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. https://doi.org/10.4007/annals.2023.197.3.3. (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. https://www.jstor.org/stable/3062125. (See section 2, up to the end of the proof of Lemma 1)

Textbooks

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

• Olds, C.D., Lax, A., Davidoff, G.P., and Davidoff, G. (2001) 'The Geometry of Numbers', Anneli Lax New Mathematical Library 41, MAA Press.
• Gruber, P.M. (2007) 'Convex and Discrete Geometry', Grundlehren der mathematischen Wissenschaften, Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71133-9
• Gruber, P.M. and Wills, J.M. (eds.) (1993) 'Handbook of Convex Geometry, Part B', North Holland. https://doi.org/10.1016/C2009-0-15706-9.
• Siegel, C.L. (1989) 'Lectures on the Geometry of Numbers', ed. Chandrasekharan, K., Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-662-08287-4.
• Cassels, J.W.S. (1959) 'An Introduction to the Geometry of Numbers', Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-662-08287-4.
• Nguyen, P.Q., and Vallée, B. (eds.) (2010) 'The LLL Algorithm', Springer-Verlag Berlin Heidelberg. https://www.ionica.nl/wp-content/uploads/2019/04/the-LLL-Algorithms.pdf. (We will follow chapter 2 pretty closely)

Works we might discuss

• Banaszczyk, W. (1993) 'New bounds in some transference theorems in the geometry of numbers', Mathematische Annalen, 296, pp. 625-635. https://doi.org/10.1007/BF01445125.
• Barroero, F., and Widmer, M. (2018) 'Counting Lattice Points and O-Minimal Structures', Journal of the European Mathematical Society, 20(4), pp. 925-965. https://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=20&iss=4&rank=7.
• Chen, C., Kerr, B., Maynard, J., and Shparlinski, I.E. (2021) 'Simultaneous Small Fractional Parts of Polynomials', Geom. Funct. Anal., 31, pp. 150-179. https://doi.org/10.1007/s00039-021-00559-3.
• 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. https://arxiv.org/abs/2212.03381.
• Davenport, H. (1951) 'On a Principle of Lipschitz', Journal of the London Mathematical Society, s1-26(3), pp. 179-183. https://doi.org/10.1112/jlms/s1-26.3.179.
• 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. http://arxiv.org/abs/2306.10677v1.
• Maynard, J. (2019) 'Primes in Arithmetic Progressions to Large Moduli', presented at the Second Symposium in Analytic Number Theory, Cetraro, July 2019. https://www.dima.unige.it/ant/symposium/slides/maynard.pdf.
• 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. https://doi.org/10.1017/fmp.2019.8.
• Maynard, J. (2022) 'Primes in Arithmetic Progressions to Large Moduli III: Uniform Residue Classes', Memoirs of the American Mathematical Society. https://ora.ox.ac.uk/bookmarks/uuid:da3b7279-d413-4ab1-962e-afa5d67507c2.
• Schmidt, W.M. (1966) 'Asymptotic Formulae for Point Lattices of Bounded Determinant and Subspaces of Bounded Height', Acta Mathematica, 117, pp. 169-228. https://doi.org/10.1007/BF02395039.
• Soundararajan, K. (2022) 'The Work of James Maynard', Proceedings of the International Congress of Mathematicians 2022, Vol. 1, pp. 66-80. http://dx.doi.org/10.4171/ICM2022/212.
• Thomas, K. (2024) 'Bounding the Number of Linear Solutions to Diophantine Equations in an Application to PDEs', URSS Showcase. https://urss.warwick.ac.uk/items/show/251.
• Thunder, J.L. (1993) 'Asymptotic Estimates for Rational Points of Bounded Height on Flag Varieties', Compositio Mathematica, 88(2), pp. 155-186. http://www.numdam.org/item?id=CM_1993__88_2_155_0.

Some other perspectives

• The classical perspective
  • Davenport, H. (1947) 'The Geometry of Numbers', The Mathematical Gazette, 31(296), pp. 206-210. https://www.jstor.org/stable/3608159.
  • 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
  • Cohen, H., Diaz y Diaz, F., and Olivier, M. (2024) 'Enumerating Quartic Dihedral Extensions of Q', Cambridge University Press. https://doi.org/10.1023/A:1016310902973.
  • Couveignes, J.-M. (2020) 'Enumerating Number Fields', Annals of Mathematics, 192(2), pp. 487-497. https://doi.org/10.4007/annals.2020.192.2.4.
  • Davenport, H., and Heilbronn, H. (1971) 'On the Density of Discriminants of Cubic Fields II', Proc. Roy. Soc. London Ser. A. https://doi.org/10.1098/rspa.1971.0075.
  • Ellenberg, J.S., Venkatesh, A., 2006. The number of extensions of a number field with fixed degree and bounded discriminant, Annals of Mathematics, 163 (2006), 723–741.
  • Lemke Oliver, R.J., and Thorne, F. (2022) 'Upper Bounds on Number Fields of Given Degree and Bounded Discriminant', Duke Mathematical Journal, 171(15), pp. 4049-4102. https://doi.org/10.1215/00127094-2022-0046.
  • Odlyzko, A.M. (1990) 'Bounds for Discriminants and Related Estimates for Class Numbers, Regulators and Zeros of Zeta Functions: A Survey of Recent Results', Journal de Théorie des Nombres de Bordeaux, 2(1), pp. 119-141. http://www.numdam.org/item?id=JTNB_1990__2_1_119_0.
  • Schmidt, W.M. (1995) 'Number Fields of Given Degree and Bounded Discriminant', Astérisque, 228, pp. 189-195. http://www.numdam.org/item?id=AST_1995__228__189_0.
• 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
  • Roy, D. (2015) 'On Schmidt and Summerer Parametric Geometry of Numbers', Annals of Mathematics, 182, pp. 739-786. http://dx.doi.org/10.4007/annals.2015.182.2.9.
• Miscellaneous
  • Gruber, P. (1978) 'Geometry of Numbers, Contributions to Geometry', in Proc. Geometry Symposium in Siegen.
  • Tyler, M. (undated) 'Discriminant Bounds', notes. https://web.stanford.edu/~mttyler/notes/discriminant_bounds.pdf.