# Vignettes in Physical Mathematics Natalie Paquette

Fundamental physics generates new mathematics. String theory and quantum field theory leverage exact and infrared dualities to provide equivalent descriptions of physical objects, some of which admit rigorous definitions; dual pairs of objects are sometimes even formalized using two a priori distinct areas of mathematics. Conversely, deep ideas in modern mathematics have been brought to bear on physical questions, yielding beautiful and unexpected results. `Physical mathematics' is the moniker sometimes given to the field which generates this cross-departmental exchange of techniques and results. I will describe new advances emerging from physical mathematics on two complementary fronts: 1.) new symmetries of supersymmetric string vacua motivated by the mathematical subject of moonshine and 2.) the action of duality on boundary conditions in (2+1)d quantum field theories, and some of its applications to geometry.