Geometry of curved Yang-Mills-Higgs gauge theories

This is my Ph.D. thesis defended at 31 May 2021, and it is devoted to the study of the geometry of curved Yang-Mills-Higgs gauge theory (CYMH GT), a theory introduced by Alexei Kotov and Thomas Strobl. This theory reformulates classical gauge theory, in particular, the Lie algebra (and its action) is generalized to a Lie algebroid $E$, equipped with a connection $\nabla$, and the field strength has an extra term $\zeta$. In the classical situation $E$ is an action Lie algebroid, $\nabla$ is then the canonical flat connection with respect to such an $E$, and $\zeta\equiv 0$. The shortened main results of this Ph.D.thesis are the following; see the abstract in the thesis itself for more information: 1. Reformulating curved Yang-Mills-Higgs gauge theory, also including a thorough introduction and a coordinate-free formulation. Especially the infinitesimal gauge transformation will be generalized to a derivation on vector bundle $V$-valued functionals, induced by a Lie algebroid connection. 2. We will discuss what type of connection for the definition of the infinitesimal gauge transformation should be used, and this is argued by studying the commutator of two infinitesimal gauge transformations, viewed as derivations on $V$-valued functionals. We take the connection on $W$ then in such a way that the commutator is again an infinitesimal gauge transformation. 3. Defining an equivalence of CYMH GTs given by a field redefinition. In order to preserve the physics, this equivalence is constructed in such a way that the Lagrangian of the studied theory is invariant under this field redefinition. It is then natural to study whether there are equivalence classes admitting representatives with flat $\nabla$ and/or zero $\zeta$, and we will do so especially for Lie algebra bundles, tangent bundles and their direct products as Lie algebroids.Comment: This is my Ph.D. thesis which I have defended on 31 May 2021, in total 293 pages; the thesis lists a more detailed abstract about my main results; newer versions with several corrected typos, clarified texts et

Scalable visualisation methods for modern Generalized Additive Models

In the last two decades the growth of computational resources has made it possible to handle Generalized Additive Models (GAMs) that formerly were too costly for serious applications. However, the growth in model complexity has not been matched by improved visualisations for model development and results presentation. Motivated by an industrial application in electricity load forecasting, we identify the areas where the lack of modern visualisation tools for GAMs is particularly severe, and we address the shortcomings of existing methods by proposing a set of visual tools that a) are fast enough for interactive use, b) exploit the additive structure of GAMs, c) scale to large data sets and d) can be used in conjunction with a wide range of response distributions. All the new visual methods proposed in this work are implemented by the mgcViz R package, which can be found on the Comprehensive R Archive Network

Hypercontractive semigroups and two dimensional self-coupled Bose fields

AbstractWe present an abstract perturbation theory for operators of the form H0 + V obeying four properties: (1) H0 is a positive self-adjoint operator on L2(M, μ) with μ a probability measure so that e−tH0 is a contraction on L1 for each t > 0; (2) e−TH0 is a bounded map of L2 to L4 for some T; (3) V ϵ Lp(M, μ) for some p > 2; (4) e−tV ϵ L1 for all t > 0. We then show that spatially cutoff Bose fields in two-dimensional space-time fit into this framework. Finally, we discuss some details of two-dimensional Bose fields in the abstract including coupling constant analyticity in the spatially cutoff case