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Thanks to Sam Chow and Julia Brandes for suggestions.

Quick Introductions/Key Ideas
  • Browning, T., Le Boudec, P., and Sawin, W. (2023) 'The Hasse Principle for Random Fano Hypersurfaces', Ann. of Math., 197(3), pp. 1115-1203. (See section 3.1 up to the end of Lemma 3.3, and also section 3.2)
  • Heath-Brown, D.R. (2002) 'The Density of Rational Points on Curves and Surfaces', Annals of Mathematics, Second Series, 155(2), pp. 553-598. (See section 2, up to the end of the proof of Lemma 1)

I have only ever used the last three, so I can't vouch for the first ones!

Works we might discuss
  • Banaszczyk, W. (1993) 'New bounds in some transference theorems in the geometry of numbers', Mathematische Annalen, 296, pp. 625-635.
  • Barroero, F., and Widmer, M. (2018) 'Counting Lattice Points and O-Minimal Structures', Journal of the European Mathematical Society, 20(4), pp. 925-965.
  • Chen, C., Kerr, B., Maynard, J., and Shparlinski, I.E. (2021) 'Simultaneous Small Fractional Parts of Polynomials', Geom. Funct. Anal., 31, pp. 150-179.
  • Dartyge, C., and Maynard, J. (2023) 'On the Largest Prime Factor of Quartic Polynomial Values: The Cyclic and Dihedral Cases', Journal of the European Mathematical Society.
  • Davenport, H. (1951) 'On a Principle of Lipschitz', Journal of the London Mathematical Society, s1-26(3), pp. 179-183.
  • Davenport, H. (1963) 'Cubic forms in sixteen variables', Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 272(1350), pp.285-303.
  • Heath-Brown, D.R. (1984) 'Diophantine approximation with square-free numbers', Mathematische Zeitschrift, 187, pp.335-344.
  • Katznelson, Y.R. (1993) 'Singular Matrices and a Uniform Bound for Congruence Groups of SLn(Z)', Duke Mathematical Journal, 69(1), pp. 159-179.
  • Katznelson, Y.R. (1994) 'Integral Matrices of Fixed Rank', Proceedings of the American Mathematical Society, 120(3), pp. 721-728.
  • Kerr, B., Mohammadi, A., and Shparlinski, I.E. (2023) 'Additive Energy of Polynomial Images', arXiv.
  • Maynard, J. (2019) 'Primes in Arithmetic Progressions to Large Moduli', presented at the Second Symposium in Analytic Number Theory, Cetraro, July 2019.
  • Maynard, J. (2019) 'Primes with restricted digits', Inventiones mathematicae, 217, pp.127-218.
  • Maynard, J. (2020) 'Primes Represented by Incomplete Norm Forms', Forum of Mathematics, Pi, 8, e3.
  • Maynard, J. (2022) 'Primes in Arithmetic Progressions to Large Moduli III: Uniform Residue Classes', Memoirs of the American Mathematical Society.
  • Schmidt, W.M. (1966) 'Asymptotic Formulae for Point Lattices of Bounded Determinant and Subspaces of Bounded Height', Acta Mathematica, 117, pp. 169-228.
  • Soundararajan, K. (2022) 'The Work of James Maynard', Proceedings of the International Congress of Mathematicians 2022, Vol. 1, pp. 66-80.
  • Thomas, K. (2024) 'Bounding the Number of Linear Solutions to Diophantine Equations in an Application to PDEs', URSS Showcase.
  • Thunder, J.L. (1993) 'Asymptotic Estimates for Rational Points of Bounded Height on Flag Varieties', Compositio Mathematica, 88(2), pp. 155-186.
Some other perspectives
  • The classical perspective
    • Davenport, H. (1947) 'The Geometry of Numbers', The Mathematical Gazette, 31(296), pp. 206-210.
    • Mulholland, H.P. (1960) 'On the Product of n Complex Homogeneous Linear Forms', Journal of the London Mathematical Society, 35, pp. 241-250.
    • Rogers, C.A. (1950) 'The Product of n Real Homogeneous Linear Forms', Acta Mathematica, 82(1), pp. 185-208.
  • Quantitative results about number fields
  • From the work of Bhargava
    • Bhargava, M. (2005) 'The Density of Discriminants of Quartic Rings and Fields', Annals of Mathematics, 162, pp. 1031-1063.
    • Bhargava, M. (2010) 'The Density of Discriminants of Quintic Rings and Fields', Annals of Mathematics, 172, pp. 1559-1591.
  • The parametric geometry of numbers
  • Miscellaneous