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 define two-parameter generalizations of Andrews' (k+1)-marked odd Durfee symbols and 2kth symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When k = 0 we obtain families of modular forms and mock modular forms. When k ≥ 1, we find quasimodular forms and quasimock modular forms.

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G2 developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors and the latter K3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.

In this paper, we give a large new class of suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducible G2-manifolds. ‘Geography’ of the values of Betti numbers b2, b3 for the new (and previously known) examples of irreducible G2 manifolds is also discussed.

We show that the abelianization of the kernel of the Magnus representation of the automorphism group of a free group is not finitely generated.

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

Let m ≥ 2 be an integer, and π an irreducible unitary cuspidal representation for GLm(), whose attached automorphic L-function is denoted by L(s, π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s, π) in the half-plane ℜs > 1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then the first sign change in the sequence {λπ(n)}n=1∞ occurs at some n ≪ Qπ1 + ϵ, where Qπ is the conductor of π, and the implied constant depends only on m and ϵ. This improves the previous bound with the above exponent 1 + ϵ replaced by m/2 + ϵ. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −(L′/L)(s, π) in the half-plane ℜs > 1.

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.

We study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.

In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

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).

Let X be a compact Riemann surface X of genus at–least two. Fix a holomorphic line bundle L over X. Let be the moduli space of Hitchin pairs (E, φ ∈ H0(End0(E) ⊗ L)) over X of rank r and fixed determinant of degree d. The following conditions are imposed:

1. (i) deg(L) ≥ 2g−2, r ≥ 2, and L⊗rKX⊗r;

2. (ii) (r, d) = 1; and

3. (iii) if g = 2 then r ≥ 6, and if g = 3 then r ≥ 4.

Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature σ, then the n–iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than 2n−1σ. The same result holds with σ replaced by 2τ, twice the Ozsváth–Szabó knot concordance invariant.

We present a method to construct examples of K3 surfaces of geometric Picard rank 1. Our approach is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on étale cohomology. This allows us to abandon the original limitation to cases of Picard rank 2 after reduction modulo p. Furthermore, the use of Galois data enables us to construct examples that require significantly less computation time.

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.

Will J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

In this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

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.

A chain is a configuration in ℝd of segments of length ℓ1, . . ., ℓn−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ℓn. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters ℓ1, . . ., ℓn. This result is analogous to the conjecture of K. Walker which concerns the special case d=2.

We show that, for commutative hypergroups, the spectrum of all L1-convolution operators on Lp is independent of p ∈ [1, ∞] exactly when the Plancherel measure is supported on the whole character space χb(K), i.e., exactly when L1(K) is symmetric and for every α ∈ Reiter's condition P2 holds true. Furthermore, we explicitly determine the spectra σp(Tϵ1) for the family of Karlin–McGregor polynomial hypergroups, which demonstrate that in general the spectra might even be different for each p.