Mathematical physics

Quantum group and its application to knot polynomial was the theme of Witten-Reshtikhin-Turaev theory. The categorfication of quantum group and application to knot homology has been discovered and developed by Khovanov-Lauda, Rouquier, and Webster, in the form of KLRW algebra. With Mina Aganagic and collaborators, we developed a new approach using Fukaya category and relate to KLRW algebra. The geometric origin allows us to give a geometric construction for the categorified coproduct. The talk is based on past work and work in progress with Mina Aganagic, Vivek Shende, Elise LePage, Ivan Danilenko and Yixuan Li.

Discs are among the simplest manifolds, but their groups of diffeomorphisms can be very complicated. I will describe the techniques from geometry, topology, and dynamics that were used to understand these groups in low dimensions, the relationship of these groups to stable homotopy theory and number theory in high dimensions, and recent breakthroughs in understanding their rational homotopy type. This talk will be aimed at a broader audience.

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture the cubic sector, they fail to encode higher-valence phenomena, for which a homotopy-theoretic extension becomes necessary. This work introduces a conceptual notion of homotopy coexact BV-algebra, defined through the homotopical interplay of commutative and BV structures, and provides a concrete model in terms of generators and relations, obtained through an extension the theory of Koszul duality for operads.