Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

Indexed squares

We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1-extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square. First author partially supported by NSF grants DMS-9703945 and DMS-0070549. Second author partially supported by NSF Grants DMS-9305990, DMS-9712580, DMS-9996280 and DMS-0088948.     

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.     

Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type

We prove the hypoellipticity for systems of H?rmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set. Received October 23, 2000 / final version received February 5, 2002?Published online May 15, 2002 The work of the first author was partially supported by NSF Grant No. DMS-9800794 The work of the second author was partially supported by NSF Grants No. DMS-9706892 and DMS-0070492     

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

Area minimizers in a K3 surface and holomorphicity

No Abstract. . Received: October 2004 Revision: May 2005 Accepted: July 2005 The second author was partially supported by NSF grant DMS-0304587.     

Poincaré domains and quasiconformal mappings

The second author is partially supported by NSF (Grant DMS-9305742).     

Complete Hartogs domains in C2 have regular Bergman and Szegő projections

The first author partially supported by NSF grant DMS-8701038     

Manifolds with quadratic curvature decay and fast volume growth

 We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms. Received: 21 March 2001 / Revised version: 21 March 2002 / Published online: 10 February 2003 Research partially supported by NSF grant DMS-0072154 and MSRI     

Cartan determinants and Shapovalov forms

We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of p-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity. Received: 20 November 2001 / Revised version: 7 February 2002 / Published online: 27 June 2002 Second author partially supported by the NSF (grant no. DMS-9900134).     

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     

Flexible piecewise approximations based on partition of unity

In this paper, we study a flexible piecewise approximation technique based on the use of the idea of the partition of unity. The approximations are piecewisely defined, globally smooth up to any order, enjoy polynomial reproducing conditions, and satisfy nodal interpolation conditions for function values and derivatives of any order. We present various properties of the approximations, that are desirable properties for optimal order convergence in solving boundary value problems. AMS subject classification 65N30, 65D05Weimin Han: Corresponding author. The work of this author was partially supported by NSF under grant DMS-0106781.Wing Kam Liu: The work of this author was supported by NSF.     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments. The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed. R. Jones is partially supported by NSF Grant DMS-9531526. J. Rosenblatt is partially supported by NSF Grant DMS-9705228. M. Wierdl is partially supported by NSF Grant DMS-9801602.     

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.     

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based. The first author was partially supported by the grant # 201/02/0594 of the Grant Agency of the Czech Republic, and by the Institutional grant MSM0021620839; the second author was partially supported by the grant No. Tn37877 of the Hungarian National Foundation for Scientific Research (OTHA); the third author was supported by the NSF grant # DMS-9971352.     

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index.?In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits. Oblatum 11-V-2001 & 13-XI-2002?Published online: 24 February 2003 RID="*" ID="*"The first author was supported by NSF DMS-9971629 and NSF DMS-0134408. The second author was supported by an EPSRC Fellowship. The third author was supported by NWO Vidi-grant 639.032.202.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Model completeness for trivial, uncountably categorical theories of Morley rank 1

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.The second author was partially supported by NSF grant DMS-0300080.