Let L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r), ), where L is totally real, r < 0 is odd and is a maximal order containing ℤ[]G, and will also provide some evidence for our conjecture.

We give a classification of all the cycle types occurring in the automorphism groups of countable homogeneous graphs.

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S•(V) and the exterior algebra ∧•(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S•(V) ⊗ ∧•(V), twisted by an arbitrary determinant power.

Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H∗(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H∗(Xn) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of Sn corresponding to the partition (n/2, n/2).

We use the Jones–Wenzl idempotents to construct a basis of the Temperley–Lieb algebra TLn. This allows a short calculation for a Gram determinant of Lickorish's bilinear form on the Temperley–Lieb algebra.

We study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as ‘Rees criteria’. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

Let M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.

Theorem 1. The topological entropy of ϕt is positive.

This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.

Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.

For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.

Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.

In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.

Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

In [4] I extended the Plücker formulae to curves in Pn for any n, under fairly strong genericity hypotheses. In this paper we seek to weaken these hypotheses using the theory of excess intersections. The method applies to just one of the formulae (so the results are complete only for n=3), and for this we obtain a number which is a sum of local contributions, given by an explicit rule. We evaluate these contributions in the cases when they are determined by the terms of least order in the respective Taylor expansions, thus obtaining explicit formulae in numerous cases.

We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.

This paper is devoted to the asymptotic behaviour of individual eigenvalues of truncated Wiener–Hopf integral operators over increasing intervals. The kernel of the operators is complex-symmetric and has a rational Fourier transform. Under additional hypotheses, the main result describes the location of the eigenvalues and provides their asymptotic expansions in terms of the reciprocal of the length of the truncation interval. Also determined is the structure of the eigenfunctions.