Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Jonas T. Hartwig We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type A, and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if q is not a root of unity, the Gelfand-Zeitlin subalgebra of Uq(gln) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Yuliang Shen, Shuan Tang Given a continuous vector field λ(t,⋅) of Sobolev class H32 on the unit circle S1, the flow maps η=g(t,⋅) of the differential equation{dηdt=λ(t,η)η(0,ζ)=ζ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [15] conjectured that the flow curve g(t,⋅) is in the Weil-Petersson class WP(S1) and is continuously differentiable with respect to the Hilbert manifold structure of WP(S1) introduced by Takhtajan-Teo [40]. The first assertion had already been demonstrated in our previous paper [36]. In this sequel to [36], we will continue to deal with the Weil-Petersson class WP(S1) and completely solve this conjecture in the affirmative.

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Ebrahim Samei, Matthew Wiersma A locally compact group G is Hermitian if the spectrum SpL1(G)(f)⊆R for every f∈L1(G) satisfying f=f⁎, and quasi-Hermitian if SpL1(G)(f)⊆R for every f∈Cc(G) satisfying f=f⁎. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of “spectral interpolation of triple Banach ⁎-algebras” and applying it to a family PFp⁎(G) (1≤p≤∞) of Banach ⁎-algebras related to convolution operators that lie between L1(G) and Cr⁎(G), the reduced group C⁎-algebra of G. We show that if G is quasi-Hermitian, then PFp⁎(G) and Cr⁎(G) have the same spectral radius on Hermitian elements in Cc(G) for p∈(1,∞), and then deduce that G must be amenable. We also give an alternative proof to Jenkin's result in [23] that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non quasi-Hermitian or they have subexponential growth. Finally, for a non-amenable group G with either rapid decay or Kunze-Stein property, we prove the stronger statement that PFp⁎(G) is not “quasi-Hermitian relative to Cc(G)” unless p=2.

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): F. Andreatta, L. Barbieri-Viale, A. Bertapelle, B. Kahn We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti–de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Thélène–Raskind).

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Friedrich Götze, Denis Koleda, Dmitry Zaporozhets We count the algebraic numbers of fixed degree by their w-weighted lp-norm which generalizes the naïve height, the length, the Euclidean and the Bombieri norms. For non-negative integers k,l such that k+2l≤n and a Borel subset B⊂R×C+l denote by Φp,w,k,l(Q,B) the number of ordered (k+l)-tuples in B of conjugate algebraic numbers of degree n and w-weighted lp-norm at most Q. We show thatlimQ→∞Φp,w,k,l(Q,B)Qn+1=Voln+1(Bp,wn+1)2ζ(n+1)∫Bρp,w,k,l(x,z)dxdz, where Voln+1(Bp,wn+1) is the volume of the unit w-weighted lp-ball and ρp,w,k,l will denote the correlation function of k real and l complex zeros of the random polynomial ∑j=1nηjwjzj, where ηj are i.i.d. random variables with density

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Graziano Crasta, Ilaria Fragalà We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in Rn satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) p-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope method by Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of positive viscosity eigenfunctions.

Abstract: Publication date: 7 January 2020Source: Advances in Mathematics, Volume 359Author(s): Roman Bessonov, Sergey Denisov We characterize even measures μ=wdx+μs on the real line R with finite entropy integral ∫Rlogw(t)1+t2dt>−∞ in terms of 2×2 Hamiltonians generated by μ in the sense of the inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Benjamin Sambale We present a strong upper bound on the number k(B) of irreducible characters of a p-block B of a finite group G in terms of local invariants. More precisely, the bound depends on a chosen major B-subsection (u,b), its normalizer NG(〈u〉,b) in the fusion system and a weighted sum of the Cartan invariants of b. In this way we strengthen and unify previous bounds given by Brauer, Wada, Külshammer–Wada, Héthelyi–Külshammer–Sambale and the present author.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Eduardo Hoefel, Muriel Livernet, Alexandre Quesney An affine action of an associative algebra A on a vector space V is an algebra morphism A→V⋊End(V), where V is a vector space and V⋊End(V) is the algebra of affine transformations of V. The one dimensional version of the Swiss-cheese operad, denoted sc1, is the operad whose algebras are affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by (sc1)∞. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an A∞-morphism A→V⋊End(V), where A is an A∞-algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an SC2 operad. That structure is naturally compatible with the E2 structure on the deformation complex of the A∞-algebra.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Ana Garcia Elsener, Philipp Lampe, Daniel Smertnig Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-1 prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss–Leclerc–Schröer.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Nadia Mazza, Peter Symonds We construct a well-behaved stable category of modules for a large class of infinite groups. We then consider its Picard group, which is the group of invertible (or endotrivial) modules. We show how this group can be calculated when the group acts on a tree with finite stabilisers.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Jason Lo On the product elliptic threefold X=C×S where C is an elliptic curve and S is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform Φ on Db(X) induced by the classical Fourier-Mukai transform on Db(C), a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on X is taken by Φ to a slope semistable sheaf, up to modification by the transform of a codimension 2 sheaf.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Fei Han, Varghese Mathai In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in [25] without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of putative S1-equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Xuezhang Chen, Yuping Ruan, Liming Sun The Han-Li conjecture states that: Let (M,g0) be an n-dimensional (n≥3) smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and c be any real number, then there exists a conformal metric of g0 with scalar curvature 1 and boundary mean curvature c. Combining with Z.C. Han and Y.Y. Li's results, we answer this conjecture affirmatively except for the case that n≥8, the boundary is umbilic, the Weyl tensor of M vanishes on the boundary and has an interior non-zero point.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Nicholas M. Katz, Pham Huu Tiep For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A1 in characteristic p>0 if 2 n and on the affine plane A2 if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp2n(q). When n≥3 is odd, we show that the explicit rigid local systems on the affine line in characteristic p>0 constructed in [11] do have the special unitary groups SUn(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Kelei Wang, Juncheng Wei In this paper we establish a uniform C2,θ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions n≤10 (which is optimal). The proof combines two ingredients: one is a reverse application of the infinite dimensional Lyapunov-Schmidt reduction method which enables us to reduce the C2,θ estimate for these level sets to a corresponding one on solutions of Toda system; the other one uses a small regularity theorem on stable solutions of Toda system to establish various decay estimates, which gives a lower bound on distances between different sheets of solutions to Toda system or level sets of solutions to Allen-Cahn equation.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Marcus Michelen, Julian Sahasrabudhe For each n, let Xn∈{0,…,n} be a random variable with mean μn, standard deviation σn, and letPn(z)=∑k=0nP(Xn=k)zk, be its probability generating function. We show that if none of the complex zeros of the polynomials {Pn(z)} is contained in a neighborhood of 1∈C and σn>nε for some ε>0, then Xn⁎=(Xn−μn)σn−1 is asymptotically normal as n→∞: that is, it tends in distribution to a random variable Z∼N(0,1). On the other hand, we show that there exist sequences of random variables {Xn} with σn>Clogn for which Pn(z) has no roots near 1 and Xn⁎ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of Pn(z) and the distribution of the random variable Xn.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Holger Brenner, Jack Jeffries, Luis Núñez-Betancourt The F-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong F-regularity. However, it is very difficult to compute.Motivated by different aspects of the F-signature, we define a numerical invariant for rings of characteristic zero or p>0 that exhibits many of the useful properties of the F-signature. We also compute many examples of this invariant, including cases where the F-signature is not known.We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas–Li–Maskit.

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Steven Dale Cutkosky Suppose that R is an excellent local domain with maximal ideal mR. The theory of multiplicities and mixed multiplicities of mR-primary ideals extends to (possibly non Noetherian) filtrations of R by mR-primary ideals, and many of the classical theorems for mR-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for mR-primary ideals do not hold for filtrations.In this article, we consider multiplicities and mixed multiplicities of R by mR-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of mR-primary ideals, that are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of mR-primary ideals I⊂I′ with the same multiplicity then I and I′ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. The classical Minkowski inequalities for mR-primary ideals I1 and I2 hold quite generally for filtrations. If R has dimension two and there is equality in the Minkowski inequalities, then Teissier and Rees and Sharp have shown that there are powers I1a and I2b that have the same integral closure. This theorem does not hold for arbitrary filtrations. The Teissier-Rees-Sharp theorem has been extended by Katz to mR-primary ideals in arbitrary dimension. We show that the Teissier-Rees-Sharp theorem does hold for divisorial filtrations in an excellent local domain of dimension two.We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme X birationally dominating R, when R is an algebraic local domain (essentially of finite type over a field).

Abstract: Publication date: 15 December 2019Source: Advances in Mathematics, Volume 358Author(s): Eric A. Carlen, Maria C. Carvalho, Michael P. Loss We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup PN,t, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the N particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study of the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.

Abstract: Publication date: Available online 5 August 2019Source: Advances in MathematicsAuthor(s): Kathrin Bringmann, Ben Kane, Steffen Löbrich, Ken Ono, Larry Rolen