Volume 26, Issue 4

Abstract. In his Inventiones paper, Ziller (1977) computed the integral homology as a graded abelian group of the free loop space of compact, globally symmetric spaces of rank 1. Chas and Sullivan (1999) showed that the homology of the free loop space of a compact closed orientable manifold can be equipped with a loop product and a BV-operator making it a Batalin–Vilkovisky algebra. Cohen, Jones and Yan (2004) developed a spectral sequence which converges to the loop homology as a spectral sequence of algebras. They computed the algebra structure of the loop homology of spheres and complex projective spaces by using Ziller's results and the method of Brown–Shih (1959, 1962). In this note we compute the loop homology algebra by using only spectral sequences and the technique of universal examples. We therefore not only obtain Ziller's and Brown–Shih's results in an elementary way, but we also replace the roundabout computations of Cohen, Jones and Yan (2004) making them independent of Ziller's and Brown–Shih's work. Moreover we offer an elementary technique which we expect can easily be generalized and applied to a wider family of spaces, not only the globally symmetric ones.

Abstract. We prove, with a short argument based on projective algebraic geometry, that all linear symmetries of the cone of positive semidefinite quadratic forms come from linear transformations of the variables.

Abstract. Using the theory of metaplectic forms, we study the asymptotic behavior of cubic exponential sums over the ring of Eisenstein integers. In the first part of the paper, some non-trivial estimates on average over arithmetic progressions are obtained. In the second part of the paper, we prove that the sign of cubic exponential sums changes infinitely often, as the modulus runs over almost prime integers.

Abstract. We study the tempered spectrum of special orthogonal groups of odd degree over p -adic fields of characteristic zero. Residues of intertwining operators for representations parabolically induced from arbitrary maximal parabolic subgroups are determined in terms of non-vanishing of weighted sums of twisted orbital integrals. This is connected with the theory of twisted endoscopy, and poles of the Langlands L -functions, in particular, with the Rankin product L -function between irreducible (generic) supercuspidal representations of the classical group SO 2 m +1 ( F ) and those of GL k ( F ). These poles are described for, any m and k , explicitly in terms of twisted endoscopy. Moreover, we show these poles describe precisely when a supercuspidal representation of GL k ( F ) is the local component of the transfer from a special odd orthogonal group.

Abstract. We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for q -zeta functions at positive integers.

Abstract. We give a simple sufficient condition for Quinn's “bordism-type spectra” to be weakly equivalent to strictly associative ring spectra. We also show that Poincaré bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's functor from bordism-type theories to spectra lifts to the category of symmetric spectra. We also give a new account of the foundations.

Abstract. The concept of adequacy of a link was introduced to determine its crossing number. We give crossing number estimates for the (much larger) class of semiadequate links, in particular improving the previously known estimates for positive links. We also give examples showing that semiadequacy may not be attained in minimal crossing number diagrams. Our approach uses the description of links of a given canonical genus, and is related to skein invariants, the signature, the new concordance invariants, and hyperbolic volume. It allows to settle the A / B -adequacy status of many examples, including all knots up to 10 crossings. As two applications, we describe generic alternating and positive knots of given genus, determining the asymptotical behaviour of the number of such knots, and classify Seifert fibered, and therewith also hyperbolic, Montesinos links.

Abstract. Bose, Anil Kumar, Krishnan and Madan (2010) showed that “Tiling implies Spectral” holds for a union of three intervals and the reverse implication was studied under certain restrictive hypotheses on the associated spectrum. In this paper, we reinvestigate the “Spectral implies Tiling” part of Fuglede's conjecture for the three interval case. We first show that the “Spectral implies Tiling” for two intervals follows from the simple fact that two distinct circles have at most two points of intersections. We then attempt this for the case of three intervals and except for one situation are able to prove “Spectral implies Tiling”. Finally, for the exceptional case, we show a connection to a problem of generalized Vandermonde varieties.

Abstract. We construct and study several infinite grids of mock modular forms. The “rows” of each grid encode an infinite family of forms, while the “columns” encode a second infinite family. Each of these grids contains a generating function of interest as its first entry (for example, the “smallest parts” function of Andrews, or a mock theta function of Ramanujan). We show that the Hecke operators transform these grids in systematic ways, and from this we deduce many corollaries for the generating functions under consideration. For example, we recover the systematic families of congruences for the smallest parts function which have recently been discovered by various authors, and we obtain families of Hecke relations in the grid associated to the mock theta function f ( q ).