FRG: Obstructions to Local-Global Principles and Applications to Algebraic Structures

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The interplay between number theory and algebraic geometry has been a source of inspiration in modern mathematics. Having led to the solution of a number of outstanding conjectures, such as Fermat's Last Theorem and the Mordell Conjecture, it continues to give rise to deep and important problems in algebra. Local-global principles are a central theme in this interplay of subjects, and many important outstanding problems can be expressed in terms of such principles. This project has the objective of understanding local-global principles and their obstructions, in contexts that are broader than those considered in number theory. The project will also support and enhance the training of graduate students and postdoctoral researchers through seminars, conferences and workshops, and mentoring activities.The Focused Research Group will focus on local-global principles for algebraic structures defined over function fields of curves over base fields such as p-adic fields, with a longer term goal of treating the case of function fields of curves over global fields. The obstructions to such local-global principles can often be formulated in terms of cohomology. Our project aims to study the finiteness of these obstructions and determine criteria for them to vanish. The resulting understanding will be applied to proving conjectures and solving open problems concerning algebraic structures such as quadratic forms and associative algebras. This will include situations that have been studied by many researchers but where solutions had previously seemed out of reach. Research methods will include field patching, cohomological methods including residues and duality, and approaches from geometry.