We announce Hodge theoretic formulae of Atiyah-Meyer type for genera and characteristic classes of complex algebraic varieties. Our results are formulated in terms of the generalized (motivic) Hirzebruch characteristic classes, and the arguments used in the proofs rely in an essential way on Saito’s theory of algebraic mixed Hodge modules.
We consider an ergodic invariant measure mu for a smooth action alpha of Z(k), k >= 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R-k, k >= 2, on a (2k + 1)-dimensional manifold. If mu is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z(k) has positive entropy, then mu is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups. We also consider actions on the torus T-N with induced action on the first homology corresponding to a finite index subgroup of a maximal semisimple abelian subgroup of SL(N, R). Such an action has a unique invariant measure, called large measure, which projects to the Lebesgue measure under the semiconjugacy with the linear action and this measure is absolutely continuous. Finally, we consider cocycles over an action on the torus with Cartan homotopy data. Every cocycle which is Holder with respect to a Lyapunov Riemannian metric a.e. for the large invariant measure is cohomologous to a constant cocycle via a Lyapunov-Holder transfer function.
In this note we give a modification theorem for a compact homogeneous solvmanifold such that a certain Mostow type condition will be satisfied. An application of this result is a simpler way to calculate the cohomology groups of compact quotients of real solvable Lie group over a cocompact discrete subgroup. Furthermore, we apply the second result to obtain a splitting theorem for compact complex homogeneous manifolds with symplectic structures. In particular, we are able to classify compact complex homogeneous spaces with pseudo-Kählerian structures.
We show the global existence of weak solutions for a free-boundary problem arising in the non-isothermal crystallization of polymers. In particular, the free interface is shown to be of codimension one for every time t in two space dimensions; Holder continuity of the temperature u is proven.
The space of smooth genus 0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such natural smooth model is expected, as the space satisfies ``Murphy's Law''. In genus 1, however, the situation remains beautiful. We give a natural smooth compactification of the space of elliptic curves in projective space, and describe some of its properties. This space is a blow up of the space of stable maps. It can be interpreted as blowing up the most singular locus first, then the next most singular, and so on, but with a twist -- these loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and Gromov-Witten theory. The proof that this construction indeed gives a desingularization will appear in math.AG/0603353v2 (currently under revision).
In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.
The set of x 2-invariant measures can be equipped with the partial order of majorization, describing relative dispersion. The minimal elements for this order are precisely the Sturmian measures of Morse and Hedlund. This yields new characterisations of Sturmian measures, and has applications to the ergodic optimization of convex functions.
In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.
Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M of dimension n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of M. But what can be said about the length of the other geodesics? We conjecture that for every k there are k geodesics between x and y of length not exceeding kd, where d denotes the diameter of M.This conjecture is obviously true for round spheres and it is not difficult to prove it for all closed Riemannian manifolds with non-trivial torsion-free fundamental groups. In this paper we announce two further results in the direction of this conjecture. Our first result is that the length of the second shortest geodesic between x and y does not exceed 2nd. Our second result is that if n=2 and M is diffeomorphic to the two-dimensional sphere, then for every k every two points on M can be connected by k geodesics of length not exceeding (k^2/2 + 3k/2 +2)d.
In this research announcement we present a new q q -analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q q -Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q q -Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker; (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev, and Lunts; (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville, and Vaughan and by Dollhopf, Goulden, and Greene; and (4) Stanley’s chromatic symmetric functions.
In this paper we give necessary and sufficient conditions for the representations of quadratic lattices over arbitrary dyadic fields. Our result is given in terms of Bases of Norm Generators (BONGs, for short). However, they can be translated in terms of the more traditional Jordan decompositions.
This document describes the authors' current research project: the evaluation of a tower of Rankin-Selberg integrals on the group E_6. We recall the notion of a tower, and two known towers, making observations about how the integrals within a tower may be related to one another via formal manipulations, and offering a heuristic for how the L-functions should be related to one another when the integrals are related in this way. A detailed description of the E_6 tower is then given.
We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries.
We give a very short self-contained combinatorial proof of the Babson-Kozlov conjecture, by presenting a cochain whose coboundary is the desired power of the characteristic class.
We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic p p -Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear p p -Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.
We introduce the notion of entropy pseudonorm for an action of R-n and prove that it vanishes for the group actions associated with a large class of integrable Hamiltonian systems.
In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π1(M). The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G 'almost' have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings.
We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.
