This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.

A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r^-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.

This thesis explores and proves the theorem of De Rham using sheaf cohomology. The theorem of De Rham states that for a smooth manifold, the singular cohomology is isomorphic to the De Rham cohomology. Intuitively, this theorem states that the number of holes in a smooth manifold coincides with the failure of the fundamental theorem of calculus for closed differential forms defined on the manifold. This theorem allows us to make analytical conclusions based on the geometry of a surface.

Initially, the homological algebra of abelian groups is studied to quantify the holes in a smooth manifold and the failure of the fundamental theorem of calculus. An isomorphism of these two cohomologies is locally established. Subsequently, sheaves and their corresponding cohomology theory are introduced to lift this local isomorphism to a global one. Ultimately, using a double diagram of sheaves, the singular and De Rham cohomologies are shown to be isomorphic, thereby proving the theorem of De Rham.

In this thesis, we derive a lower bound on a quantity appearing in a Fourier multiplier inequality on solvable Lie groups.
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$

Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\

Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes.

First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes.

Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode.

Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide.

Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

In the last decades there has been an increasing interest in computing the local strain at the atomic scale of materials. By knowing aspects of the local strain in a lattice, one has information about measurements of distortions of lattice parameters concerning shifts, deformations and defects computed with respect to a smooth, defect-free reference region. Multiple methods have been implemented so far in order to map the strain of two-dimensional lattice patterns, which are obtained through means of a High Resolution Electron Microscope (HRTEM). The functioning of a HRTEM is based on the same principles as an optical microscope, but it uses a beam of electrons instead of visible light. One of the computational methods which then processes the obtained two-dimensional images is called the Geometrical Phase Analysis (GPA) and makes use of a very important mathematical tool, the Fourier transform. The GPA method lies at the center of this project and consists of several steps. First, the Fourier transform of the lattice image is plotted and two Bragg peaks corresponding to two linearly independent frequency vectors in the power spectrum are chosen. Next, a mask is applied around these peaks, separately. In my project I have chosen to apply the Hann smoothing filter. Then, the inverse Fourier transform is applied to the masked image and the phase (also called the raw phase) is plotted. The next step is to compute the reduced phase, which is defined at a local pixel as being the raw phase from which the following product is subtracted: 2𝜋 ⃗𝑔 ⋅ ⃗𝑟, where ⃗𝑟 is the vector corresponding to a pixel in the real space and ⃗𝑔 the frequency vector corresponding to the Bragg peak around which one has applied the mask. At this point the reference region is computed by choosing a smooth, homogeneous area in the reduced phase image. In order to obtain the strain, one needs an optimal frequency vector ⃗𝑔 defined at every pixel of lattice. In order to do so, a minimization process defined in the context of a computer algorithm in the programming language Python has been implemented. These computations should lead to obtaining the lattice strain, which is calculated by taking the symmetric part of the derivation of the displacement obtained from the two linearly independent Fourier components, which is in turn called the distortion. The antisymmetric part of the distortion is the rotation component and serves as a check for the correctness of the computational method. The goal of my project is not only to provide a solid theoretical background for the GPA method and to discuss the strain at atomic level in several lattice patterns, but to also provide a rigorous computer algorithm that makes these computations reality. This algorithm, opposed to pre-existing software, facilitates the reader’s process immensely by the large amount of detail which is given at every step, detail which easily motivates and supports the reader in potential side-steps they would want to take in order to make the method their own.

This thesis is divided into five parts (covering chapter 1–5), that together try to give the reader a basic understanding of the symmetries of and the mathematical structure behind the standard model. The first two chapters cover some Lie and representation theory of the Lie groups U(1), SU(2) and SU(3), which play a central role in the standard model. In the third chapter, the structure of the standard model is given, based purely on observed symmetries and (quantum numbers of) particles. In this chapter, the mathematics of the first two chapters is not used yet. In the fourth chapter, the Lie theory of chapter 1 is used to treat the gauge theory of U(1), SU(2) and SU(3), where for SU(2) also spontaneous symmetry breaking is incorporated. Finally, the fifth and last theoretical chapter gives examples of how representation theory and gauge theory can together structure the particles of the standard model and can dictate which particles and particle interactions are physically allowed. The thesis ends with some recommendations for further study of the topic at hand and for further research to symmetries beyond the standard model.

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.

Recent progress on the representation theory of certain infinite dimensional gauge groups has raised an interest in the strongly continuous unitary representations of groups of a specific form that satisfy a certain positive energy condition. An equivalent formulation of the positive energy condition is obtained, allowing for a geometrical interpretation of this condition and which yields necessary conditions for satisfying this condition. By the theory of the Mackey machine, the strongly continuous unitary representations of such groups that are of positive energy are classified by corresponding stabilizer subgroups. In a specific case, these are fully determined up to equivalence.\\ Finally, a method is developed that embeds homogeneous bundles as eigenspace subbundles of trivial bundles that in particular applies to the bundles obtained through the representation theory of groups mentioned above. The eigenspace subbundles thus obtained allow for a more detailed understanding of the induced representation and moreover resemble various theories in particle physics.

In this project, quantum cloning machines are analyzed that take in N quantum systems in the same unknown pure state and output M quantum systems with M > N, such that the output best resembles the ideal, but impossible output of an M-fold tensor product of the pure input state. The proof of Keyl and Werner is reviewed in which a unique optimal solution is constructed, both in the case where the quality of the cloning is determined by the entire output, and in the case where the quality is determined by measurements on a single clone. Furthermore, it is shown that previous work on qubit cloning machines is a special case of the presented optimal solution.

In deze scriptie volgen we de lijn die in W. van Est in “A group theoretic interpretation of area in the elementary geometries” heeft uitgezet, maar we gaan grondiger in op de stof en bewijzen de meeste claims die door Van Est worden gedaan. We kijken naar wat triviale en nontriviale centrale groepsuitbreidingen zijn, wat die te maken hebben met cocykels en coranden en cohomologieklassen. We zien dat iedere cocykel een corresponderende homogene cocykel heeft. Aan de hand van het oppervlaktebegrip in een tweedimensionale ruimte kunnen we een homogene cocykel definiëren en daarmee kunnen we dan een centrale groepsuitbreiding construeren van de symmetriegroepen op het tweedimensionale hyperbolische, euclidische en elliptische vlak. We zien dat die uitbreidingen niet triviaal zijn, maar dat wanneer we ieder van deze uitbreidingen verheffen naar hun universele overdekking, de uitbreiding op de symmetriegroep van het hyperbolische vlak triviaal wordt, in tegenstelling tot die van het euclidische vlak.