## Reading seminar on the circle method, Fall 2021For three introductory lectures on the circle method for Waring's problem see also Kevin Hughes' online course starting on October 11, 2021. ## ProgramWaring's problem - Oct. 4 at 14:00 (Boaz)
Introduction to Waring's problem and the circle method ([N, §I.5.1], [D, Foreword, §§1-2], [V, §1]). Decomposition into major and minor arcs ([N, §I.5.3], [D, §4]). The minor arcs ([N, §I.5.4], [D, Lemma 4.1]). Weyl's inequality ([N, Theorem I.4.3], [D, Lemma 3.1], [V, Lemma 2.4]). - Oct. 11 at 14:00 (Rosa)
Dirichlet's theorem ([N, Theorem I.4.1], [V, Lemma 2.1]). Hua's Lemma ([N, Theorem I.4.6], [D, Lemma 3.2], [V, Lemma 2.5]). Auxiliary functions for the major arcs ([N, §I.5.5], [D, §4 till Lemma 4.3]). - Oct. 18 at 14:00 (Miriam)
The singular integral ([N, §I.5.6], [D, Theorem 4.1]). The singular series ([N, §I.5.7], [D, §§5-6]). - Oct. 25 at 14:00 (Francesca)
Waring's problem vs solution of equations ([D, §7-10]).
Homogeneous equations and Birch's theorem - Nov. 1 at 14:00 (Marta)
Introduction to the problem for homogeneous forms ([D, §§11, 19 (only overview)], [B, §1]). Introduction to the problem for cubic forms ([D, §§13, 14]). - Nov. 8 at 14:00 (Judith)
Cubic forms via the circle method ([D, §§15-17]). - Nov. 15 at 14:00 (Miriam)
p-adic questions for cubic forms ([D, §18]) p-adic questions for Birch's theorem ([B, §7]) - Nov. 22 at 14:00 (Julia)
Birch's theorem ([D, §19], [B, §§2-6], [V, §9], [Browning, §8 till §8.2.3])
Heath-Brown's delta method - Nov. 29 at 14:00 (Rosa)
Introduction, weight functions, and theorems 1 and 2 ([HB, §§1-3]). - Dec. 6 at 14:00 (Judith)
Description of the singular integral: Theorem 3 ([HB, §§4-6]). - Dec. 13 at 14:00 (Shuntaro)
Singular integral estimates and the singular series ([HB, §§7-9]). - Dec. 20 at 14:00 (Francesca)
Applications to quadratic forms ([HB, §§10-14]).
Possible expansions - Van Valckenborgh,
*Squareful numbers in hyperplanes*, arXiv link - Browning-Yamagishi,
*Arithmetic of higher-dimensional orbifolds and a mixed Waring problem*, arXiv link - Shute,
*Sums of four squareful numbers*, arXiv link
## References [B] B. J. Birch, [Browning] T. D. Browning, [D] H. Davenport, [HB] D. R. Heath-Brown, [N] M. B. Nathanson, [V] R. C. Vaughan, |