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Explore the Grunwald problem for solvable groups in this mathematical lecture that examines what constitutes the closure of the image of H^1(K,G) → ∏_{v ∈ M_K} H^1(K_v,G) for a finite group scheme G over a number field K. Discover how the Brauer-Manin obstruction affects this problem and learn about the conjecture that it represents the only obstruction for general groups G. Examine the 2017 breakthrough technique by Harpaz and Wittenberg that successfully demonstrated the Brauer-Manin obstruction as the sole obstruction for supersolvable groups. Understand a new fibration theorem over quasi-trivial tori and see how combining this theorem with the Harpaz-Wittenberg approach extends the positive resolution to all solvable groups, representing a significant advancement in algebraic number theory and group cohomology.
