Professor Shende works on mathematics inspired by fundamental questions about quantum phenomena in the physical world. He is presently studying topological and geometric structures arising from string theory, especially in the context of symplectic geometry.
Symplectic geometry orginated from classical mechanics: the possible configurations of, say the planets in the solar system, form the points of a symplectic manifold, and the formalism of symplectic geometry is set up to translate a single function – the potential energy – into the vector field describing how the planets move.
New inputs from physics – most recently, string theory – continue to drive the subject. Now the orbits are the stationary points where a string might rest, and we consider trajectories traveling from one orbit to another. Surfaces traced out by the strings (termed ‘Floer trajectories’ or ‘pseudoholomorphic curves’) are the solutions of a certain nonlinear partial differential equation. They are very hard to find explicitly. One deforms the geometry to eliminate some possible solutions and make others apparent. These solutions organize themselves into algebraic structures—Floer homologies, Fukaya categories. Many open conjectures connect these structures with a priori unrelated areas of mathematics, which makes Fukaya categories particularly interesting.
One flagship conjecture is Kontsevich’s ‘homological mirror symmetry’, which states that the Fukaya category is equivalent to a more algebraic category, namely the derived category of coherent sheaves in algebraic geometry. Shende has made significant contributions to our understanding of this huge challenge and he has proved it in a number of cases. In short, computing Fukaya categories is a fundamental and central problem in modern mathematics.