Description
Mathematics Colloquium.
Speaker: Swarnendu Sil.
Affiliation: ETH Zurich.
Date and Time: Wednesday 11 March, 04:00 pm - 05:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Approximation questions in higher dimensional gauge theory.
Abstract: Although Yang-Mills functional and gauge theory originated in physics, its Riemannian or elliptic version has already influenced a vast territory of mathematics as a whole, ranging from real and complex differential geometry and differential topology of vector and principal bundles over manifolds to symplectic geometry and topology and even complex algebraic geometry. Nearly all of these spectacular developments followed our improved understanding of gauge theory over four manifolds. However, very little is known about gauge theory on manifolds of dimension five and higher. In this talk, we shall visit some of the major open questions in higher dimensional gauge theory and their connections to other branches of mathematics. We shall see why in this setting, we are forced to work with certain ‘weak’ notions of principal bundles and connections. Finally, we shall discuss how approximation questions, i.e. understanding when these weak classes of bundles and connections can be approximated by smooth bundles and connections, enters the fray and present a few recent results in this direction.