Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement

Reverse mathematics and equivalents of the axiom of choice

We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal subfamily with the finite intersection property and the principle asserting that if $P$ is a property of finite character then every set has a $\subseteq$-maximal subset of which $P$ holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to $\mathsf{Z}_2$ to being weaker than $\mathsf{ACA}_0$ and incomparable with $\mathsf{WKL}_0$. In particular, we identify a choice principle that, modulo $\Sigma^0_2$ induction, lies strictly below the atomic model theorem principle $\mathsf{AMT}$ and implies the omitting partial types principle $\mathsf{OPT}$

Spatially independent martingales, intersections, and applications

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is H\"older continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. {\L}aba in connection to the restriction problem for fractal measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in Section 8. Polishing notation and other small changes. All main results unchange

Complex algebraic compactifications of the moduli space of Hermitian-Yang-Mills connections on a projective manifold

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.Comment: minor changes to the exposition based on referee's comments; final version to appear in Geometry & Topology; 95 page

Hilbert C*-modules over a commutative C*-algebra

Peer reviewedPreprin

Uncountable families of prime z-ideals in C_0(R)

Denote by \continuum=2^{\aleph_0} the cardinal of continuum. We construct an intriguing family (P_\alpha: \alpha\in\continuum) of prime $z$-ideals in \C_0(\reals) with the following properties: If $f\in P_{i_0}$ for some i_0\in\continuum, then $f\in P_i$ for all but finitely many i\in \continuum; \bigcap_{i\neq i_0} P_i \nsubset P_{i_0} for each \i_0\in \continuum. We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type $\kappa$ of prime $z$-ideals in \C_0(\reals) for any ordinal $\kappa$ of cardinality \continuum.Comment: 12 page

Clique trees of infinite locally finite chordal graphs

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new result