In this paper, we show that the deformed Hermitian Yang–Mills (dHYM) equation on a rational homogeneous variety, equipped with any invariant Kähler metric, always admits a solution. Unlike the known existence results for projective varieties, our result shows that in the homogeneous setting no additional hypotheses on the phase angle (such as the supercritical condition) is necessary to solve the dHYM equation. Moreover, we describe the Lagrangian phase, with respect to any invariant Kähler metric, of every closed invariant (1,1) -form in terms of Lie theory. Additionally, we provide an explicit formula, in terms of Lie theory, for the slope of torsion-free coherent sheaves on rational homogeneous varieties. Using this formula, we establish a new criterion for the slope semistability of holomorphic vector bundles over these varieties by restricting the bundle to the generators of the associated cone of curves. Furthermore, we provide a new characterization of slope (semi)stability for holomorphic vector bundles over rational homogeneous varieties in terms of central charges defined by rational curves. As a byproduct, we introduce a constructive method to obtain non-trivial examples of Hermitian–Einstein metrics on certain holomorphic vector bundles over \mathbb{P}(T_{\mathbb{P}^2}) from solutions of linear Diophantine equations. Also, motivated by the problem of constructing line bundles with prescribed slope, we present some new insights that explore the interplay between elementary number theory and combinatorics via Schubert calculus.

In this article, we give a characterization of Fourier–Mukai transforms on algebraic spaces that commute with the Frobenius thereby extending the work of Daniel Bragg [Bull. Lond. Math. Soc. 56 (2024), 3477–3483]. We also treat the case of twisted sheaves on algebraic spaces.

Initiated by a result of Gorin and Marcus (2020) and an observation of Steinerberger (2019), there has been a recent growing body of literature connecting repeated differentiation of real rooted polynomials to free additive convolution semigroups in free probability. Roughly, this connection states that in the large degree limit the empirical measure of the roots after many derivatives is, up to a rescaling, the original empirical measure of the roots raised to a free additive convolution power. If the original roots satisfy some bounds and the number of derivatives is such that the remaining degree is fixed, then it has been shown in various contexts that these high derivatives converge to the Hermite polynomials. In the context of convolution semigroups and finite free probability, where Hermite polynomials are the analogue of the Gaussian distribution, these results have a natural interpretation as a central limit theorem for repeated differentiation.We consider the case when these root bounds are removed and identify the potential limits of repeated differentiation as the real rooted Appell sequences. We prove that a sequence of polynomials is in the domain of attraction of an Appell sequence exactly when the empirical measures of the roots are, up to a rescaling, in the domain of attraction of a free infinitely divisible distribution naturally associated to the Appell sequence. We consider the limits of Appell sequences, generalizing the well known fact that the roots of a Hermite polynomial, after being appropriately normalized, are asymptotically distributed according to the semicircle distribution. We additionally extend these notions of infinite divisibility and fractional convolution semigroups to rectangular finite free probability.Our approach is based on the finite free R -transform of the polynomials, providing a step towards an analytic theory of finite free probability. These transforms provide a clear connection between Appell polynomials and free infinitely divisible distributions, where the finite free R -transform of a real rooted Appell polynomial is a truncated version of the R -transform of an infinitely divisible distribution.

We develop the theory of costratification in the setting of relative tensor-triangular geometry, in the sense of Stevenson, providing a unified approach to classification results of Neeman and Benson–Iyengar–Krause. In addition, we introduce and study prime localizing submodules and prime colocalizing \mathrm{hom} -submodules, in the first case, generalizing objectwise-prime localizing tensor-ideals. We apply our results to show that the derived category of quasi-coherent sheaves over a noetherian separated scheme is costratified. An application of the results proved in this paper, which appears in separate work, is that the singularity category of a locally hypersurface ring (more generally a noetherian separated scheme with hypersurface singularities) is costratified.

We show that a plt surface singularity (P\in X,B) is F -liftable if and only if it is F -pure and is not a rational double point of type E_{8}^{1} in characteristic p=5 . As a consequence, we prove the logarithmic extension theorem for F -pure surface pairs and Bogomolov–Sommese vanishing for globally F -split surface pairs. These results were previously known to hold only in characteristic p>5 .

We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work of Cléry and van der Geer in the setting of Siegel modular forms, but it also requires overcoming challenges that do not arise in the Siegel setting.

We perform a systematic study of the base change conductor for Jacobians. Through the lens of intersection theory and Deligne’s Riemann–Roch theorem, we present novel computational approaches for both the tame and wild parts of the base change conductor. Our key results include a general formula of the tame part, as well as a computation of the wild part in terms of Galois quotients of semi-stable models of the curves. We treat in detail the case of potential good reduction when the quotient only has weak wild quotient singularities, relying on recent advances by Obus and Wewers.

The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A , with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki–Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.

We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple “naïve” extension to commuting tuples in a general Banach algebra. The approach is naïve in the sense that the naïvely defined joint spectrum maybe too big. The advantage of the approach is that the functional calculus then is given by a simple concrete formula from which all its continuity properties can easily be derived.We apply this framework to multivariate functions arising as divided differences of a univariate function. This provides a rich set of examples to which our naïve calculus applies. Foremost, we offer a natural and straightforward proof of the Connes–Moscovici Rearrangement Lemma in the context of the multivariate holomorphic functional calculus. Secondly, we show that the Daletski–Krein type noncommutative Taylor expansion is a natural consequence of our calculus. Also Magnus’ Theorem which gives a nonlinear differential equation for the \log of the solutions to a linear matrix ODE follows naturally and easily from our calculus. Finally, we collect various combinatorial related formulas.

Following the ideas of Flach and Morin (2018), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function \zeta(X,s) at s=n<0 , where X is a separated scheme of finite type over \operatorname{Spec}\mathbb{Z} . We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases.