UBC Math Department Colloquium: Ingmar Saberi

Pure mathematics and theoretical physics have a long history of mutual influence and cross-pollination. In recent times, many exciting developments have to do with the effort to better understand quantum field theory. From the physicist's perspective, quantum field theory is a powerful framework for understanding many-body systems whose degrees of freedom are naturally parameterized by a geometric space, and which are equivariant with respect to the symmetries of that space; its applications range from elementary particles to condensed-matter theory and beyond. From the perspective of a mathematician, such theories provide powerful sets of organizing principles for topological or geometric invariants and give rise to new and unexpected conjectures about those invariants; modern approaches, using ideas from derived geometry, extend these principles to fully incorporate spacetime locality. After a bit of overview, I will discuss some recent results that construct new symmetry algebras related to higher-dimensional field theories. These provide new (derived) generalizations of loop algebras, such as the Kac-Moody and Virasoro algebras, and give rise to a surprising relation between certain variants of conformal structures and complex structures in dimensions higher than two.