Projective free algebras of continuous functions on compact abelian groups

It is proved that the Wiener algebra of functions on a connected compact abelian group whose Bohr-Fourier spectra are contained in a fixed subsemigroup of the (additive) dual group, is projective free. The semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite. The projective free property is proved also for the algebra of continuous functions with the same condition on their Bohr-Fourier spectra. As an application, the connected components of the set of factorable matrices are described. The proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration.     

A Problem on Zero Subsums in Abelian Groups

Let a₁, a₂, ... be elements of an abelian group such that a_mhas order larger than m^m. Then the multiset of the a_m's can be partitioned into two parts which are free of zero subsums. The result was motivated by a question of Pál Erds.     

非极大交换子群皆正规的有限群

设H是有限群G的一个交换子群.如果H在G中的中心化子正是它本身,则称H为G的极大交换子群.本文主要研究每一非极大交换子群都正规的有限群的结构,对这类有限群给出其完全分类.     

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.     

Measures with natural spectra on locally compact abelian groups

Every bounded regular Borel measure on noncompact LCA groups is a sum of an absolutely continuous measure and a measure with natural spectrum. The set of bounded regular Borel measures with natural spectrum on a nondiscrete LCA group whose Fourier-Stieltjes transforms vanish at infinity is closed under addition if and only if is compact.

     

Special ideals in partial abelian monoids

Under some conditions, the special congruences of partial abelian monoid are those induced by the special ideals, and a class of special ideals of partial abelian monoid has some upper and lower bound properties.     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

Uniqueness of uniform decompositions in exact categories

Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.     

At the Dawn of a Unifying Theory

I present a brief summary of the recent advancements on the unification of abelian quantum Hall states that covers the effects of disorder as well as the Coulomb interactions. I also present some exact results on the CP^N–1 model at large N. In contrast to the previous expectations, this theory shows all the basic features of the quantum Hall effect, including the massless bulk excitations at =.