Abstract: “I will give an overview and some of the highlights of a research program initiated in 2013 with Gadde and Putrov, the main subject of which is the study of 2d-3d combined systems that involve 3d N=2 theories with 2d N=(0,2) boundary conditions. We will see how holomorphic-topological twists of such systems lead to boundary chiral algebras, somewhat similar to Beem-Rastelli chiral algebras of 4d N=2 theories. In particular, just like the characters of Beem-Rastelli chiral algebras are computed by the so-called Schur index of 4d N=2 theories, characters of our boundary chiral algebras are computed by the “half-index” of 2d-3d combined systems, also introduced in the above-mentioned work with Gadde and Putrov. It was originally motivated by applications to topology, which are the subject of the math colloquium on Thursday.”