaces with more interesting and complex categories.
Through this research, Cautis hopes to develop a deeper understanding of mathematical theories and what they can tell us about our universe.