We show that for general linear groups GLn(q) as well as for q-Schur algebras the knowledge of the modular Alvis–Curtis duality over fields of characteristic ℓ, ℓ ∤ q, is equivalent to the knowledge of the decomposition numbers.

Given two integrally equivalent integral quadratic forms in at least three variables and with cube-free determinant, we establish an upper bound on the smallest unimodular matrix transforming one of the forms into the other. This bound is polynomial in the height of the two forms involved, confirming a conjecture of Masser for the class of forms considered.

In this paper, we look at the Diophantine equation(0.1)Under the ABC conjecture, we show that it has only finitely many nontrivial solutions. Unconditionally, we show that the set of n for which the above equation admits an integer solution a1,. . .,at is of asymptotic density zero.

Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program.

The paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on and for a suitable geometric collection of coherent sheaves on both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in .

This paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four . In fact, if U ⊂ X is the open subset formed by deleting the lines from X, and H is the usual projective height function on , then the height zeta function is analytically continued to the half-plane ℜe(s) > 17/20.

Let H denote a subgroup of the unitary group U(R, Λ) which is normalized by EU(J, ΓJ) for some form ideal (J, ΓJ) of a commutative form ring (R, Λ). We prove that H satisfies a “sandwich” property, i.e., there exists a form ideal (I ΓI) such thatfor some form ideal (I : J7, Ω). This answers a conjecture of Bak (1967) in the stable case.

We provide a new self contained proof of the following result of H. Schubert: If K is a (p,q)-torus knot, then the bridge number of K is min{p, q}.

The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n, 1), n > 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

If two permutations are strongly conjugate, then their corresponding positive permutation braids (also called simple braids) are conjugate. In this paper we exhibit two conjugate simple braids whose associated permutations are not strongly conjugate. In terms of the grey and black graphs with vertices in the ultra summit set defined in [1], this result can be reformulated by saying that there are ultra summit sets with simple braids (hence with canonical length k = 1) in which the grey graph is not connected. Birman, Gebhardt and González Meneses have given similar examples, but with k ≥ 2 [1]. Recall that the set of simple braids on n strings is a basis of the Hecke algebra Hn. If two simple braids on n strings are conjugate, the associated permutations are centrally conjugate, which means that the coefficients of any central element of Hn corresponding to these simple braids are equal. Working on topological dynamics, Hall and Carvalho [8] have discovered two braids on 12 strings (one and its reverse) which are not conjugate. Since one braid is the reverse of the other, their corresponding permutations are centrally conjugate. For n ≤ 6 we checked that central conjugacy implies conjugacy of the corresponding simple braids.

We find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus over given rings of integers. One basis is a variation on the bases defined in [GMW] for the lattices of the SO(3)-TQFT-theory modules of the torus. Moreover, we discuss the quantization functors (Vp, Zp) for p = 1, and p = 2. Then we give concrete bases for the lattices of the modules in the 2-theory. We use the above results to discuss the ideal invariant defined in [FK]. The ideal can be computed for all the 3-manifolds using the 2-theory, and for all 3-manifolds with torus boundary using the SU(2)-TQFT-theory. In fact, we show that this ideal using the SU(2)-TQFT-theory is contained in the product of the ideals using the 2-theory and the SO(3)-TQFT-theory under a certain change of coefficients, and with equality in the case of torus boundary.

We use the theory of sectorial sesquilinear forms to characterize the closure of the Creation Operator of Quantum Mechanics in the classical set-up. Further, we bring that theory to bear upon the class of unbounded cyclic subnormal operators that admit analytic models; in particular, we provide a sufficient condition for the existence of complete sets of eigenvectors for certain sectorial operators related to unbounded subnormals. The relevant theory is illustrated in the context of a class of analytic models of which the classical Segal–Bargmann space is a prototype. The framework of sectorial sesquilinear forms is also shown to be specially useful for treating questions related to the existence, uniqueness and stability of certain parabolic evolution equations naturally associated with such analytic models.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

We study equilibrium states in relativistic galactic dynamics which are described by stationary solutions of the Einstein–Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses.