@article{14986,
  abstract     = {We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k). Let k be an algebraically closed field of characteristic p>n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bunn,P the moduli stack of (quasi-)parabolic vector bundles on X, and by Locn,P the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category Db(Qcoh(Loc0n,P)) of quasi-coherent sheaves on an open substack Loc0n,P⊂Locn,P, and the bounded derived category Db(D0Bunn,P-mod) of D0Bunn,P-modules, where D0Bunn,P is a localization of DBunn,P the sheaf of crystalline differential operators on Bunn,P. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles .},
  author       = {Shen, Shiyu},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  keywords     = {General Mathematics},
  publisher    = {Oxford University Press},
  title        = {{Tamely ramified geometric Langlands correspondence in positive characteristic}},
  doi          = {10.1093/imrn/rnae005},
  year         = {2024},
}

@article{14737,
  abstract     = {John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^{d}$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called Löwner’s ellipsoid, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the Löwner problems is more intricate than it is in the setting of convex bodies.},
  author       = {Ivanov, Grigory and Naszódi, Márton},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  keywords     = {General Mathematics},
  number       = {23},
  pages        = {20613--20669},
  publisher    = {Oxford University Press},
  title        = {{Functional John and Löwner conditions for pairs of log-concave functions}},
  doi          = {10.1093/imrn/rnad210},
  volume       = {2023},
  year         = {2023},
}

@article{9034,
  abstract     = {We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of P3 outside certain planes using universal torsors.},
  author       = {Wilsch, Florian Alexander},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  number       = {8},
  pages        = {6780--6808},
  publisher    = {Oxford Academic},
  title        = {{Integral points of bounded height on a log Fano threefold}},
  doi          = {10.1093/imrn/rnac048},
  volume       = {2023},
  year         = {2023},
}

@article{10867,
  abstract     = {In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.},
  author       = {Akopyan, Arseniy and Karasev, Roman},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  keywords     = {General Mathematics},
  number       = {3},
  pages        = {669--697},
  publisher    = {Oxford University Press},
  title        = {{Waist of balls in hyperbolic and spherical spaces}},
  doi          = {10.1093/imrn/rny037},
  volume       = {2020},
  year         = {2020},
}

@article{9576,
  abstract     = {In 1989, Rota made the following conjecture. Given n bases B1,…,Bn in an n-dimensional vector space V⁠, one can always find n disjoint bases of V⁠, each containing exactly one element from each Bi (we call such bases transversal bases). Rota’s basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (e.g., the conjecture was recently the subject of the collaborative “Polymath” project). In this paper we prove that one can always find (1/2−o(1))n disjoint transversal bases, improving on the previous best bound of Ω(n/logn)⁠. Our results also apply to the more general setting of matroids.},
  author       = {Bucić, Matija and Kwan, Matthew Alan and Pokrovskiy, Alexey and Sudakov, Benny},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  number       = {21},
  pages        = {8007--8026},
  publisher    = {Oxford University Press},
  title        = {{Halfway to Rota’s basis conjecture}},
  doi          = {10.1093/imrn/rnaa004},
  volume       = {2020},
  year         = {2020},
}

@article{9577,
  abstract     = {An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clogn⁠. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. Motivated by an old problem of Erd̋s and McKay, recently Narayanan, Sahasrabudhe, and Tomon conjectured that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of Θ(n2) different sizes. In this paper we prove this conjecture.},
  author       = {Kwan, Matthew Alan and Sudakov, Benny},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  number       = {6},
  pages        = {1621–1638},
  publisher    = {Oxford University Press},
  title        = {{Ramsey graphs induce subgraphs of quadratically many sizes}},
  doi          = {10.1093/imrn/rny064},
  volume       = {2020},
  year         = {2020},
}