Yangians and Cohomological Hall algebras of curves

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During the FY2022, I focused on the definition of double (2-dimensional) cohomological Hall algebras (COHAs) and categorified ones (CatHAs) in a general framework. Given a torsion pair (T,F) on the heart of a fixed t-structure of a dg category C (satisfying certain technical conditions), in [arXiv:2207.08926], Diaconescu, Porta, and I introduce a general formalism to associate a COHA (resp. CatHA) to T together with a left and right module (resp. categorified module) associated to F. The full Yangian (resp. categorified Yangian) of T is defined as the monoidal (resp. associative) algebra generated by these two module structures. This framework provides for example a categorification of the Ding-Iohara-Miki algebra of a smooth projective complex surface S introduced by Negut in [arXiv:1703.02027] and a generalization of it including operators of "Hecke modifications along curves"; it allows the definition of actions on the homology, K-theory, etc, of the moduli space of PT stable pairs on S.In [arXiv:2303.17154], Diaconescu, Porta, and I studied the geometry of the moduli space of PT stable pairs and Hilbert schemes of points on a space curve singularity, using motivic Hall algebras, providing an explicit formula of the generating function of their Euler numbers when the space curve singularity is locally complete intersection.Finally, Diaconescu, Porta, Schiffmann, Vasserot, and I are finalizing the project about a description of the COHA of the minimal resolution of a type ADE singularity in terms of the affine Yangian of the same type.