Singularities of admissible normal functions

In a recent paper, M. Green and P. Griffiths used R. Thomas’ work on nodal hypersurfaces to sketch a proof of the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using Morihiko Saito’s mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest. P. Brosnan’s research was supported in part by an NSERC discovery grant. H. Fang’s research was supported in part by NSF grant number DMS 0606721. G. Pearlstein’s research was supported in part by NSF grant number DMS 0703956. N. Fakhruddin School of Mathematics, Tata Institue of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: naf@math.tifr.res.in     

Non-amenable finitely presented torsion-by-cyclic groups

Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x,y] ⁿ = 1. Manuscrit reĉu le 8 février 2001. RID="*" ID="*"Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 99-01-00894 and by the INTAS grant, the research of the second author was supported in part by the NSF grant DMS 9978802.     

Nevanlinna-Pick Interpolation for {\mathbb{C}} + BH^\infty

We study the Nevanlinna-Pick problem for a class of subalgebras of H ^∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ^∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C ^*-envelopes. This research was partially supported by the NSF grant DMS 0300128. This research was completed as part of my Ph.D. dissertation at the University of Houston.     

On the codimensions of the verbally prime P.I. algebras

Razmyslov’s theory of trace identities for the prime P.I. algebrasM _{k, l} is applied to give bounds for the cocharacters and the codimensions of these algebrasM _{k, l}, as well as for the matrix algebrasM _k(E) over the Grassmann algebraE. These bounds are easier to obtain and are better (tighter) than earlier obtained bounds. Work supported by NSF grant DMS 9100258. Work supported by NSF grant DMS 9101488.     

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.     

Tiny models of categorical theories

Summary We explore the existence and the size of infinite models of categorical theories having cardinality less than the size of the associated Tarski-Lindenbaum algebra. Restricting to totally transcendental, categorical theories we show that Every tiny model is countable is independent of ZFC. IfT is trivial there is at most one tiny model, which must be the algebraic closure of the empty set. We give a new proof that there are no tiny models ifT is not totally transcendental and is non-trivial.Research supported by NSF grant DMS #8505550Research partially supported by NSF grant DMS #8505550Research partially supported by NSF grant DMS #8505550 and the Deutsche Forschungsgemeinschaft. This author would like to thank Eli Danbru at Paris for helpful discussions     

The domestic levels ofK c are iterable

We show that the models produced by theK ^c construction before (if ever) it reaches a non-domestic premouse are all iterable. As a corollary we get thatPFA plus the existence of a measurable cardinal implies the existence of a non-domestic premouse. Partially supported by the Italian MURST. Research partially supported by NSF grant DMS 98-03292. Research partially supported by NSF grant DMS 98-03611.     

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ ₁ ∪…∪J _r andJ ₁ …J _r, where eachJ _i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer. Partially supported by NSF grant DMS 9303230. Partially supported by NSF grant DMS 9101488.     

Examples of expandingC 1 maps having no σ-finite invariant measure equivalent to Lebesgue

In this paper we construct aC ¹ expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on higher dimensional tori and the Riemann sphere. We also discuss recurrence of Lebesgue measure for the family of tent maps. Supported by the Deutsche Forschungsgemeinschaft (DFG). The research was carried out while HB was employed at the University of Erlangen-Nürnberg, Germany. Partially supported by NSF grant DMS # 9203489.     

Ehrhart polynomials and stringy Betti numbers

We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes. The first author was partially supported by NSF grant DMS 0500127 and the second author was supported by a Graduate Research Fellowship from the NSF     

A class of real cocycles having an analytic coboundary modification

We define here a certain class of procedures (a.a.c.c.p.) for constructing real valued cocycles over irrational rotations. Each such procedure is realizable over a residual set of possible rotations, and we prove that each such cocycle is cohomologous to a real analytic cocycle. The procedure in Section 3 of [10] is seen to be of this type and hence not only is cohomologous toC ^∞ as is shown there, but is actually cohomologous to a real analytic cocycle. We also show that following the method of [6] a procedure can be given to obtain rank-1 Anzai skew products of mixed spectral type that are real analytic. Research supported by KBN grant 512/2/91. Research supported by KBN grant 512/2/91. Research supported by NSF grant DMS 01524351.     

On formulas of Ramanujan and Evans

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank. Research partially supported by NSF grant DMS 99-70865. 2000 Mathematics Subject Classification Primary—39A70, 11P83; Secondary—33C20     

The group SK1 for simple algebras

We prove that the reduced Whitehead group SK ₁(A) is generically nontrival for any central simple algebra A of index divisible by 4. This work has been supported by the NSF grant DMS #0355166.     

Orbit cardinals: On the effective cardinalities arising as quotient spaces of the formX/G whereG acts on a Polish spaceX

We prove an Ulm-type classification theorem for actions inL(ℝ), thereby answering a question of Becker and Kechris, and investigate the effective cardinalities which can be induced by various classes of Polish groups. Research partially supported by NSF grant DMS 96-22977.     

Infinite joins that are finitely join-irreducible

Complete lattices are studied which contain an element u which is not the join of a finite set of smaller elements, but is the join of all elements <u.This work was done while the first author was partly supported by NSF contract DMS 85-02330; during the completion of the article the second author was partly supported by NSF grant DMS 88-07043.     

Bessel identities in the waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL ₂(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL ₂(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. Research of first author was partially supported by NSF grant DMS-0070762. Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

The space of surface group representations

In this note we prove that the number of irreducible components of Hom (π,G) is the same as π₁(G), where π is a surface group andG is complex semisimple. This is established by studying the flat bundles on Riemann surfaces. The present work is partially supported by NSF grant DMS89-04922     

Crystalline cohomology and GL(2, ℚ)

This paper applies recent advances in crystalline cohomology to the classical case of open elliptic modular curves. In so doing control is gained over the action of inertia in the Galois representations attached to modular forms. Our aim is to study the modular Galois representations attached to automorphic forms modp of weightk≥2. We generalize to higher weightk several results which were previously accessible only in the case of weight 2 where jacobian varieties can be invoked. Additionally we reconsider Gross’s theorem on companion forms in a crystalline context. Partially supported by NSF grant DMS 90-02744. Partially supported by NSA grant MDA904-90-H-4020 and by a PSC-CUNY grant.