Non-negative hereditary polynomials in a free *−algebra

We prove a non-negative-stellensatz and a null-stellensatz for a class of polynomials called hereditary polynomials in a free *-algebra.Partially supported by NSF, DARPA and Ford Motor Co.Partially supported by NSF grant DMS-0140112Partially supported by NSF grant DMS-0100367     

On extensions of the Baer-Suzuki Theorem

We find a necessary and sufficient condition for an element of prime order in a finite group to be in a normalp-subgroup. This generalizes the Baer-Suzuki Theorem. Our proof depends on a result about elements of prime order contained in a unique maximal subgroup containing a result of Wielandt. We discuss various consequences, linear and algebraic group versions of the result. For John Thompson Partially supported by NSF grant DMS-91011407. Partially supported by NSF grant DMS-9208667.     

Structure of some ℤ-graded lie superalgebras of vector fields

In this paper we classifyℤ-graded transitive Lie superalgebras with prescribed nonpositive parts listed in [K2]. The classification of infinite-dimensional simple linearly compact Lie superalgebras given in [K2] is based on this result. We also study the structure of the exceptionalℤ-graded transitive Lie superalgebras and give their geometric realization. Dedicated to the memory of Claude Chevalley Partially supported by NSC grant 88-2115-M006-013 of the ROC Partially supported by NSF grant DMS-9622870 An erratum to this article is available at .     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

The Length Spectrum of a Compact Constant Curvature Complex is Discrete

In this short note we prove that the length spectrum of a compact constant curvature complex is discrete. After recalling the relevant definitions and reducing to a relatively simple situation, the result follows easily from a foundational result about real semi-algebraic sets and the Morse–Sard theorem. We conclude with a conjecture which remains open, a few remarks and an easy application.Partially supported by NSF grant no. DMS-0123344 (N. Brady).Partially supported by NSF grant no. DMS-0101506 (J. McCammond)     

A local version of the two-circles theorem

A necessary and sufficient condition is given so that in a domain Ω there are no functions whose average over all balls contained in Ω of radiir ₁,r ₂ vanish except the zero function. Partially supported by NSF grant DMS-8401356 and by NSF grant OJR 85-OV-108 through the Systems Research Center of the University of Maryland.     

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA _n ⊃ X and μ-almost every pointx∈X the inclusionT ⁿ x∈A _n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma. Partially supported by NSF grant DMS-9732728. Partially supported by NSF grant DMS-9704489.     

Morera and mean-value type theorems in the hyperbolic disk

Some Morera and mean-value type theorems are proved in the hyperbolic disk. Partially supported by NSF grants DMS-9000619 and CDR-8803012. Partially supported by NSF grants DMS-8703072 and DMS-9000619, and DGI-CYT grant PB 89-0311.     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

On a property of special groups

We show that Conjectures I and II of Serre about the vanishing ofH ¹ cannot be strengthened.Partially supported by NSF grant DMS-9801675.     

Uniformity of stably integral points on principally polarized abelian varieties of dimension ≤2

The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤2 is used only at the last step, where we apply Pacelli’s stronger uniformity results for elliptic curves. Partially supported by NSF grant DMS-9700520 and by an Alfred P. Sloan research fellowship. Partially supported by NSA grant MDA904-96-1-0008.     

Holomorphy of Rankin tripleL-functions; special values and root numbers for symmetric cubeL-functions

In this paper we prove the holomorphy of Rankin tripleL-functions for three cusp forms on GL(2) on the entire complex plane, if at least one of them is non-monomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric powerL-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cubeL-functions. Partially supported by NSF grant DMS9610387. Partially supported by NSF grant DMS9970156.     

Unstable minimal surfaces and Heegaard splittings

Partially supported by NSF grant DMS-8701746     

The refined\mathfrak{p}-adic abelian Stark conjecture in function fields

Partially supported by NSF grant DMS-8702716     

Artin groups of finite type are biautomatic

Partially supported by NSF grant DMS-8905378     

On the main conjecture of Iwasawa theory for imaginary quadratic fields

Partially supported by NSF grant DMS-8501937 and by a Sloan fellowship     

L 2 Riemann-Roch theorem for elliptic operators

Partially supported by NSF grant DMS-9222491.     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

On fixed points of permutations

The number of fixed points of a random permutation of {1,2,…,n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial – almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of {1,2,…,n}, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results. This paper is dedicated to the life and work of our colleague Manfred Schocker. We thank Peter Cameron for his help. Diaconis was supported by NSF grant DMS-0505673. Fulman received funding from NSA grant H98230-05-1-0031 and NSF grant DMS-0503901. Guralnick was supported by NSF grant DMS-0653873. This research was facilitated by a Chaire d’Excellence grant to the University of Nice Sophia-Antipolis.