@article{17475,
  abstract     = {As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical
graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by
their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers,
is that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex
graphs which are determined by their spectrum converges to 1 as n → ∞.
In this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex
graphs which are determined by their spectrum. This improves on previous bounds (of shape e
c
√
n
). We also
propose a number of further directions of research.
},
  author       = {Koval, Illya and Kwan, Matthew Alan},
  issn         = {1464-3847},
  journal      = {Quarterly Journal of Mathematics},
  number       = {3},
  pages        = {869--899},
  publisher    = {Oxford University Press},
  title        = {{Exponentially many graphs are determined by their spectrum}},
  doi          = {10.1093/qmath/haae030},
  volume       = {75},
  year         = {2024},
}

@article{18930,
  abstract     = {We study sumsets 𝒜 + ℬ in the set of squares 𝒮 (and, more generally, in the set of kth powers 𝒮k, where k ≥2 is an integer). It is known by a result of Gyarmati that 𝒜 + ℬ ⊂ 𝒮k ∩[1,N] implies that min(|𝒜|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ| decreases, when the size of |𝒜| increases (or vice versa). In particular, if |𝒜| ≥ Ck1m m(logN)1m , then |ℬ| = Ok(m2logN), for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with m ∼ loglogN this gives: If |𝒜| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2).},
  author       = {Elsholtz, Christian and Wurzinger, Lena},
  issn         = {1464-3847},
  journal      = {The Quarterly Journal of Mathematics},
  number       = {4},
  pages        = {1243--1254},
  publisher    = {Oxford University Press},
  title        = {{Sumsets in the set of squares}},
  doi          = {10.1093/qmath/haae044},
  volume       = {75},
  year         = {2024},
}

@article{22192,
  abstract     = {In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the
distribution of primes in log-size intervals is Poissonian. He did so by computing average values
of the singular series constants over different sets of a fixed size k contained in an interval [1,h]
as h → ∞, and then using this average to compute moments of the distribution of primes. In this
paper, we study averages where k is relatively large with respect to h. We then apply these averages
to the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood
conjectures and in certain ranges of the parameters, the number of intervals [n,n + λlogx] with
n ≤ x containing at least k primes is ≪ x exp(−k/(λe)).},
  author       = {Kuperberg, Vivian Zieve},
  issn         = {1464-3847},
  journal      = {The Quarterly Journal of Mathematics},
  number       = {4},
  pages        = {1457--1479},
  publisher    = {Oxford University Press},
  title        = {{Sums of singular series with large sets and the tail of the distribution of primes}},
  doi          = {10.1093/qmath/haad030},
  volume       = {74},
  year         = {2023},
}

@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{687,
  abstract     = {Pursuing the similarity between the Kontsevich-Soibelman construction of the cohomological Hall algebra (CoHA) of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the integrality conjecture for motivic Donaldson-Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the CoHA associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of CoHAs with Yangians, and explaining how to connect the study of character varieties with the study of CoHAs.},
  author       = {Davison, Ben},
  issn         = {0033-5606},
  journal      = {Quarterly Journal of Mathematics},
  number       = {2},
  pages        = {635 -- 703},
  publisher    = {Oxford University Press},
  title        = {{The critical CoHA of a quiver with potential}},
  doi          = {10.1093/qmath/haw053},
  volume       = {68},
  year         = {2017},
}

@article{206,
  abstract     = {Let T ⊂ ℙ 4 be a non-singular threefold of degree at least four. Then we show that the number of points in T(ℚ), with height at most B, is o(B 3) or B → ∞.},
  author       = {Browning, Timothy D},
  issn         = {1464-3847},
  journal      = {Quarterly Journal of Mathematics},
  number       = {1},
  pages        = {33 -- 39},
  publisher    = {Unknown},
  title        = {{A note on the distribution of rational points on threefolds}},
  doi          = {10.1093/qjmath/54.1.33},
  volume       = {54},
  year         = {2003},
}

@article{208,
  abstract     = {For any ε > 0 and any diagonal quadratic form Q ∈ Z[x1, x2, x3, x4] with a square‐free discriminant of modulus ΔQ ≠ 0, we establish the uniform estimate ≪ε B3/2+ε + B2+ε/Δ1/6Q for the number of rational points of height at most B lying in the projective surface Q = 0.

},
  author       = {Browning, Timothy D},
  issn         = {1464-3847},
  journal      = {Quarterly Journal of Mathematics},
  number       = {1},
  pages        = {11 -- 31},
  publisher    = {Oxford Academic},
  title        = {{Counting rational points on diagonal quadratic surfaces}},
  doi          = {doi/10.1093/qjmath/54.1.11},
  volume       = {54},
  year         = {2003},
}