Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

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     

A measurable-group-theoretic solution to von Neumann’s problem

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors. D. Gaboriau’s research was supported by CNRS. R. Lyons’ research was supported partially by NSF grant DMS-0705518 and Microsoft Research.     

Averaging Special Values of Dirichlet <Emphasis Type="Italic">L</Emphasis>-Series

In this paper we derive estimates for weighted averages of the special values of Dirichlet L-series which generalize similar estimates of David and Pappalardi [1]. The author is partially supported by NSF grant DMS-0090117. 2000 Mathematics Subject Classification: Primary–11M06; Secondary–11G05     

Optimal transport and Perelman’s reduced volume

We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume. This research was partially supported by NSF grant DMS-0604829.     

On the index of fractions with square-free denominators in arithmetic progressions

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

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.     

An Extension of Schreier Unconditionality

The main result of the paper extends the classical result of E. Odell on Schreier unconditionality to arrays in Banach spaces. An application is given on the “multiple of the inclusion plus compact" problem which is further applied to a hereditarily indecomposable Banach space constructed by N. Dew. The present paper is part of the Ph.D thesis of the second author who is partially supported under Prof. Girardi’s NSF grant DMS-0306750.     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ _n arrangements. The first author was supported by NSF grant DMS-0500638. The second author was supported by NSF grant DMS-0245623.     

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

δ- and θ-modifications of generalized topologies

It is shown that the theory of δ-and θ-modifications of topological spaces (see [3]) can be generalized for the case when the topology is replaced by a generalized topology in the sense of [1]. Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.     

Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset. T. Braden’s research was supported in part by NSF grant DMS-0201823. N. Proudfoot’s research was supported in part by an NSF Postdoctoral Research Fellowship and NSF grant DMS-0738335.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far. Parts of this work have been done while the second and the third author had been enjoying the hospitality of the Department of Mathematics of the University of Pittsburgh. P.H. was supported by NSF grant DMS-0500966. P.S. was partially supported by the MNiSzW grant no 1 PO 3A 005 29. X.Z. was supported by the Academy of Finland, project 207288.     

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.     

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre. N. Garofalo is supported in part by NSF grant DMS-0701001. A. Petrosyan is supported in part by NSF grant DMS-0701015.     

Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon–Nikodym property, Clarke’s generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set. Research of the first author is supported by the Hungarian Scientific Research Fund (OKTA) under grant K62316. Research of the second author is supported by the National Science Foundation under grant DMS-0306260.