TY - JOUR AB - 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. AU - Hausel, Tamás AU - Wong, Michael Lennox AU - Wyss, Dimitri ID - 14244 IS - 4 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - Arithmetic and metric aspects of open de Rham spaces VL - 127 ER - TY - JOUR AB - 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. AU - Kwan, Matthew Alan ID - 9581 IS - 6 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - Almost all Steiner triple systems have perfect matchings VL - 121 ER - TY - JOUR AB - We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian. AU - Yang, Yaping AU - Zhao, Gufang ID - 5999 IS - 5 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - The cohomological Hall algebra of a preprojective algebra VL - 116 ER - TY - JOUR AB - Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y . We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over Q. AU - Browning, Timothy D AU - Gorodnik, Alexander ID - 270 IS - 6 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - Power-free values of polynomials on symmetric varieties VL - 114 ER -