Pencils and infinite dihedral covers of

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve , infinite dihedral covers, and pencils of curves containing .

     

-actions with -dimensional fixed point set

We describe the equivariant cobordism classification of smooth actions of the group , considered as the group generated by two commuting involutions, on closed smooth -dimensional manifolds , for which the fixed point set of the action is a connected manifold of dimension and or . For , the classification is known.

     

Polynomials with roots in for all

Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of .

     

Equipartitions of measures in

A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) on there exist hyperplanes dividing into parts of equal measure. It is known that the answer is positive in dimension (see H. Hadwiger (1966)) and negative for (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grünbaum's problem in the critical dimension by proving that each measure in admits an equipartition by hyperplanes, provided that it is symmetric with respect to a -dimensional affine subspace of . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension ; see G. C. Tootill (1956) and D. E. Knuth (2001).

     

Admissible metrics in the -Yamabe equation

In most previous works on the existence of solutions to the -Yamabe problem, one assumes that the initial metric is -admissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.

     

Wavelet multipliers on

We give results on the boundedness and compactness of wavelet multipliers on .

     

Splicing and the Casson invariant

We establish a formula for the Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the Casson invariant vanishes for spliced sums along knots in .

     

Convolution and restriction estimates for measures on curves in

We study convolution and Fourier restriction estimates for some degenerate curves in .

     

Symmetric Markov chains on with unbounded range

We consider symmetric Markov chains on where we do not assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on corresponding to an elliptic operator in divergence form.

     

A Calabi-Yau threefold with Brauer group

We compute the Brauer group of a Calabi-Yau threefold discovered by the first author and Sorin Popescu, and find it is , the largest known Brauer group of a non-singular Calabi-Yau threefold.

     

Spaces between and

In this paper we consider the spaces that lie between and . We discuss their interpolation properties and the behavior of maximal functions and singular integrals acting on them.

     

On Generic differential -extensions

Let be an algebraically closed field with trivial derivation and let denote the differential rational field , with , , , , differentially independent indeterminates over . We show that there is a Picard-Vessiot extension for a matrix equation , with differential Galois group , with the property that if is any differential field with field of constants , then there is a Picard-Vessiot extension with differential Galois group if and only if there are with well defined and the equation giving rise to the extension .

     

Orbit equivalence for Cantor minimal -systems

We show that every minimal, free action of the group on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, -actions and -actions.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

A new construction of -manifolds

This paper provides a topological method to construct all simply-connected, spin, smooth -manifolds with torsion-free homology using simply-connected, smooth -manifolds as building blocks. We explicitly determine the invariants that classify these -manifolds from the intersection form and specific homology classes of the -manifold building blocks.

     

A tensor norm preserving unconditionality for -spaces

We show that, for each , there is an -tensor norm (in the sense of Grothendieck) with the surprising property that the -tensor product has local unconditional structure for each choice of arbitrary -spaces . In fact, is the tensor norm associated to the ideal of multiple -summing -linear forms on Banach spaces.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

A Solomon descent theory for the wreath products

We propose an analogue of Solomon's descent theory for the case of a wreath product , where is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, and Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

     

Symplectic , subgroup separability, and vanishing Thurston norm

Let be a closed, oriented -manifold. A folklore conjecture states that admits a symplectic structure if and only if admits a fibration over the circle. We will prove this conjecture in the case when is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes -manifolds with vanishing Thurston norm, graph manifolds and -manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic -manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the -manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.