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MCSK Report Summary

Project ID: 320368
Funded under: FP7-IDEAS-ERC
Country: Switzerland

Final Report Summary - MCSK (Moduli of curves, sheaves, and K3 surfaces)

Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the 20th century, several fundamental links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The ERC project MCSK studied the moduli spaces of curves, sheaves, and K3 surfaces. While these moduli problems have independent roots, striking new results and relationships between them were found: the Givental-Teleman structure of semisimple CohFTs was used to constrain the algebra of tautological classes on the moduli spaces of curves, the GW/PT correspondence for 3-folds was used to prove the Kata-Klemm-Vafa conjecture for K3 surfaces in all cases, the Virasoro constraints for 3-fold stable pairs in the stationary toric case was proven via a descendent GW/PT correspondence and stable map results, the virtual class of the moduli space of stable maps was used to prove the generation of the tautological rings of the moduli spaces of K3 surfaces via Noether-Lefschetz cycles. The approaches taken used a mix of new geometries and new techniques. The results and the new directions opened are fundamental to the understanding of moduli spaces in mathematics and their interplay with topology, string theory, and classical algebraic geometry.

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