@article{21751, abstract = {We define a certain class of simple varieties over a field k by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if k = k and char k = p, the p-adic cyclotomic trace is an equivalence; (ii) if k = Q, the Goodwillie–Jones trace is an isomorphism in degree zero; (iii) we can control homotopy invariant K-theory KH, which is equivariantly formal and determined by its topological counterparts. Simple varieties are quite special, but encompass important singular examples appearing in geometric representation theory. We, in particular, show that both finite and affine Schubert varieties for GLn lie in this class, so all the above results hold for them. }, author = {Löwit, Jakub}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {7}, publisher = {Oxford University Press}, title = {{Equivariant localizing invariants of simple varieties}}, doi = {10.1093/imrn/rnag058}, volume = {2026}, year = {2026}, } @article{20222, abstract = {Let X be a smooth projective hypersurface defined over Q. We provide new bounds for rational points of bounded height on X. In particular, we show that if X is a smooth projective hypersurface in Pn with n 4 and degree d 50, then the set of rational points on X of height bounded by B have cardinality On,d,ε (Bn−2+ε ). If X is smooth and has degree d 6, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.}, author = {Verzobio, Matteo}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {16}, publisher = {Oxford University Press}, title = {{Counting rational points on smooth hypersurfaces with high degree}}, doi = {10.1093/imrn/rnaf249}, volume = {2025}, year = {2025}, } @article{20504, abstract = {Let r, k, be integers such that 0 ≤ ≤ (k/r). Given a large r-uniform hypergraph G, we consider the fraction of k-vertex subsets that span exactly edges. If is 0 or (k/r), this fraction can be exactly 1 (by taking G to be empty or complete), but for all other values of , one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if is not 0 or (k/r), then this fraction is at most (1/e) + ε, assuming k is sufficiently large in terms of r and ε > 0, and G is sufficiently large in terms of k. Previously, this was only known for a very limited range of values of r, k, (due to Kwan–Sudakov–Tran, Fox–Sauermann, and Martinsson–Mousset–Noever–Trujic). Our result answers a question of Alon–Hefetz–Krivelevich–Tyomkyn, who suggested this as a hypergraph generalization of their edge-statistics conjecture. We also prove a much stronger bound when is far from 0 and (k/r).}, author = {Jain, Vishesh and Kwan, Matthew Alan and Mubayi, Dhruv and Tran, Tuan}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {18}, publisher = {Oxford University Press}, title = {{The edge-statistics conjecture for hypergraphs}}, doi = {10.1093/imrn/rnaf273}, volume = {2025}, year = {2025}, } @article{21265, abstract = {We explain how the (shifted) Ratios Conjecture for $L(s,\chi )$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of “beyond square-root cancellation” for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.}, author = {Wang, Victor and Xu, Max Wenqiang}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {18}, publisher = {Oxford University Press}, title = {{Harper’s beyond square-root conjecture}}, doi = {10.1093/imrn/rnaf279}, volume = {2025}, year = {2025}, } @article{18900, abstract = {We prove that certain closable derivations on the GNS Hilbert space associated with a non-tracial weight on a von Neumann algebra give rise to GNS-symmetric semigroups of contractive completely positive maps on the von Neumann algebra.}, author = {Wirth, Melchior}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {14}, pages = {10597--10614}, publisher = {Oxford University Press}, title = {{Modular completely Dirichlet forms as squares of derivations}}, doi = {10.1093/imrn/rnae092}, volume = {2024}, year = {2024}, } @article{19051, abstract = {This paper corrects an error in an earlier work of the author.}, author = {Browning, Timothy D}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {13}, pages = {10165--10168}, publisher = {Oxford University Press}, title = {{The polynomial sieve and equal sums of like polynomials}}, doi = {10.1093/imrn/rnae066}, volume = {2024}, year = {2024}, } @article{19486, abstract = {Consider the family of elliptic curves En:y2=x3+n2, where n varies over positive cubefree integers. There is a rational 3-isogeny ϕ from En to E^n:y2=x3−27n2 and a dual isogeny ϕ^:E^n→En. We show that for almost all n, the rank of Selϕ(En) is 0, and the rank of Selϕ^(E^n) is determined by the number of prime factors of n that are congruent to 2mod3 and the congruence class of nmod9.}, author = {Chan, Yik Tung}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {9}, pages = {7571--7593}, publisher = {Oxford University Press}, title = {{The 3-isogeny selmer groups of the elliptic curves y2=x3+n2}}, doi = {10.1093/imrn/rnad266}, volume = {2024}, year = {2024}, } @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}, number = {7}, pages = {6176--6208}, publisher = {Oxford University Press}, title = {{Tamely ramified geometric Langlands correspondence in positive characteristic}}, doi = {10.1093/imrn/rnae005}, volume = {2024}, year = {2024}, } @article{17281, abstract = {We extend the free convolution of Brown measures of R-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.}, author = {Campbell, Andrew J and O'Rourke, Sean and Renfrew, David T}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {13}, pages = {10189--10218}, publisher = {Oxford University Press}, title = {{The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation}}, doi = {10.1093/imrn/rnae062}, volume = {2024}, 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 University Press}, title = {{Integral points of bounded height on a log Fano threefold}}, doi = {10.1093/imrn/rnac048}, volume = {2023}, year = {2023}, } @article{20617, abstract = {Our previous paper describes a geometric translation of the construction of open Gromov–Witten invariants by Solomon and Tukachinsky from a perspective of $A_{\infty }$-algebras of differential forms. We now use this geometric perspective to show that these invariants reduce to Welschinger’s open Gromov–Witten invariants in dimension 6, inline with their and Tian’s expectations. As an immediate corollary, we obtain a translation of Solomon–Tukachinsky’s open WDVV equations into relations for Welschinger’s invariants.}, author = {Chen, Xujia}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {9}, pages = {7021--7055}, publisher = {Oxford University Press}, title = {{Solomon-Tukachinsky’s versus Welschinger’s open Gromov-Witten invariants of symplectic six-folds}}, doi = {10.1093/imrn/rnaa318}, volume = {2022}, year = {2022}, } @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}, }