The non-abelian Hodge theory originates in the groundbreaking works of A. Grothendieck, P. Deligne and A. Beilinson. A major development in the field was made by C. Simpson. Together with his students and collaborators, Simpson developed the theory of geometric n-stacks in a form which is particularly well adapted to understanding non-abelian cohomology of complex algebraic varieties as well as their Hodge structures. An indication of the power of the theory is that even though the notion of a geometric n-stack was invented for the task of understanding non-abelian Hodge structures it immediately acquired a life of its own and keeps popping up in different areas of modern mathematics.

Part I (March 28-30): Higgs bundles and applications
Introduced by Hitchin in 1987, and with pioneering work by Corlette, Donaldson and Simpson, Higgs bundles and their moduli spaces have remarkably rich geometric properties, and unexpectedly far-reaching applications. The moduli spaces are simultaneously symplectic quotients, algebraic quotients and gauge theoretic moduli spaces. They have a natural hyperkahler structure, they form an algebraic completely integrable system (the Hitchin system), they have an interpretation as representation varieties for fundamental groups, and are central in the development of non-abelian Hodge theory. Stable Higgs bundles play a key role in the theory of variations of Hodge structures, while natural gauge theoretic equations on Higgs bundles are related to harmonic maps into homogeneous spaces. Constructions based on Higgs bundles have found application in topological quantum field theory.

This workshop will be the first ever to focus explicitly on the theory and applications of Higgs bundles. The participants will include experts on Higgs bundles as well as leading specialists in areas where Higgs bundles have found useful application. The workshop will be a timely opportunity to review the gains of the last 15 years and to assess future needs and opportunities.

Part II (April 1 -5): Theory of high categories and applications
Among the geometric applications of the stack part of the non-abelian Hodge theory one may mention: Simpson's fundamental restriction of the class of lattices that can be fundamental groups of smooth projective manifolds; Simpson's version of secondary Kodaira-Spencer classes taking values in non-abelian cohomology; the non-abelian analogues of Griffiths (p,p)-cycle theorem and Open Orbit theorem recently proven by L. Katzarkov, T. Pantev and C. Simpson; and the notion and theory of non-abelian MHS recently introduced by L. Katzarkov, T. Pantev and C. Simpson. All these indicate that until non-abelian Hodge theory has reached is full maturity it will provide us with new restrictions on the homotopy types and the monodromy of the projective varieties. These topics as well as new application for theory of n categories to homotopy theory will be discussed.

Group photo of participants

Funding

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