Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs (), (), , , and () and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift of every non-projective simple module is isomorphic to its third syzygy .

     

The flat model structure on complexes of sheaves

Let be the category of chain complexes of -modules on a topological space (where is a sheaf of rings on ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on . As a corollary, we have a general framework for doing homological algebra in the category of -modules. I.e., we have a natural way to define the functors and in .

     

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Maximal families of Gorenstein algebras

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring . Let be the scheme parametrizing graded quotients of with Hilbert function . We prove there is a close relationship between the irreducible components of , whose general member is a Gorenstein codimension quotient, and the irreducible components of , whose general member is a codimension Cohen-Macaulay algebra of Hilbert function related to . If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of , this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay' components of to ``Gorenstein' components of , in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay' component and a sum of two invariants of . Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.

     

Varieties with small discriminant variety

Let be a smooth complex projective variety, let be an ample and spanned line bundle on , defining a morphism and let be its discriminant locus, the variety parameterizing the singular elements of . We present two bounds on the dimension of and its main component relying on the geometry of . Classification results for triplets reaching the bounds as well as significant examples are provided.

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

Uniform asymptotics for Jacobi polynomials with varying large negative parameters-- a Riemann-Hilbert approach

An asymptotic expansion is derived for the Jacobi polynomials with varying parameters and , where and are constants. Our expansion is uniformly valid in the upper half-plane . A corresponding expansion is also given for the lower half-plane . Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve , which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of , and tend to as .

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

Open loci of graded modules

Let be an excellent homogeneous Noetherian graded ring and let be a finitely generated graded -module. We consider as a module over and show that the -loci of are open in . In particular, the Cohen-Macaulay locus    is Cohen-Macaulay is an open subset of . We also show that the -loci on the homogeneous parts of are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module over an excellent ring and for an ideal which is not contained in any minimal prime of , the -loci for the modules are eventually stable.

     

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the -dimensional Euclidean space is said to be uniformly dense in a surface of codimension if, for every small the volume of the intersection of with a ball of radius and center does not depend on for

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary , and we show that the principal curvatures of satisfy certain identities.

The case in which the surface coincides with is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if , it must be either a circle or a straight line and (ii) if it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

     

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let

and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function on the plane and a non-zero define the Hilbert transform of in the direction to be

p.v.

Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

     

-estimates for the linearized Monge-Ampère equation

Let be a strictly convex domain and let be a convex function such that    det in . The linearized Monge-Ampère equation is

where det is the matrix of cofactors of . We prove that there exist and depending only on , and such that

for all solutions to the equation .

     

Approximation and regularization of Lipschitz functions: Convergence of the gradients

We examine the possible extensions to the Lipschitzian setting of the classical result on -convergence: first (approximation), if a sequence of functions of class from to converges uniformly to a function of class , then the gradient of is a limit of gradients of in the sense that ; second (regularization), the functions can be chosen to be of class and -converging to in the sense that . In other words, the space of functions is dense in the space of functions endowed with the pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if is a locally Lipschitz function, we build a sequence of smooth functions such that

In other words, the space of functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality , which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

     

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.

     

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let be a bigraded ideal in the bigraded polynomial ring . Assume that has codimension 2. Then is a finite set of points. We prove that if is a local complete intersection, then any syzygy of the vanishing at , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

     

Layers and spikes in non-homogeneous bistable reaction-diffusion equations

We study , where , , and is sufficiently small, on an interval with boundary conditions at . All solutions with an independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to with and also to an infinite interval.