A compactification of the moduli space of polynomials

In this paper, we introduce a compactification of the moduli space of polynomial maps with a fixed degree such that the map from it to defined by using the elementary symmetric functions of multipliers at fixed points is a continuous surjection.

     

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 .

     

Distinguishing embedded curves in rational complex surfaces

We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces with the property that no self-diffeomorphism of sends one to the other. In particular, as a special case we answer a question originally posed by R. Gompf (1995) concerning genus two curves of self-intersection number 0 in .

     

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.

     

Affine pseudo-planes and cancellation problem

We define affine pseudo-planes as one class of -homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line . Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.

     

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 .

     

A new geometric proof of Jung's theorem on factorisation of automorphisms of

Building up on the classical theory of algebraic surfaces and their birational transformations we prove Jung's theorem on factorisation of automorphisms of reducing it to a simple combinatorial argument.

     

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.

     

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.

     

Traces of convex domains

Diederich and Ohsawa proved that in there exists a locally hyperconvex, Stein open subset which is not hyperconvex. In this paper we generalize their results.

     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

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 .

     

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.

     

On the Castelnuovo-Mumford regularity of connected curves

In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in having maximal regularity, but no extremal secants. We also show that any connected curve in of degree at least 5 with maximal regularity and no linear components has an extremal secant.

     

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of , and the orbifold quantum cohomology of the symmetric product of . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

     

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map and modify it into a nonharmonic biharmonic map . We show to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

     

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

Uniqueness of exceptional singular quartics

We prove that given a general collection of 14 points of ( an infinite field) there is a unique quartic hypersurface that is singular on .

This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of that are singular on a collection of general points.

     

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

     

On a class of special linear systems of

In this paper we deal with linear systems of through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.