@article{21931,
  abstract     = {In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.},
  author       = {Edelsbrunner, Herbert and Fillmore, Christopher D and Oliveira, Goncalo},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {5},
  publisher    = {Wiley},
  title        = {{Counting equilibria of the electrostatic potential}},
  doi          = {10.1112/plms.70163},
  volume       = {132},
  year         = {2026},
}

@article{22157,
  abstract     = {A graph 𝐺 is said to be Ramsey for a tuple of graphs(𝐻 1 , … , 𝐻𝑟 ) if every 𝑟-coloring of the edges of 𝐺 con-tains a monochromatic copy of 𝐻𝑖 in color 𝑖, for some 𝑖.A fundamental question at the intersection of Ramseytheory and the theory of random graphs is to deter-mine the threshold at which the binomial randomgraph 𝐺𝑛,𝑝 becomes asymptotically almost surely Ram-sey for a fixed tuple (𝐻 1 , … , 𝐻𝑟 ), and a famous conjectureof Kohayakawa and Kreuter predicts this threshold.Earlier work of Mousset–Nenadov–Samotij, Bowtell–Hancock–Hyde, and Kuperwasser–Samotij–Wigdersonhas reduced this probabilistic problem to a determinis-tic graph decomposition conjecture. In this paper, weresolve this deterministic problem, thus proving theKohayakawa–Kreuter conjecture. Along the way, weprove a number of novel graph decomposition resultsthat may be of independent interest.},
  author       = {Christoph, Micha and Martinsson, Anders and Steiner, Raphael and Wigderson, Yuval},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {1},
  publisher    = {Wiley},
  title        = {{Resolution of the Kohayakawa–Kreuter conjecture}},
  doi          = {10.1112/plms.70013},
  volume       = {130},
  year         = {2025},
}

@article{22203,
  abstract     = {We prove near-optimal upper bounds for the oddmoments of the distribution of coprime residues inshort intervals, confirming a conjecture of Montgomeryand Vaughan. As an application, we prove near-optimalupper bounds for the average of the refined singularseries in the Hardy–Littlewood conjectures concerningthe number of prime 𝑘-tuples for 𝑘 odd. The mainnew ingredient is a near-optimal upper bound for thenumber of solutions to ∑1 ⩽𝑖 ⩽𝑘𝑎 𝑖𝑞𝑖∈ ℤ when 𝑘 is odd,with gcd(𝑎𝑖 , 𝑞𝑖 ) = 1 and restrictions on the size of thenumerators and denominators, which is of indepen-dent interest.},
  author       = {Bloom, Thomas F. and Kuperberg, Vivian Zieve},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {1},
  publisher    = {Wiley},
  title        = {{Odd moments and adding fractions}},
  doi          = {10.1112/plms.70068},
  volume       = {131},
  year         = {2025},
}

@article{14244,
  abstract     = {In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank 
 bundle on P1. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four-dimensional cases are expected to be of type ALF.},
  author       = {Hausel, Tamás and Wong, Michael Lennox and Wyss, Dimitri},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {4},
  pages        = {958--1027},
  publisher    = {Wiley},
  title        = {{Arithmetic and metric aspects of open de Rham spaces}},
  doi          = {10.1112/plms.12555},
  volume       = {127},
  year         = {2023},
}

@article{9581,
  abstract     = {We show that for any  𝑛  divisible by 3, almost all order-  𝑛  Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for example, we sketch a proof of the theorem that almost all Latin squares have transversals.},
  author       = {Kwan, Matthew Alan},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {6},
  pages        = {1468--1495},
  publisher    = {Wiley},
  title        = {{Almost all Steiner triple systems have perfect matchings}},
  doi          = {10.1112/plms.12373},
  volume       = {121},
  year         = {2020},
}

@article{1464,
  abstract     = {The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle.

This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle K. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes.

The spaces of Higgs bundles with values in K(n) for n > 0 turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes. 2000 Mathematics Subject Classification 14H60 (primary), 14D20, 14H81, 32Q55, 58D27 (secondary). },
  author       = {Hausel, Tamas and Thaddeus, Michael},
  issn         = {1460-244X},
  journal      = {Proceedings of the London Mathematical Society},
  number       = {3},
  pages        = {632 -- 658},
  publisher    = {Oxford University Press},
  title        = {{Generators for the cohomology ring of the moduli space of rank 2 higgs bundles}},
  doi          = {10.1112/S0024611503014618},
  volume       = {88},
  year         = {2004},
}