A summer school and conference on Hodge theory will be held at the University of British Columbia (Vancouver, Canada) from June 10-20, 2013. This award will support 30 US participants at various stages of their careers. The 24 invited speakers are leading experts in aspects of complex and arithmetic geometry, algebraic cycles, and representation theory, which are in the early stages of a synthesis around the study of period mappings and generalized period domains. The conference will accelerate this process and give graduate students and recent Ph.D.'s an opportunity to enter an emerging discipline.

In its simplest form, Hodge theory is the study of periods -- integrals of algebraic differential forms which arise in the study of complex geometry, number theory and physics. Its difficulty and richness arise in part from the non-algebraicity of these integrals. What algebraic structure they do have is recorded by symmetry groups called Mumford-Tate groups, and according to the Hodge conjecture (and its variants) should be explained by the presence of objects called algebraic cycles. The conference will serve to disseminate recent progress on both of these fronts, create new interdisciplinary collaborations, and train the next generation of Hodge theorists.

The web site for the conference is www.pims.math.ca/scientific-event/130610-rahtpdaca

Project Report

" took place in Vancouver, Canada, from June 10-20, 2013. This award covered travel and accomodation costs for US mathematicians attending the conference. Summary: The event began with a 4-day summer school for graduate students and postdocs consisting of lectures by well-known experts. Topics covered included Mumford-Tate domains, algebraic cycles, degenerations of Hodge structures and D-modules, arithmetic of periods, automorphic cohomology. The 7-day workshop that followed featured 4-5 talks each day, followed by discussion sessions. Each day was focused on one of the specific themes: Algebraic cycles and the Hodge conjecture; arithmetic aspects of cycles and period maps; period domains and their compactifications; Mumford-Tate group and representation theory; automorphic forms and automorphic cohomology; relative completion of the fundamental group. The overall concept was to highight links between various areas of algebra and geometry, with Hodge theory and period maps at the center. Intellectual Merit: Hodge theory is a vibrant subject at the center of contemporary complex geometry. In its simplest form, Hodge theory is the study of periods - integrals of algebraic differential forms which arise in many areas, including algebraic geometry, number theory, and physics. Its difficulty and richness arises in part from the non-algebraicity of these integrals. According to the beautiful conjectures of Hodge, Bloch and Beilinson, what algebraic structure they have should be explained by (generalized) algebraic cycles. There has been much recent progress on these conjectures and on classifying spaces for periods, as well as their asymptotics and arithmetic. This conference brought together over 90 leading scholars and students from 12 countries to survey the state of the subject. Broader Impact: This award allowed significantly more US mathematicians, including many young researchers, to attend this major summer school and workshop. This was important for the future development of Hodge theory and related subjects in the US. The conference itself sparked many interesting discussions and a few new collaborations. The forthcoming proceedings volume, which has a strong expository component, should draw an even larger audience into the area.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tie Luo
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Washington University
Saint Louis
United States
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