Chern–Weil and Chern–Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeomorphism groups of manifolds.

An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C-algebra associated to a bundle E → M , equipped with a compatible connection ∇, which plays the role of the algebra of functions on the infinit...

The Bohl algebra B is the ring of linear combinations of functions te on the real line, where k is any nonnegative integer, and λ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of B (where we use the topology of uniform convergence) are infinite.

ω-powers of finitary languages are ω-languages in the form V , where V is a finitary language over a finite alphabet Σ. Since the set Σ of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [Niw90], by Simonnet [Sim92], and by Staiger [Sta97b]. It has been proved in [Fin01]...

ω-powers of finitary languages are ω-languages in the form V , where V is a finitary language over a finite alphabet Σ. Since the set Σ of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [Niw90], by Simonnet [Sim92], and by Staiger [Sta97b]. It has been proved in [Fin01]...

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.

If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Such maps appear as a first step toward superrigidity results.

In this paper we show how Rank-Crank type PDE’s (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.

Assuming that GCH holds and κ is κ+3-supercompact, we construct a generic extension W of V in which κ remains strongly inaccessible and (α+)HOD < α+ for every infinite cardinal α < κ. In particular the rank-initial segment Wκ is a model of ZFC in which (α+)HOD < α+ for every infinite cardinal α.