Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles

We show how to properly gauge fix all the symmetries of the Ponzano-Regge model for 3D quantum gravity. This amounts to do explicit finite computations for transition amplitudes. We give the construction of the transition amplitudes in the presence of interacting quantum spinning particles. We introduce a notion of operators whose expectation value gives rise to either gauge fixing, introduction of time, or insertion of particles, according to the choice. We give the link between the spin foam quantization and the hamiltonian quantization. We finally show the link between Ponzano-Regge model and the quantization of Chern-Simons theory based on the double quantum group of SU(2)Comment: 48 pages, 15 figure

On the Tautological Ring of Moduli Spaces of Riemann Surfaces.

openThis thesis "On the Tautological Ring of Moduli Spaces of Riemann Surfaces." gives an overview of known results about the relations on the tautological ring of Moduli spaces. First, we introduce the Moduli Space of Stable Curves as an Orbifold of a etale groupoid, a generalization of complex manifold and orbit space of a group. Then we define the Tautological Ring on it as a subring of the Cohomology Ring. Finally, we present the work of Pandharipande, Pixton and Dvonkine in \cite{3-spin}, they discovered a set of relations on the Tautological Ring that is, up to date, the largest known. This set is obtained by Cohomological Field Theories, a tool to compute and glue cohomology classes in a Tautological way, and a group action on CohFTs. Using Teleman’s characterization of Cohomological Field Theories in a specific case, they manage to deduce an explicit formula for a suitable modification of Witten’s 3-spin CohFT. This turns out to vanish non trivially, providing a set of relations

On the Tautological Ring of Moduli Spaces of Riemann Surfaces.

This thesis gives an overview of known results about the relations on the tautological ring of Moduli spaces. First, we introduce the Moduli Space of Stable Curves as an Orbifold of a etale groupoid, a generalization of complex manifold and orbit space of a group. Then we define the Tautological Ring on it as a subring of the Cohomology Ring. Finally, we present the work of Pandharipande, Pixton and Dvonkine in \cite{3-spin}, they discovered a set of relations on the Tautological Ring that is, up to date, the largest known. This set is obtained by Cohomological Field Theories, a tool to compute and glue cohomology classes in a Tautological way, and a group action on CohFTs. Using Teleman’s characterization of Cohomological Field Theories in a specific case, they manage to deduce an explicit formula for a suitable modification of Witten’s 3-spin CohFT. This turns out to vanish non trivially, providing a set of relations. Davide Accadi

Sheaf Theory as a Foundation for Heterogeneous Data Fusion

A major impediment to scientific progress in many fields is the inability to make sense of the huge amounts of data that have been collected via experiment or computer simulation. This dissertation provides tools to visualize, represent, and analyze the collection of sensors and data all at once in a single combinatorial geometric object. Encoding and translating heterogeneous data into common language are modeled by supporting objects. In this methodology, the behavior of the system based on the detection of noise in the system, possible failure in data exchange and recognition of the redundant or complimentary sensors are studied via some related geometric objects. Applications of the constructed methodology are described by two case studies: one from wildfire threat monitoring and the other from air traffic monitoring. Both cases are distributed (spatial and temporal) information systems. The systems deal with temporal and spatial fusion of heterogeneous data obtained from multiple sources, where the schema, availability and quality vary. The behavior of both systems is explained thoroughly in terms of the detection of the failure in the systems and the recognition of the redundant and complimentary sensors. A comparison between the methodology in this dissertation and the alternative methods is described to further verify the validity of the sheaf theory method. It is seen that the method has less computational complexity in both space and time

On moduli spaces of Riemann surfaces : new generators in their unstable homology and the homotopy type of their harmonic compactification

By M_g(m,n) we denote the moduli space of conformal structures on an oriented compact cobordism S_g(m,n) of genus g greater or equal 0 and with m+n greater or equal 0 enumerated, parametrized boundary components of which n are incoming and m are outgoing. The study of these spaces reflects a strong relationship between geometry, topology and mathematical physics. In this thesis, we study (1) the homotopy type of Bödigheimer's harmonic compactification of these moduli spaces (and of variations of these) and (2) the unstable homology of these moduli spaces (and of variations of these)

Enumerative geometry via the moduli space of super Riemann surfaces

In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces $\overline{\cal M}_{g,n}$. This allows us to use a recursion between the super volumes recently proven by Stanford and Witten to deduce recursion relations of a natural collection of cohomology classes $\Theta_{g,n}\in H^*(\overline{\cal M}_{g,n})$. We give a new proof that a generating function for the intersection numbers of $\Theta_{g,n}$ with tautological classes on $\overline{\cal M}_{g,n}$ is a KdV tau function. This is an analogue of the Kontsevich-Witten theorem where $\Theta_{g,n}$ is replaced by the unit class $1\in H^*(\overline{\cal M}_{g,n})$. The proof is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem replacing volumes of moduli spaces of hyperbolic surfaces with volumes of moduli spaces of super hyperbolic surfaces.Comment: 65 page

Topics in Topology and Homotopy Theory

This book is an account of certain topics in general and algebraic topology

Design and optimisation of scientific programs in a categorical language

This thesis presents an investigation into the use of advanced computer languages for scientific computing, an examination of performance issues that arise from using such languages for such a task, and a step toward achieving portable performance from compilers by attacking these problems in a way that compensates for the complexity of and differences between modern computer architectures. The language employed is Aldor, a functional language from computer algebra, and the scientific computing area is a subset of the family of iterative linear equation solvers applied to sparse systems. The linear equation solvers that are considered have much common structure, and this is factored out and represented explicitly in the lan-guage as a framework, by means of categories and domains. The flexibility introduced by decomposing the algorithms and the objects they act on into separate modules has a strong performance impact due to its negative effect on temporal locality. This necessi-tates breaking the barriers between modules to perform cross-component optimisation. In this instance the task reduces to one of collective loop fusion and array contrac