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Explore a lecture on the $P=W$ conjecture in algebraic geometry, focusing on its connection to the Lie algebra $H_2$ of polynomial Hamiltonian vector fields on the plane. Delve into the moduli space of stable twisted Higgs bundles on algebraic curves, examining two natural filtrations on its cohomology: the weight filtration W from the Betti realization and the perverse filtration P induced by the Hitchin map. Discover how computations in Khovanov-Rozansky homology of links motivate the search for an $H_2$ action on the cohomology of the moduli space. Investigate the algebra generated by tautological classes and Hecke operators, and learn how both P and W coincide with the filtration associated with an $sl_2$ subalgebra of $H_2$. Gain insights into this joint work with Hausel, Minets, and Schiffmann, building upon the proof by De Cataldo, Hausel, and Migliorini for rank 2 cases and exploring the conjecture for arbitrary ranks.
