@article{14717,
  abstract     = {We count primitive lattices of rank d inside Zn as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subspace that a lattice spans, namely its projection to the Grassmannian; its homothety class and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude the joint equidistribution of these parameters. In addition to the primitive d-lattices Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic formulas for the number of primitive lattices include an explicit bound on the error term.},
  author       = {Horesh, Tal and Karasik, Yakov},
  issn         = {1464-3847},
  journal      = {Quarterly Journal of Mathematics},
  number       = {4},
  pages        = {1253--1294},
  publisher    = {Oxford University Press},
  title        = {{Equidistribution of primitive lattices in ℝn}},
  doi          = {10.1093/qmath/haad008},
  volume       = {74},
  year         = {2023},
}

@article{13091,
  abstract     = {We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.},
  author       = {Browning, Timothy D and Sawin, Will},
  issn         = {1944-7833},
  journal      = {Algebra and Number Theory},
  number       = {3},
  pages        = {719--748},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Free rational curves on low degree hypersurfaces and the circle method}},
  doi          = {10.2140/ant.2023.17.719},
  volume       = {17},
  year         = {2023},
}

@article{9199,
  abstract     = {We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano
varieties.},
  author       = {Browning, Timothy D and Horesh, Tal and Wilsch, Florian Alexander},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  number       = {10},
  pages        = {2385--2407},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Equidistribution and freeness on Grassmannians}},
  doi          = {10.2140/ant.2022.16.2385},
  volume       = {16},
  year         = {2022},
}

@article{9260,
  abstract     = {We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ) when Δ is a Q-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.},
  author       = {Browning, Timothy D and Yamagishi, Shuntaro},
  issn         = {1432-1823},
  journal      = {Mathematische Zeitschrift},
  pages        = {1071–1101},
  publisher    = {Springer Nature},
  title        = {{Arithmetic of higher-dimensional orbifolds and a mixed Waring problem}},
  doi          = {10.1007/s00209-021-02695-w},
  volume       = {299},
  year         = {2021},
}