Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [−P,P]s, provided only that s > (3r+1)2k−2. By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2k.

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d-dimensional packing lattice of balls one can find a four- dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

An explicit, asymptotically good, tower of curves over the field with eight elements is constructed. The genus and the number of rational points are calculated explicitly.

Let p and q be odd prime numbers. It is shown that all quasitriangular Hopf algebras of dimension pq over an algebraically closed field k of characteristic zero are semisimple and therefore isomorphic to a group algebra.

Results of the Landesman–Lazer type provide necessary and sufficient conditions for the existence of periodic solutions of certain nonlinear differential equations with forcing. Typically, they deal with scalar problems. This paper presents a discussion of possible extensions to systems. The emphasis is placed on the new phenomena produced by the increase of the dimension.

This paper provides sufficient conditions that guarantee the global attractivity of the zero solution of a delay population model under impulsive perturbations.

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

Hyperbolic invariant sets Λ of C1+γ diffeomorphisms where either the stable or unstable leaves are 1-dimensional are considered in this paper. Under the assumption that the Λ has local product structure, the authors prove that the holonomies between the 1-dimensional leaves are C1+α for some 0 < α < 1.

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

This paper provides a new, simpler proof of the fact that the smallest volume hyperbolic 3-manifold has betti number at least 3. In the process, the best known lower bound on the volume of such a manifold is improved.

A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar sets in higher-dimensional Euclidean spaces are also presented.

This paper deals with an operator theory approach to the corona conjecture for H∞([ ]n). Treil gave a counter-example to this conjecture in the case where n = 1 for operator-valued functions; thus one might hope to use this to disprove the corona conjecture for H∞([ ]n) (for n [ges ] 2). This paper shows that this natural approach towards a negative answer fails. On the other hand, the second result here shows that ‘commutant lifting’ cannot be true for more than two contractions for any constant. This obstructs a natural attempted proof of the corona conjecture for H∞([ ]n) (for n [ges ] 2) by our previous result.