From physics perspective, Gukov(-Pei)-Putrov-Vafa have introduced a non-perturbative definition of complex Chern-Simons invariants of a 3-manifold, which is called BPS q-series. Although its mathematical definition is not given yet, it exhibit remarkable mathematical properties: (non-semisimple) TQFT structure, modularity, cyclotomic expansions. In this talk, I will survey the recent development of the BPS q-series.
I am interested in mathematical structure hidden behind dynamics of M5-branes. Although the world-volume theory on M5-branes remains mysterious, it generates huge classes of quantum field theories in various dimensions. I would like to understand intriguing connections of supersymmetric gauge theories, topological quantum field theories, mathematical physics and string theory via dynamics of M5-branes.