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Rough schedule

A possible schedule is as follows; reality will certainly diverge from this.

Week 1. Definition of a lattice and its determinant. History and origins of the geometry of numbers. The dual lattice. Bases, quadratic forms, the Gaussian volume heuristic. Sublattices and projections. Primitive sublattices. The shape of a lattice. Minkowski minima.
Week 2. Orthogonalisation, existence of a reduced basis.
Week 3. How to use reduced bases. Minima with respect to a norm. Minkowski’s theorems. Lenstra–Lenstra–Lovász algorithm. Historical questions of interest in the geometry of numbers. More recent questions about lattice reduction.
Week 4. Counting points of bounded norm in a lattice: asymptotic and upper bounds. Orthogonal sublattices. Covolume of orthogonal lattices.
Week 5. Manin’s conjecture. The equation $x_1y_1+\dots+x_ny_n=0$. Basic idea of Thunder’s proof. Assignment check-in deadline and lecture.
Week 6. Manin’s conjecture for $x_1y_1+\dots+x_ny_n=0$: completion of the proof. Review of course.
Week 7. Primes with missing digits: why geometry of numbers is relevant. Vaughan’s identity. The circle method. Statement of Maynard’s Proposition 9.3, Lemma 13.1, Proposition 13.3 and Proposition 13.4.
Week 8. Selected proofs from Maynard.
Week 9. Assignment final deadline.