Hodge theory originated in the work of Hodge, Kodaira, and Weyl, who introduced analytic methods, particularly techniques from partial differential equations, to study the cohomology of compact Riemannian manifolds. Since then, Hodge theory has developed into a central and unifying framework in both complex and algebraic geometry. It plays a fundamental role in the study of algebraic cycles and moduli spaces, and it has deep connections with many other areas of mathematics, including model theory, symplectic geometry, singularity theory, D-modules and perverse sheaves, derived categories, representation theory, as well as arithmetic geometry and number theory. Over the past two decades, the field has witnessed remarkable progress, driven by the interaction of Hodge theory with tools and ideas from diverse areas.
Long-standing questions about period mappings, dating back to the 1970s, have been resolved by combining Hodge theory with techniques from model theory and o-minimal geometry. This has led to natural compactifications of images of period maps, with significant applications to moduli theory (notably for Calabi–Yau varieties) and birational geometry (including results related to the b-semiampleness conjecture).
Major progress has been made on the Hodge conjecture. In particular, the conjecture has been proved for all four-dimensional abelian varieties. At the same time, the integral Hodge conjecture for very general abelian varieties of dimension at least five has been disproved, using methods that reveal an unexpected connection with matroid theory.
By combining Hodge-theoretic techniques with ideas from Gromov–Witten theory and mirror symmetry, new birational invariants have been constructed. These have led to the resolution of the long-standing problem of the irrationality of very general cubic fourfolds.
The P = W conjecture in non-abelian Hodge theory has been proved, establishing a deep link between the topology of Hitchin systems and the mixed Hodge structure of character varieties. This breakthrough has already led to substantial new progress on questions involving Hodge theory and algebraic cycles on abelian fibrations.
The goal of the proposed program is to provide a comprehensive and coherent overview of these developments by bringing together leading experts, early-career researchers, and graduate students from the United States and abroad for a semester-long program at SLMath. Such a program is timely: despite the vitality of the field, there has been no recent large-scale event dedicated to Hodge theory as a whole. With many new techniques now in place and numerous open problems emerging at their intersections, the field is well positioned for further advances. We hope that this program will both catalyze new collaborations and inspire the next generation of mathematicians to contribute to these developments.
Hodge theory originated in the work of Hodge, Kodaira, and Weyl, who introduced analytic methods, particularly techniques from partial differential equations, to study the cohomology of compact Riemannian manifolds. Since then, Hodge theory has developed into a central and unifying framework in both complex and algebraic geometry. It plays a fundamental role in the study of algebraic cycles and moduli spaces, and it has deep connections with many other areas of mathematics, including model theory, symplectic geometry, singularity theory, D-modules and perverse sheaves, derived categories, representation theory, as well as arithmetic geometry and number theory. Over the past two decades, the field has witnessed remarkable progress, driven by the interaction of Hodge theory with tools and ideas from diverse areas.
- Long-standing questions about period mappings, dating back to the 1970s, have been resolved by combining Hodge theory with techniques from model theory and o-minimal geometry. This has led to natural compactifications of images of period maps, with significant applications to moduli theory (notably for Calabi–Yau varieties) and birational geometry (including results related to the b-semiampleness conjecture).
- Major progress has been made on the Hodge conjecture. In particular, the conjecture has been proved for all four-dimensional abelian varieties. At the same time, the integral Hodge conjecture for very general abelian varieties of dimension at least five has been disproved, using methods that reveal an unexpected connection with matroid theory.
- By combining Hodge-theoretic techniques with ideas from Gromov–Witten theory and mirror symmetry, new birational invariants have been constructed. These have led to the resolution of the long-standing problem of the irrationality of very general cubic fourfolds.
- The P = W conjecture in non-abelian Hodge theory has been proved, establishing a deep link between the topology of Hitchin systems and the mixed Hodge structure of character varieties. This breakthrough has already led to substantial new progress on questions involving Hodge theory and algebraic cycles on abelian fibrations.
The goal of the proposed program is to provide a comprehensive and coherent overview of these developments by bringing together leading experts, early-career researchers, and graduate students from the United States and abroad for a semester-long program at SLMath. Such a program is timely: despite the vitality of the field, there has been no recent large-scale event dedicated to Hodge theory as a whole. With many new techniques now in place and numerous open problems emerging at their intersections, the field is well positioned for further advances. We hope that this program will both catalyze new collaborations and inspire the next generation of mathematicians to contribute to these developments.
Show less
