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Auslander and Bridger [ Stable module theory , Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969] introduced the notions of n-spherical modules and n-torsionfree modules. In this paper, we construct an equivalence between the stable category of n-spherical modules and the category of modules of grade at least n, and provide its Gorenstein analogue. As an application, we prove that if R is a Gorenstein local ring of Krull dimension d>0, then there exists a stable equivalence between the category of (d-1)-torsionfree R-modules and the category of d-spherical modules relative to the local cohomology functor. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

Let s(n)=\sum _{d\mid n,~d

A classical observation of Deligne shows that, for any prime p \geq 5, the divisor polynomial of the Eisenstein series E_{p-1}(z) mod p is closely related to the supersingular polynomial at p, \begin{align*} S_p(x) ≔\prod _{E/\overline {\mathbb {F}}_p \text { supersingular}}(x-j(E))\,\, \in \mathbb {F}_p[x]. \end{align*} Deuring, Hasse, and Kaneko and Zagier [ Supersingular j-invariants, hypergeometric series, and Atkin's orthogonal polynomials , Providence, RI, 1998, pp. 97–126] found other families of modular forms which also give the supersingular polynomial at p. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms T_k(z) defined by the Hecke action on the space of cusp forms S_k. We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p, and then relate their divisor polynomials to S_p(x) using Deligne's observation. [ABSTRACT FROM AUTHOR]

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity |u|α−1u, where α ∈ [1, d/d−2], in dimension d ≥ 3 with initial data in Mp, p'1(Rd) × Mp,p'(Rd) for p sufficiently close to 2. The proof is an application of the high-low method described by Bourgain in [ Global solutions of nonlinear Schrödinger equations , American Mathematical Society, Providence, RI, 1999] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces. [ABSTRACT FROM AUTHOR]

We resolve an open problem due to B. Simon [ Trace ideals and their applications , Cambridge University Press, Cambridge–New York, 1979; Trace ideals and their applications , American Mathematical Society, Providence, RI, 2005] concerning optimal constants in Hölder inequality for certain quasi-normed principal ideals of importance in Mathematical Physics and Noncommutative Geometry. [ABSTRACT FROM AUTHOR]

Given a countable group G, we say that a metrizable flow Y is model-universal if by considering the various invariant measures on Y, we can recover every free measure-preserving G-system up to isomorphism. Weiss in[ Dynamical systems and group actions , American Mathematical Society, Providence, RI, 2012, pp. 249-264] constructs a minimal model-universal flow. In this note, we provide a new, streamlined construction, allowing us to show that a minimal model-universal flow is far from unique. [ABSTRACT FROM AUTHOR]

We prove that it is relatively consistent with ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice (AC)) that the Axiom of Countable Choice (AC0) is true, but the Urysohn Lemma (UL), and hence the Tietze Extension Theorem (TET), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin [Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998]. We also prove that in Läuchli's permutation model of ZFA + -UL, AC0 is false. This fills the gap in information in the above monograph of Howard and Rubin. [ABSTRACT FROM AUTHOR]