Organizers: Arkadij Bojko

Let $C$ be a compact Riemann surface, and $\Gamma=\pi_1(C)$ its fundamental group. While enumerating the representations of $\Gamma$ over finite fields, Hausel and Rodrigues-Villegas have noticed that they always obtained palindromic polynomials. One way to interpret this symmetry is to say that the cohomology of the variety $\mathcal{M}_B$ of representations of $\Gamma$ over complex numbers (so called character variety) admits a "curious" Poincaré duality. Similar dualities were previously studied by de Cataldo and Migliorini, but their theory could only be applied to the homeomorphic variety $\mathcal{M}_D$ of Higgs bundles on $C$. Matching up these two dualities (the "curious" one being conjectural) boils down to an equality of two filtrations (perverse and weight) on $H^*(\mathcal{M}_B)$ of very different natures. This equality became known as $P=W$ conjecture. In my talk, I will explain this story in more detail, and give some indication of how this conjecture was eventually proved. Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.

The main technical tool in our proof of $P=W$ conjecture was the construction of an action of a certain algebra, which we called deformed $W_{1+\infty}$-algebra, on the cohomology of the moduli space of stable Higgs bundles on the curve $C$. It turns out that this algebra is nothing else than the cohomological Hall algebra of finite length sheaves on $T^*C$ (see my talk at Academia Sinica in 2019). This isomorphism can be extended to any (cohomologically pure) surface $S$, which opens the way for algebraic study of cohomology of various moduli spaces of sheaves. I will explain the proof of this isomorphism, and speculate about applications to $P=C$ phenomena for $K3$ surfaces and computation of ring structure of $H^*(Hilb S)$ and its cousins. Based on joint work with A. Mellit, O. Schiffmann, E. Vasserot.

We propose a theory of enumerative invariants for structure groups of type B/C/D, that is, for the orthogonal and symplectic groups. For example, we count orthogonal or symplectic principal bundles on projective varieties, or a quiver analogue called self-dual quiver representations. We also discuss algebraic structures arising from the relevant moduli spaces. In type A, Joyce constructed a somewhat mysterious vertex algebra structure on the homology of moduli spaces, and in type B/C/D, we construct a twisted module for this vertex algebra. We then use these algebraic structures to write down wall-crossing formulae for our invariants.

Following our reformulation (joint with Lim-Moreira) of sheaf-theoretic Virasoro constraints with applications to curves and surfaces, I will talk about the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I will sketch their proof for any finite quiver with relations, with frozen vertices, but without cycles. I will use partial flag varieties as a guiding example throughout, following our reformulation (joint with Lim-Moreira) of sheaf-theoretic Virasoro constraints with applications to curves and surfaces, I will talk about the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I will sketch their proof for any finite quiver with relations, with frozen vertices, but without cycles. I will use partial flag varieties as a guiding example throughout but the most exciting upshot is an independent, self-contained proof of Virasoro constraints for Mumford (semi)stable torsion-free sheaves on the complex projective plane but the most exciting upshot is an independent, self-contained proof of Virasoro constraints for Mumford (semi)stable torsion-free sheaves on the complex projective plane.

The Atiyah-Bott localization theorem says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. We discuss a categorified version of this result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology). This is based on a joint work with Adeel Khan.

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The canonical 4-term sequence on moduli spaces of quiver representations is an important tool to study their geometry. First I will use it to describe the Chow ring in more detail. Next, I will use it to prove the rigidity of quiver moduli, and describe their infinitesimal symmetries. Finally, I will use it to establish Schofield's partial tilting conjecture, and obtain a fully faithful Fourier-Mukai functor. The stacky perspective of these moduli spaces, and Teleman quantization, plays an important role. This surveys several joint works with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke.

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