Robinson's conjecture on Abelian groups

Let x₁,…,x_n be elements of a finite abelian group G, having respective orders k₁,…,k_n such that (x₁−1)(x₂−1)⋯(x_n−1)=0 in $\text{Z}\text{(G)}$, where n⩽1. We prove that min k_i≤n−1 with equalityossible if only if n−1 is prime. If all k_i are equal, and not divisible by the cube of a prime, we prove $\text{n⩾k}{}_1\text{(1+}\text{1}\text{r)}$ where r is the least prime dividing k₁. We also establish an inequality concerning coverings of a set by subsets.     

Abelian varieties and the general Hodge conjecture

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.     

Latin squares over Abelian groups

In this paper, parametric families of Latin squares over Boolean vectors and prime fields constructed earlier are generalized to the case of Abelian groups. Some criteria for realizability of this construction are presented. Some classification results are also given. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 65–71, 2006.     

Connes' embedding conjecture and sums of hermitian squares

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic certificates involve sums of hermitian squares and commutators. We prove that they always exist for a similar nonnegativity condition where elements of separable II₁-factors are considered instead of matrices. Under the presence of Connes' conjecture, we derive degree bounds for the certificates.     

Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ₁ and φ₂ of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [M,φ₁]=[M,φ₂] in the oriented bordism group Ω₂(G) of the group G. In this paper, we compute the bordism class [M,φ] for any such action of G on M and we determine for a given M, the bordism classes in Ω₂(G) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form n(Z/pαZ)).     

Analytic hypoellipticity for sums of squares and the Treves conjecture

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson–Treves stratification are symplectic.We produce a model operator, $P_1$, having a single symplectic stratum and prove that it is Gevrey $s_0$ hypoelliptic and not better. See Theorem 2.1 for a definition of $s_0$. We also show that this phenomenon has a microlocal character.We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.     

The Atiyah-Jones conjecture for rational surfaces

We show that if the Atiyah-Jones conjecture holds for a surface X, then it also holds for the blow-up of X at a point. Since the conjecture is known to hold for P² and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

Filling area conjecture and ovalless real hyperelliptic surfaces

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics.Received: February 2004 Revised: February 2005 Accepted: November 2004     

Matrices as sums of squares: a conjecture of Griffin and Krusemeyer

Griffin and Krusemeyer have cinjecturted that if M is a square matrix with entries in a field of characteristic different from two and if M is not a scalar multiple of the indentity then M is expressible as a sum of two squares. In this paper it is shown that that conjecture is true.     

Matrices as sums of squares: a conjecture of Griffin and Krusemeyer

Griffin and Krusemeyer have cinjecturted that if M is a square matrix with entries in a field of characteristic different from two and if M is not a scalar multiple of the indentity then M is expressible as a sum of two squares. In this paper it is shown that that conjecture is true.     

Tate conjecture for twisted Picard modular surfaces

In this paper we prove Tate conjecture for twisted Picard modular surfaces.