Abstract We construct the quantum double ramification (DR) hierarchy associated with the Gromov–Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the DR cycle, the Gromov–Witten classes of the elliptic curve, and the Hodge class , together with vanishing results for to produce a closed, modular expression for the resulting integrable hierarchy. It is the first explicit nontrivial example of a quantum integrable hierarchy from a cohomological field theory containing fermionic fields, which correspond to the odd classes in the cohomology of the elliptic curve.
Abstract In this paper, we provide an alternative proof of Chandee and Li's result on the second moment of special ‐values. Our method is conceptually more direct as it neither detects the “Eisenstein–Kloosterman” cancelation nor uses the Poisson summation formula.
Hölder–Brascamp–Lieb inequalities have become a ubiquitous tool in Fourier analysis in recent years, due in large part to a theorem of Bennett et al. characterizing finiteness of the Hölder–Brascamp–Lieb constant. Here we provide a new characterization of a substantially different nature involving directed graphs of subspaces. Its practical value derives from its complementary nature to the Bennett et al. conditions: it creates a means by which one can establish finiteness of the Hölder–Brascamp–Lieb constant by analysis of a well‐chosen, finite list of subspaces rather than by checking conditions on all subspaces of the underlying vector space. The proof is elementary and is essentially an “explicitization” of the semi‐explicit factorization algorithm of Carbery et al.
Abstract Ideals of minors arising from minimal free resolutions—equivalently, the Fitting ideals of syzygy modules—are natural invariants of interest in commutative algebra and algebraic geometry. A surprising observation in recent years by Brown–Dao–Sridhar is that for many nice classes of rings, such as complete intersection rings and Golod rings, ideals of minors tend to become eventually periodic. In this article, we establish similar periodicity phenomena for further classes of rings, namely fiber products and artinian stretched Gorenstein rings. In fact, we show that when the embedding dimension is at least 3, under a mild characteristic assumption, the ideals of minors stabilize to powers of the maximal ideal, exhibiting extremal behavior. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring and a super‐regular element , if the ideals of minors of an ‐module have extremal behavior, then so do the ideals of minors of over .
Abstract We prove that level one cyclotomic Khovanov–Lauda–Rouquier (KLR) algebras in type are graded Morita equivalent to level two cyclotomic KLR algebras in type . We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type .
Abstract We show that Iida–Taniguchi's ‐valued slice‐torus invariant cannot be realized as a linear combination of Rasmussen's ‐invariant, Ozsváth–Szabó's ‐invariant, the ‐concordance invariants (), Baldwin–Sivek's instanton ‐invariant, Daemi–Imori–Sato–Scaduto–Taniguchi's instanton ‐invariant and Sano–Sato's Rasmussen‐type invariants .
Let be an odd prime, and suppose that and are weight two newforms sharing the same irreducible Galois representation modulo . We establish a transition formula relating the ‐invariants and in the case where their underlying modular varieties have the same dimension. For abelian extensions , this essentially removes the condition present in the earlier work of Greenberg–Vatsal and Emerton–Pollack–Weston.
We prove that the Heisenberg group admits infinitely many inequivalent equivariant compactifications into for all . This result provides a non‐commutative analog of Hassett–Tschinkel's classical result, which shows that there exist infinitely many inequivalent equivariant compactifications of vector groups into projective spaces of dimension at least 6.
Let be a cyclic ‐group or generalized quaternion group, be a virtual ‐set, and be a fixed point free complex ‐representation. Under conditions depending on the sizes of , , and , we construct a self map on the cofiber of which induces an equivalence in ‐equivariant ‐theory. These are transchromatic ‐self maps, in the sense that they are lifts of classical ‐self maps for which the telescope can have nonzero rational geometric fixed points.
In this paper, we construct a model structure for ‐categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of ‐categories given by complete Segal spaces and Segal categories. We furthermore prove that this model structure has desirable properties: it is cartesian closed and left proper. As applications, we get a simple description of the inclusion of categories into ‐categories and of homotopy limits of ‐categories.
Fix a prime number . The post-critically finite polynomials of the form play a fundamental role in polynomial dynamics. While many results are known in the complex dynamical setting, much less is understood about the arithmetic properties of these polynomials. In this paper, we describe the factorization of the iterates of post-critically finite polynomials over their fields of definition. As a consequence, we prove new cases of a conjecture of Andrews and Petsche on abelian arboreal Galois representations.
Let denote the Araki-Woods factor-the unique separable injective type factor. For extremal almost periodic states , we show that if and have the same point spectrum, then for some . Consequently, the extremal almost periodic states on are parameterized by countable dense subgroups of , up to precomposition by automorphisms. As an application, we show that KMS states for generalized gauge actions on Cuntz algebras agree (up to an automorphism) with tensor products of Powers states on their von Neumann completions.
We prove, under certain assumptions, the algebraicity of the ratio , where is a cuspidal automorphic cohomological unitary representation of , and , are finite-order Hecke characters such that , and are specific positive integers which depend only on . The methods in this paper generalize those in the work of Mahnkopf [Cohomology of arithmetic groups, parabolic subgroups and the special values of -functions of , J. Inst. Math. Jussieu. 4 (2005)].
We show that the universal Teichm & uuml;ller family of compact Riemann surfaces of genus with punctures is a Stein manifold. We describe its basic function-theoretic properties and pose some challenging questions. We show, in particular, that the space of fibrewise algebraic functions on the universal family is dense in the space of holomorphic functions, and that there is a fibrewise algebraic map of the universal family to a Euclidean space which restricts to a proper embedding on any fibre. We also obtain a relative Oka principle for holomorphic fibrewise algebraic maps of the universal family to any flexible algebraic manifold.
We show that Iida-Taniguchi's -valued slice-torus invariant cannot be realized as a linear combination of Rasmussen's -invariant, Ozsv & aacute;th-Szab & oacute;'s -invariant, the -concordance invariants (), Baldwin-Sivek's instanton -invariant, Daemi-Imori-Sato-Scaduto-Taniguchi's instanton -invariant and Sano-Sato's Rasmussen-type invariants .
We establish vanishing results for the first cohomology group of nilpotent groups and Lie rings when the submodule of invariants is trivial. Our results are obtained within a model-theoretic setting, namely for structures that are definable in a finite-dimensional theory, which encompasses algebraic groups over algebraically closed fields, real semialgebraic groups, and finite-dimensional Lie algebras over an algebraically or real closed field. Since classical tools-such as computations with spectral sequences and rigidity of the linear dimension-are not available in our setting, we develop an elementary algebraic approach. As applications, we derive a form of Frattini's argument for Cartan subrings and a definable version of Maschke's theorem for actions of definable connected -divisible abelian groups, with a view toward the ongoing study of soluble finite-dimensional Lie rings.
We develop a potential theory for the Wess-Zumino-Witten (WZW) equation in the space of K & auml;hler potentials, which is parallel to the potential theory for the Hermitian-Yang-Mills equation. A concept called -harmonicity on graphs is introduced, which characterizes the WZW equation. We also show that, with respect to a Banach-Mazur-type distance function, the distance between two solutions of the WZW equation is subharmonic. The harmonic map into the space of K & auml;hler potentials, as a special case of the WZW equation, is also investigated. In particular, we show the solvability of the Dirichlet problem for the harmonic map, and the approximation/quantization by its finite-dimensional counterparts.
In this paper, we establish asymptotic estimates for the rational lines on diagonal cubic hypersurfaces defined by with , provided that . This improves the previously known bound required to obtain such asymptotic estimates. Our approach develops a multidimensional shifting variables argument together with a pruning argument, and exploits recent progress on the Parsell-Vinogradov system.
Let We show that if is close to then essentially , thereby showing that sign-changes in the average give power-saving cancellation past the expected . This therefore for the first time provides unconditional evidence for a deep conjecture of Montgomery and Soundararajan.
The purpose of this note is to address the gap in the stably finite/purely infinite dichotomy of selfless C^*-probability spaces. In particular, we show that nonfaithful selfless C^*-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless C^*-algebra is either purely infinite or stably finite. Notably, this shows that every selfless C^*-algebra is pure. To accomplish this, we show that infinite reduced free products of C^*-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having resolved the selfless dichotomy, we improve existing permanence properties of selfless C^*-probability spaces, make progress on a conjecture of Choda and Dykema, and produce several new isomorphisms arising from reduced free products.
