On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

boundedness of discrete singular Radon transforms

We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator

extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

     

Links in an open book decomposition and in the standard contact structure

We study a relationship between arc presentations of links in and Legendrian links in with the standard tight contact structure. We determine the arc indices of torus knots.

     

Fuglede's conjecture fails in dimension 4

In this note we modify a recent example of Tao and give an example of a set such that admits an orthonormal basis of exponentials for some set , but which does not tile by translations. This shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.

     

The Kowalevski top as a reduction of a Hamiltonian system on

We show that the Kowalevski top and Kowalevski gyrostat are obtained as a reduction of a Hamiltonian system on . Therefore the Lax-pair representations for the Kowalevski top and Kowalevski gyrostat are obtained via a direct method by transforming the canonical Lax-pair representation of a system on . Also we show that the nontrivial integral of motion of the Kowalevski top comes from a Casimir function of the Lie-Poisson algebra .

     

The size of the Dini subdifferential

Given a lower semicontinuous function , we prove that the points of , where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on . Consequently, the set of all these points has a null Lebesgue measure.

     

Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphere correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are , where , with . When the ambient space is the four-sphere , the regular homotopy class of immersions of the two-sphere is determined by the self-intersection number ; here we shall prove that the possible critical values are , where . Moreover, if , the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration , from a rational curve in and, if , via stereographic projection, from a minimal surface in with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some or (equivalently) when the minimal surface of is complex with respect to a suitable complex structure of .

     

Gradient ranges of bumps on the plane

For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

     

On unboundedness of maximal operators for directional Hilbert transforms

We show that for any infinite set of unit vectors in the maximal operator defined by

is not bounded in .

     

A uniform Fourier restriction theorem for surfaces in

We establish a uniform Fourier restriction estimate for certain hypersurfaces in .

     

Rigidity of some Weyl manifolds with nonpositive sectional curvature

We provide a list of all locally metric Weyl connections with nonpositive sectional curvatures on two types of manifolds, -dimensional tori and with the standard conformal structures. For we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. In the case of we prove the same in the more narrow class of integrable connections.

     

Porous measures on : Local structure and dimensional properties

We study dimensional properties of porous measures on . As a corollary of a theorem describing the local structure of nearly uniformly porous measures we prove that the packing dimension of any Radon measure on has an upper bound depending on porosity. This upper bound tends to as porosity tends to its maximum value.

     

Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations

We consider divergence form elliptic operators , defined in , where the coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in , then for any vector-valued we have the bilinear estimate

where and where is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients and generalizes an analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. We also identify the domain of the generator of the Poisson semigroup for the equation in

     

Exact multiplicity result for the perturbed scalar curvature problem in

Let denote the closure of in the norm Let and define the constants and Let We consider the following problem for

We show an exact multiplicity result for for all small .

     

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space . For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone , we construct a unique ``symbol valued trace', which extends the -trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space . Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over . The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

Comments about the Steinhaus tiling problem

Recently, using Fourier transform methods, it was shown that there is no measurable Steinhaus set in , a set which no matter how translated and rotated contains exactly one integer lattice point. Here, we show that this argument cannot generalize to any lattice and, on the other hand, give some lattices to which this method applies. We also show there is no measurable Steinhaus set for a special honeycomb lattice, the standard tetrahedral lattice in

     

Quantum cohomology of Hilb and enumerative applications

We compute the small quantum cohomology of Hilb and determine recursively most of the big quantum cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in . This extends the results obtained by Graber (2001) for Hilb and hyperelliptic curves in .

     

A topological property of integrable systems

If we are given real-valued smooth functions on which are in involution, then, under some mild hypotheses, the subset of where these functions are linearly independent is not simply connected.

     

Toric hypersymplectic quotients

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space by the action of a compact Abelian group. These -dimensional quotients carry a multi-Hamilitonian action of an -torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in . We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on and metrics on complements of hypersurfaces in compact manifolds.