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The geometry of numbers in its canonical form comes from work of Minkowski in the late 19th century. The general idea is to give estimates for the cardinality $N(B,M) = |B\cap M\mathbb Z^n|$ when $M$ varies over some set of real matrices, $B$ is a fixed domain containing the origin, and we want our bounds to have explicit dependence on both quantities. For example, if $M \in \operatorname{SL}_n$, we may ask what conditions on $B$ always guarantee a nonzero point in $B\cap M\mathbb Z^n$. Alternatively, if $M = mI_n$ is a dilation, we may ask for simple-looking upper and lower bounds with the same order of magniture for $N(B,M)$. To answer the latter question, we need a 'reduced basis' for $\mathbb Z^n$.

Classical applications of geometry of numbers include Minkowski's results on discriminants and class numbers of number fields, the study of lattice packings, and a simple proof of the four-squares theorem. Historically, there was a lot of interest in the case $B = \{x:\prod x_i \leq 1\}$, in applying the 'reduced basis' idea to study 'reduction of quadratic forms', and in getting very good constants in bounds on discriminants of number fields, culminating perhaps in the work of Rogers and Mulholland.

More recent developments in the field include the LLL algorithm and algorithmic geometry of numbers, counting number fields with bounded discriminant and specified Galois group, and the reduction theory of quadratic forms as a key component of Bhargava's work ('Bhargavology'). The 'parametric geometry of numbers' goes beyond the cases when $M$ varies over dilations, to a two-parameter family, which in the 2D case boils down to continued fractions, providing a way to extend the concept of continued fractions to vectors. Some reading on these ideas is available, but they aren't the focus of this module.

Instead we will focus on another direction, anticipated by Davenport: the geometry of numbers is a powerful tool for estimating the number of solutions to systems of linear and multilinear Diophantine equations and inequalities. This leads to many applications in modern number theory, where such quantities may control error terms even when they are not the main quantity of interest.

Topics include Minkowski minima, reduced bases, LLL algorithm, orthogonal and dual lattices, and point-counting results. The course will cover the proof of Manin’s conjecture for $x_1y_1+\dots+x_ny_n=0$, which is a special case of a result of J L Thunder, and then examine a more advanced application of geometry of numbers, possibly in J Maynard's work on primes with missing digits or small fractional parts of polynomials.