IGAP/IFPU Seminar Room (Room 205, new SISSA Building)
Venue: For in-person attendeesIGAP/IFPU Lecture Room (New SISSA Building, room 205)
For virtual attendees: Zoom
Please use the links below (one per lecture) to register.
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This short series of four lectures by T. Hausel (IST, Austria) and A. Mellit (Univ. of Vienna) will discuss the recent proof of the P=W conjecture in the theory of the moduli space of Higgs bundles on a curve.
The following seminars will be given on Monday 5 December and Tuesday 6 December in the IGAP/IFPU Seminar Room (room number 205, new SISSA Building), but interested people can also attend via zoom (see links below - one link per lecture!):
Tamas Hausel Introduction to P=W and mirror symmetry
Motivated by topological mirror symmetry for Langlands dual Higgs bundle moduli spaces, one can study the arithmetic of the corresponding character varieties to gain information on the weight filtration on their cohomology. In turn this lead to the P=W conjecture which identifies the weight filtration on the cohomology of the character variety with the perverse filtration of the Hitchin system on the cohomology of the Higgs moduli space.
Mirror symmetry and big algebras
First we discuss the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we explain a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.
P=W via H_2
By H_2 we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of H_2 on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P
and W coincide with the filtration canonically associated to the sl_2 subalgebra of H_2.