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Rough syllabus

We introduce the geometry of numbers with a focus on applications in analytic number theory, including Manin’s conjecture and recent work in prime number theory. Topics covered include:

  • Definition of lattices and their properties, including the dual lattice, Minkowski minima, reduced basis, sublattices, and projections, with an emphasis on solving problems related to point counting and lattice reduction.
  • Explanation of the Lenstra-Lenstra-Lovász algorithm and its application to lattice reduction.
  • Application of asymptotic and upper bounds to count points of bounded norm in a lattice, and to count integral lattices with a given shape.
  • Explanation of Manin's conjecture and its significance in the study of Diophantine and polynomial equations. The spirit of Thunder's proof for bilinear varieties.
  • Application of the geometry of numbers to reduce problems in exponential sums over primes to geometric problems, and use of this approach to bound expressions related to prime numbers. In particular we will discuss J Maynard's work on primes with missing digits (time permitting).