Mathematics of the topological open string

String theory has introduced into mathematics many new ideas, and perhaps even more remarkably, new relationships between old ideas. For instance, it turns out that we learn new things about knots in ordinary three-dimensional space by considering this space as a boundary condition for the surface that a string-theorist's string traces out in a six dimensional space; the knot being how the string arrives at the boundary. Another remarkable relationship concerns how this kind of geometry of strings and boundaries can be translated into the geometry of an entirely different kind - that concerned with polynomial equations - on an entirely different space, related to the original by a process in which, rather mysteriously, short distances are exchanged for long! These are the sort of phenomena Shende will pursue as a Villum Investigator.