(Homological) Knot Invariants From Mirror Symmetry
- Speaker
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Mina Aganagic, Ph.D.University of California, Berkeley
Presidential Lectures are free public colloquia centered on four main themes: Biology, Physics, Mathematics and Computer Science, and Neuroscience and Autism Science. These curated, high-level scientific talks feature leading scientists and mathematicians and are intended to foster discourse and drive discovery among the broader NYC-area research community. We invite those interested in the topic to join us for this weekly lecture series.
Quantum invariants of knots (also known as Chern-Simons knot invariants) have many applications in mathematics and physics. For example, Khovanov showed in 1999 that the simplest such invariant, the Jones polynomial, arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: What are they homologies of?
Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to the fore. Typically, though, it leads to invariants that have no particular interest outside of the problem at hand.
In this lecture, Mina Aganagic will present how she recently showed there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants. In particular, they solve the knot categorification problem.
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About the Speaker
Aganagic is a professor of mathematics and physics at the University of California, Berkeley. She applies string theory to problems in pure mathematics, including knot theory, enumerative geometry, representation theory and geometric Langlands. She received a Department of Energy Outstanding Junior Investigator Award, a Sloan Fellowship, a Simons Investigator Award, and she was elected a fellow of the American Physical Society. She is an invited speaker at the 2022 ICM in Saint Petersburg.