Using non-A(1)-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic K-theory of singular rings, such as rings of the form Z/p(n), Z[x]/(x(e)), and C(X; C) for X a compact Hausdorff space. These refinements are made possible by the use of integral p-adic Hodge theory, as a replacement for the standard use of trace methods in K-theory.
We study equivalences of the form \Sigma^{V}X\simeq\Sigma^{W}X , where G is a compact Lie group, X is a G -spectrum, and V and W are G -representations. These equivalences encode a periodicity phenomenon in G -equivariant homotopy theory which generalizes the classical James periodicity for G=C_{2} .In the case where X=C(a_{\lambda}) is the cofiber of an Euler class, we construct an \mathit{RO}(G) -graded J -homomorphism J\colon\pi_{\lambda}\mathit{KO}_{G}\rightarrow\pi_{\star}^{G} C(a_{\lambda})^{\times} which gives control over these periodicities. It also produces infinite periodic families in the G -equivariant stable stems. We illustrate this with several explicit examples.More generally, our work gives information about \mathit{RO}(G) -graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the G -homotopy fixed point spectral sequence, and other equivariant Atiyah–Hirzebruch spectral sequences.
The Hessian map is the rational map that sends a homogeneous polynomial to the determinant of its Hessian matrix. We prove that the Hessian map is birational on its image for ternary forms of degree d >= 4, d not equal 5, by considering the action of the orthogonal group. In a previous paper, we proved the analogous result for binary forms, with more geometric techniques.
We compute the Bessel models of every irreducible representation of the finite group \operatorname{GSp}(4,q) .
In this paper, we introduce the notion of a bi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of $1$-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $\mathbb{A}_g$. We propose an Analytic Subspace Conjecture, which is the analogue of the Wüstholz's Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $\mathbb{A}_g$, implies that all quadratic $\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones.
This paper presents a rigorous analysis for a degenerate diffusion Keller-Segel type system featuring rotational flux terms in even-dimensional spaces. The proposed model synthesizes fundamental principles from mathematical biology, fluid dynamics, and electrokinetic phenomena. Through discussion of the diffusion exponent m is an element of [2n/n+2, 2-2/n) and rotation angle alpha is an element of (2k pi-pi; 2k pi+pi], k is an element of Z, we obtain results on the existence and blow-up of solutions. In particular, we derive a sharp initial criterion to distinguish the global existence and the finite-time blow-up of solutions for the case alpha is an element of (2k pi-pi/2, 2k pi+pi/2), k is an element of Z and m is an element of (2n/n+2, 2-2/n), and show the existence of solutions under any initial condition for the case alpha is an element of (2k pi-pi; 2k pi-pi/2] U [2k pi+pi/2, 2k pi+pi],k is an element of Z. This result reveals that repulsive effects caused by rotation effectively prevent solution blow-up.
We correct an error in the paper [Geom. Topol. 11 (2007), 315–427], and take the opportunity to examine in more detail the derived functors of inverse limits over orbit categories of (infinite) locally finite groups. The main results show how to reduce this in many cases to limits over orbit categories of finite groups, but we also look at generalizations of the Lyndon–Hochschild–Serre spectral sequence for higher limits over orbit categories for an extension of locally finite groups.
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 study $\mathbb{E}_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect, in the non-group-complete situation. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group-completion. This gives an alternative description of the $p$-inverted higher algebraic $K$-theory of a ring.
Let p be a prime number. For a field F containing a root of unity of order p , let H^{\bullet}(F)=H^{\bullet}(F,\mathbb{F}_{p}) be the mod- p Galois cohomology graded \mathbb{F}_{p} -algebra of F . By the Norm Residue Theorem, H^{\bullet}(F) is a purely quadratic graded-commutative algebra, and is therefore determined by the cup product \cup\colon H^{1}(F)\times H^{1}(F)\to H^{2}(F) . We prove that the class of all Galois cohomology algebras H^{\bullet}(F) is cofinal in the class of all purely quadratic graded-commutative \mathbb{F}_{p} -algebras A_{\bullet} , in the following sense: For every A_{\bullet} there exists F such that the bilinear map A_{1}\times A_{1}\to A_{2} , which determines A_{\bullet} , embeds in the cup product bilinear map \cup\colon H^{1}(F)\times H^{1}(F)\to H^{2}(F) .We further provide examples of \mathbb{F}_{p} -bilinear maps which are not realizable by fields F in this way. These are related to recent results by Snopce–Zalesskii and Blumer-Quadrelli–Weigel on the Galois theory of pro- p right-angled Artin groups, as well as to a conjecture by Marshall on the possible axiomatization of quadratic form theory of fields.
We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal or a weakly formal scheme, or for the convergent or the overconvergent de Rham-Witt complex on a smooth scheme over a perfect field of positive characteristic. This enables us to give a new description of convergent and overconvergent isocrystals with a Frobenius structure.
We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple "na & iuml;ve" extension to commuting tuples in a general Banach algebra. The approach is na & iuml;ve in the sense that the na & iuml;vely 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 na & iuml;ve 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.
We initiate the study of Prym-Brill-Noether theory for ramified double covers, extending several key results from classical Prym-Brill-Noether theory to this new framework. In particular, we improve Kanev's results on the dimension of pointed Prym-Brill-Noether loci for ramified double covers. Additionally, we compute the dimension of twisted Prym-Brill-Noether loci with vanishing conditions at points, thus extending the results of Tarasca. Furthermore, we compute the class of the twisted Prym-Brill-Noether loci inside (a translation of) the Prym variety, thus extending the results of de Concini and Pragacz to ramified double covers. Finally, we prove that a generic Du Val curve is Prym-Brill-Noether general.
We study a class of triangulated categories obtained as Verdier quotients of 3-Calabi-Yau categories combinatorially described by quivers with potential from (decorated) marked surfaces. We study their bounded t-structures and consider in particular the exchange graphs of hearts and silting objects, and show that the Koszul isomorphism between these graphs is preserved under Verdier quotient.
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 semistable 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.
We introduce the notion of (strong) subexponential growth for etale groupoids and study its basic properties. In particular, we show that the K-groups of the associated reduced groupoid L-p-operator algebras are independent of p is an element of[1,infinity) whenever the groupoid has strong subexponential growth. Several examples are discussed. Most significantly, we apply classical tools from analytic number theory to exhibit an example of an etale groupoid associated with a shift of infinite type which has strong subexponential growth, but not polynomial.
We study torus-equivariant algebraic K-theory of affine Schubert varieties in the perfect affine Grassmannians over F-p. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that F-p-linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these K-theory rings. We establish various structural results for equivariant perfect algebraic K-theory on the way; we believe these are of independent interest.
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.
