Math 8440: Combinatorial Number Theory

More specifically I plan to discuss the following topics:

Waring's problem: sums of k-th powers.
Weyl's inequality and Hua's mean value theorem.
The singular integrals and the singular series.
Vinogradov's main conjecture and mean value theorem.
Efficient congruencing on the model case k=2.
Efficient congruencing for k=3 and the general case.
Non-diagonal cubic forms and Davenport's theorem.
Birch's theorem on general diophantine systems.

I. Waring's problem: the minor and major arcs

We discuss the asymtotics for the number of solutions to Waring's problem dute to Hardy-Littlewood and Hua.
Notes: H. Davenport: Diophantine equations and inequalities

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10

II. Vinogradov's mean value theorem and efficient congruencing

We discuss Vinogradov's improvement on Waring's problem through his mean value theorem. We also discuss the recent solution to Vingradov's cojecture through the so-called efficient congruencing method.
Notes: Heath-Brown: The cubic case of Vinogradov's Mean Value Theorem: A simplified approach to Wooley's ``efficient congruencing"

Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18

III. Systems of non-diagonal diophantine equations

We introduce a general form of the circle-method to treat systems of non-diagonal equations, due the Birch and Davenport.
Notes: H. Davenport: Diophantine equations and inequalities

Lecture 19 Lecture 20 Lecture 2 Lecture 22 Lecture 23 Lecture 24 Lecture 25

Analytic Number Theory: Diophantine Equations