Induction theorems on the stable rationality of the center of the ring of generic matrices

Following Procesi and Formanek, the center of the division ring of generic matrices over the complex numbers is stably equivalent to the fixed field under the action of , of the function field of the group algebra of a -lattice, denoted by . We study the question of the stable rationality of the center over the complex numbers when is a prime, in this module theoretic setting. Let be the normalizer of an -sylow subgroup of . Let be a -lattice. We show that under certain conditions on , induction-restriction from to does not affect the stable type of the corresponding field. In particular, and are stably isomorphic and the isomorphism preserves the -action. We further reduce the problem to the study of the localization of at the prime ; all other primes behave well. We also present new simple proofs for the stable rationality of over , in the cases and .

     

On transversality with deficiency and a conjecture of Sard

Let be a map between manifolds and a manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of maps which satisfy a certain transversality condition with respect to in a weak sense. As an application, by considering the case where is a point, we obtain some new results about the topological properties of , where is the set of points of where the rank of the differential of is less than or equal to . In particular, we show a result about the topological dimension of , which is closely related to a conjecture of Sard concerning the Hausdorff measure of .

     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

     

Trace on the boundary for solutions of nonlinear differential equations

Let be a second order elliptic differential operator in with no zero order terms and let be a bounded domain in with smooth boundary . We say that a function is -harmonic if in . Every positive -harmonic function has a unique representation

where is the Poisson kernel for and is a finite measure on . We call the trace of on . Our objective is to investigate positive solutions of a nonlinear equation

for [the restriction is imposed because our main tool is the -superdiffusion which is not defined for ]. We associate with every solution a pair , where is a closed subset of and is a Radon measure on . We call the trace of on . is empty if and only if is dominated by an -harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.

     

Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Let be a second order elliptic differential operator on a Riemannian manifold with no zero order terms. We say that a function is -harmonic if . Every positive -harmonic function has a unique representation

where is the Martin kernel, is the Martin boundary and is a finite measure on concentrated on the minimal part of . We call the trace of on . Our objective is to investigate positive solutions of a nonlinear equation

for [the restriction is imposed because our main tool is the -superdiffusion, which is not defined for ]. We associate with every solution of (*) a pair , where is a closed subset of and is a Radon measure on . We call the trace of on . is empty if and only if is dominated by an -harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when is a second order elliptic differential operator in and is a bounded smooth domain in . We obtained necessary and sufficient conditions for a pair to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators with bounded coefficients in an arbitrary bounded domain of , assuming only that the Martin boundary and the geometric boundary coincide.

     

Universal maps on trees

A map of continua and is called a universal map from to if for any map , for some point . When and are trees, we characterize universal maps by reducing to the case of light minimal universal maps. The characterization uses the notions of combinatorial map and folded subedge of .

     

A -deformation of a trivial symmetric group action

Let be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system . Let be the Varchenko matrix for this arrangement with all hyperplane parameters equal to . We show that is the matrix with rows and columns indexed by permutations with entry equal to where is the number of inversions of . Equivalently is the matrix for left multiplication on by

Clearly commutes with the right-regular action of on . A general theorem of Varchenko applied in this special case shows that is singular exactly when is a root of for some between and . In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the -module structure of the nullspace of in the case that is singular. Our first result is that

in the case that where Lie denotes the multilinear part of the free Lie algebra with generators. Our second result gives an elegant formula for the determinant of restricted to the virtual -module with characteristic the power sum symmetric function .

     

Baire and

Let be a locally compact Hausdorff space and let be the Banach space of all bounded complex Radon measures on . Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called Baire sets of and those of are called -Borel sets of (since they are precisely the -bounded Borel sets of ). Identifying with the Banach space of all Borel regular complex measures on , in this note we characterize weakly compact subsets of in terms of the Baire and -Borel restrictions of the members of . These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

     

Test ideals in quotients of -finite regular local rings

Let be an -finite regular local ring and an ideal contained in . Let . Fedder proved that is -pure if and only if . We have noted a new proof for his criterion, along with showing that , where is the pullback of the test ideal for . Combining the the -purity criterion and the above result we see that if is -pure then is also -pure. In fact, we can form a filtration of , that stabilizes such that each is -pure and its test ideal is . To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let , where is either a polynomial or a power series ring and is generated by monomials and the are regular. Set . Then .

     

The metric projection onto the soul

We study geometric properties of the metric projection of an open manifold with nonnegative sectional curvature onto a soul . is shown to be up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of , and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of .

     

Characterizations of weakly compact operators on

Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

     

A construction of homologically area minimizing hypersurfaces with higher dimensional singular sets

We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if is smooth, compact dimensional manifold, , and if is an embedded, orientable submanifold of dimension , then we construct metrics on such that the homologically area minimizing hypersurface , homologous to , has a singular set equal to a prescribed number of spheres and tori of codimension less than . Near each component of the singular set, is isometric to a product , where is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.

     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

Embeddings in generalized manifolds

We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

     

The Stable Homotopy Types of Stunted Lens Spaces mod

Let be the mod stunted lens space . Let denote the exponent of in , and the number of integers satisfying , and . In this paper we complete the classification of the stable homotopy types of mod stunted lens spaces. The main result (Theorem 1.3 (i)) is that, under some appropriate conditions, and are stably equivalent iff , where or .

     

Lower bounds for dimensions of representation varieties

The set of -dimensional complex representations of a finitely generated group form a complex affine variety . Suppose that is such a representation and consider the associated representation on complex matrices obtained by following with conjugation of matrices. Then it is shown that the dimension of at is at least the difference of the complex dimensions of and . It is further shown that in the latter cohomology may be replaced by various proalgebraic groups associated to and .

     

The structure of indecomposable injectives in generic representation theory

This paper considers the structure of the injective objects in the category of functors between -vector spaces. A co-Weyl object is defined, for each simple functor in . A functor is defined to be -good if it admits a finite filtration of which the quotients are co-Weyl objects. Properties of -good functors are considered and it is shown that the indecomposable injectives in are -good. A finiteness result for proper sub-functors of co-Weyl objects is proven, using the polynomial filtration of the shift functor . This research is motivated by the Artinian conjecture due to Kuhn, Lannes and Schwartz.