Volume 2020, Issue 766

We show that if a noncollapsed CD ⁢ ( K , n ) {\mathrm{CD}(K,n)} space X with n ≥ 2 {n\geq 2} has curvature bounded above by κ in the sense of Alexandrov, then K ≤ ( n - 1 ) ⁢ κ {K\leq(n-1)\kappa} and X is an Alexandrov space of curvature bounded below by K - κ ⁢ ( n - 2 ) {{K}-\kappa(n-2)} . We also show that if a CD ⁢ ( K , n ) {\mathrm{CD}(K,n)} space Y with finite n has curvature bounded above, then it is infinitesimally Hilbertian.

In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

We show that the Bowl soliton in ℝ 3 {\mathbb{R}^{3}} is the unique translating solution of the mean curvature flow whose tangent flow at - ∞ {-\infty} is the shrinking cylinder. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.

We generalize the cosection localized Gysin map to intersection homology and Borel–Moore homology, which provides us with a purely topological construction of the Fan–Jarvis–Ruan–Witten invariants and some GLSM invariants.

We prove seven of the Rogers–Ramanujan-type identities modulo 12 that were conjectured by Kanade and Russell. Included among these seven are the two original modulo 12 identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level 2 modules of A 9 ( 2 ) {A_{9}^{(2)}} . We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

We construct new examples of solutions of the Hull–Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection ∇ {\nabla} on the tangent bundle is Hermite–Yang–Mills. With this ansatz for the connection ∇ {\nabla} , we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull–Strominger system on compact non-Kähler manifolds with different topology.

In the mid 1980s it was conjectured that every bispectral meromorphic function ψ ⁢ ( x , y ) {\psi(x,y)} gives rise to an integral operator K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions ψ ⁢ ( x , y ) {\psi(x,y)} where the commuting differential operator is of order ≤ 6 {\leq 6} . We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

We correct a heavy misprint in our paper.