8 research outputs found
Glicci simplicial complexes
One of the main open questions in liaison theory is whether every homogeneous
Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the
G-liaison class of a complete intersection. We give an affirmative answer to
this question for Stanley-Reisner ideals defined by simplicial complexes that
are weakly vertex-decomposable. This class of complexes includes matroid,
shifted and Gorenstein complexes respectively. Moreover, we construct a
simplicial complex which shows that the property of being glicci depends on the
characteristic of the base field. As an application of our methods we establish
new evidence for two conjectures of Stanley on partitionable complexes and on
Stanley decompositions
Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs
The collapsibility number of simplicial complexes was introduced by Wegner in
order to understand the intersection patterns of convex sets. This number also
plays an important role in a variety of Helly type results. There are only a
few upper bounds for the collapsibility number of complexes available in
literature. In general, it is difficult to establish such non-trivial upper
bounds. In this article, we construct a sequence of upper bounds
for the collapsibility number of a simplicial complex . We also show that
the bound given by is tight if the underlying complex is -vertex
decomposable. We then give an upper bound for and therefore for the
collapsibility number of the non-cover complex of a hypergraph in terms of its
covering number
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi