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# A gentle introduction to non-abelian Hodge and P=W conjecture

## June 12 at 16:00 – 18:00 CEST

Oberseminar Algebra und Geometrie

Alexandre Minets (MPI Bonn)

Abstract:

Title: A gentle introduction to non-abelian Hodge and P=W conjecture

Abstract: Let C be a Riemann surface, and M_B the character variety of C; in other words, the points of M_B parameterise representations of the fundamental group of C. Up to a _non-algebraic_ homeomorphism, M_B can also be realized as the variety M_H of algebraic vector bundles on C equipped with a twisted endomorphism (otherwise known as Higgs bundles). The P=W conjecture of de Cataldo-Hausel-Migliorini postulates that some algebraic data survives the homeomorphisms above; more precisely, the number of F_q-points of M_B is controlled by an algebraic integrable system on M_H. I will give a gentle introduction to these objects, focusing on one explicit example. If time permits, I will sketch some ideas that went into our recent proof of P=W (joint with T. Hausel, A. Mellit, O. Schiffmann).

Abstract: Let C be a Riemann surface, and M_B the character variety of C; in other words, the points of M_B parameterise representations of the fundamental group of C. Up to a _non-algebraic_ homeomorphism, M_B can also be realized as the variety M_H of algebraic vector bundles on C equipped with a twisted endomorphism (otherwise known as Higgs bundles). The P=W conjecture of de Cataldo-Hausel-Migliorini postulates that some algebraic data survives the homeomorphisms above; more precisely, the number of F_q-points of M_B is controlled by an algebraic integrable system on M_H. I will give a gentle introduction to these objects, focusing on one explicit example. If time permits, I will sketch some ideas that went into our recent proof of P=W (joint with T. Hausel, A. Mellit, O. Schiffmann).