Homology of generalized Steinberg varieties and Weyl group invariants

Let be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and anti-invariants of the cohomology of the Springer fibres of the cone of nilpotent elements, , of for the Steinberg variety of triples.

Using a general specialization argument we show that for a parabolic subgroup of the space of -invariants and the space of -anti-invariants of are isomorphic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced by Douglass and Röhrle (2004).

The rational group algebra of the Weyl group of is isomorphic to the opposite of the top Borel-Moore homology of , where . Suppose is a parabolic subgroup of . We show that the space of -invariants of is , where is the idempotent in the group algebra of affording the trivial representation of and is defined similarly. We also show that the space of -anti-invariants of is , where is the idempotent in the group algebra of affording the sign representation of and is defined similarly.

     

Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Let be a differentiably simple Noetherian commutative ring of characteristic (then is local with ). A short proof is given of the Theorem of Harper (1961) on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation of the ring such that if , then for some . The derivation is given explicitly, and it is unique up to the action of the group of ring automorphisms of . Let be the set of all such derivations. Then . The proof is based on existence and uniqueness of an iterative -descent (for each ), i.e., a sequence in such that , and for all . For each , and .

     

Fixed point property and the Fourier algebra of a locally compact group

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) spaces and use this for the Fourier algebra of a locally compact group In particular we show that if is an IN-group, then has the weak fpp if and only if is compact. We also show that if is any locally compact group, then has the fixed point property (fpp) if and only if is finite. Furthermore if a nonzero closed ideal of has the fpp, then must be discrete.

     

The growth of iterates of multivariate generating functions

The vector-valued function of a -vector has components . For each , is the (multivariate) Laplace transform of a discrete measure concentrated on with only a finite number of atoms. The main objective is to give conditions for the functional iterates of to grow like for a suitable . The initial stimulus was provided by results of Miller and O'Sullivan (1992) on enumeration issues in `context free languages', results which can be improved using the theory developed here. The theory also allows certain results in Jones (2004) on multitype branching to be proved under significantly weaker conditions.

     

Bounded -calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

Operators of the form with a pseudodifferential symbol belonging to the Hörmander class , , , and certain perturbations are shown to possess a bounded -calculus in Besov-Triebel-Lizorkin and certain subspaces of Hölder spaces, provided is suitably elliptic. Applications concern pseudodifferential operators with mildly regular symbols and operators on manifolds of low regularity. An example is the Dirichlet-Neumann operator for a compact domain with -boundary.

     

Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function , the number of bound states of the operator in below . Here is a bounded potential behaving asymptotically like where is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator on the sphere has negative eigenvalues less than , we prove that may be estimated as

Thus, in particular, if there are no such negative eigenvalues, then has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that and that there is exactly one eigenvalue less than , with all others , we show that the negative spectrum is asymptotic to a geometric progression with ratio .

     

Nonabelian cohomology with coefficients in Lie groups

In this paper we prove some properties of the nonabelian cohomology of a group with coefficients in a connected Lie group . When is finite, we show that for every -submodule of which is a maximal compact subgroup of , the canonical map is bijective. In this case we also show that is always finite. When and is compact, we show that for every maximal torus of the identity component of the group of invariants , is surjective if and only if the -action on is -semisimple, which is also equivalent to the fact that all fibers of are finite. When , we show that is always surjective, where is a maximal compact torus of the identity component of . When is cyclic, we also interpret some properties of in terms of twisted conjugate actions of .

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

On the Clifford algebra of a binary form

The Clifford algebra of a binary form of degree is the -algebra , where is the ideal generated by . has a natural homomorphic image that is a rank Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit -divisor in , where is the curve and is the genus of .

     

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with being a critical frequency in the sense that We show that if the zero set of has isolated connected components such that the interior of is not empty and is smooth, has isolated zero points, , , and has critical points such that , then for small, there exists a standing wave solution which is trapped in a neighborhood of Moreover the amplitudes of the standing wave around , and are of a different order of . This type of multi-scale solution has never before been obtained.

     

Bimodules and -rationality of vertex operator algebras

This paper studies the twisted representations of vertex operator algebras. Let be a vertex operator algebra and an automorphism of of finite order For any , an - -bimodule is constructed. The collection of these bimodules determines any admissible -twisted -module completely. A Verma type admissible -twisted -module is constructed naturally from any -module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra is -rational if and only if its twisted associative algebra is semisimple and each irreducible admissible -twisted -module is ordinary.

     

Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space with and . On the scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation:

for , , and . We deduce global well-posedness for , and real valued initial data.

     

Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.

     

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.

     

Prime specialization in genus 0

For a prime polynomial , a classical conjecture predicts how often has prime values. For a finite field and a prime polynomial , the natural analogue of this conjecture (a prediction for how often takes prime values on ) is not generally true when is a polynomial in ( the characteristic of ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values as varies. We prove the surprising fact that this ``Möbius average,' which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve .

The periodic Möbius average behavior implies in specific examples that a polynomial in does not take prime values as often as analogies with suggest, and it leads to a modified conjecture for how often prime values occur.

     

Profinite and pro- completions of Poincaré duality groups of dimension 3

We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic).

We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .

     

Structural interactions of conjugacy closed loops

We study conjugacy closed loops by means of their multiplication groups. Let be a conjugacy closed loop, its nucleus, the associator subloop, and and the left and right multiplication groups, respectively. Put . We prove that the cosets of agree with orbits of , that and that one can define an abelian group on . We also explain why the study of finite conjugacy closed loops can be restricted to the case of nilpotent. Group is shown to be a subgroup of a power of (which is abelian), and we prove that can be embedded into . Finally, we describe all conjugacy closed loops of order .

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.