# Applications of additive combinatorics to Diophantine equations

 Joint IAS/Princeton University Number Theory Seminar Topic: Applications of additive combinatorics to Diophantine equations Speaker: Alexei Skorobogatov Affiliation: Imperial College London Date: Thursday, April 10 Time/Room: 4:30pm - 5:30pm/S-101 Video Link: http://video.ias.edu/jointiasnts/2014/0410-AlexeiSkorobogatov

The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the Brauer-Manin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over $\mathbb Q$ when the degenerate fibres are all defined over $\mathbb Q$. In this setting the result of Green, Tao and Ziegler replaces Schinzel's Hypothesis (H) used in earlier papers of Colliot-Thélène and Sansuc. I will give an overview of recent work in this direction due to L. Matthiesen, T. Browning, Y. Harpaz, O. Wittenberg and the speaker.