@article{20249,
  abstract     = {We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.},
  author       = {Browning, Timothy D and Wilsch, Florian Alexander},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica New Series},
  number       = {4},
  publisher    = {Springer Nature},
  title        = {{Integral points on cubic surfaces: heuristics and numerics}},
  doi          = {10.1007/s00029-025-01074-1},
  volume       = {31},
  year         = {2025},
}

@article{12312,
  abstract     = {Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface
$A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.},
  author       = {Lombardo, Davide and Verzobio, Matteo},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{On the local-global principle for isogenies of abelian surfaces}},
  doi          = {10.1007/s00029-023-00908-0},
  volume       = {30},
  year         = {2024},
}

@article{14930,
  abstract     = {In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.},
  author       = {Hausel, Tamás and Letellier, Emmanuel and Rodriguez-Villegas, Fernando},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{Locally free representations of quivers over commutative Frobenius algebras}},
  doi          = {10.1007/s00029-023-00914-2},
  volume       = {30},
  year         = {2024},
}

@article{9998,
  abstract     = {We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.},
  author       = {Koroteev, Peter and Pushkar, Petr and Smirnov, Andrey V. and Zeitlin, Anton M.},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica},
  number       = {5},
  publisher    = {Springer Nature},
  title        = {{Quantum K-theory of quiver varieties and many-body systems}},
  doi          = {10.1007/s00029-021-00698-3},
  volume       = {27},
  year         = {2021},
}

@article{7683,
  abstract     = {For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an AHa0C-action. These triples can be interpreted as certain sheaves on PC(ωC⊕OC). In particular, we obtain an action of AHa0C on the cohomology of Hilbert schemes of points on T∗C.},
  author       = {Minets, Sasha},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica, New Series},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces}},
  doi          = {10.1007/s00029-020-00553-x},
  volume       = {26},
  year         = {2020},
}