5,156 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
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Valuing Low Carbon Energy - Insights for Fusion Commercialisation
The proponents of nuclear fusion believe that a small modular approach has the potential to achieve a viable source of energy in timescales smaller than those projected for the large scale multinational ITER/DEMO programme. If the numerous technical challenges can be overcome, the question still remains as to whether fusion small modular reactors (SMRs) will be commercially viable. This thesis aims to provide insight into this question and to identify whether approaches other than the generation of electricity to the grid have the potential to increase the value of a fusion SMR or a fleet of SMRs to a developer.
The work has three main components. Firstly, the Net Present Value (SMR) of a fusion SMR supplying electricity for sale to the grid in the UK was evaluated. This showed that there are combinations of electricity prices, capital cost and discount rates that will result in positive NPVs.
In the second component of the work, an existing approach to engineering flexibilities / real options has been extended and applied to the production of hydrogen from methane with carbon capture and storage. The results of this work demonstrate that the application of engineering flexibilities / real options has the potential to increase the value of a project.
In the final stage of the thesis, an engineering flexibility / real options approach has been combined with a portfolio approach to a fleet of fusion SMRs. This demonstrated that this approach has the potential to increase the value of a fleet of fusion SMRs to a developer.
The thesis has demonstrated that it is possible that fusion SMRs may be commercially viable. It has also demonstrated that the use of techniques such as engineering flexibilities and portfolio theory has the potential to increase the value to a developer of a fleet of fusion SMRs based on a tokomak design.</br
On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy
We discuss aspects of the global topology of moduli spaces of hyperkähler metrics.
If the second Betti number is larger than , we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\
An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer surface.\\
We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than . For a compact simply connected manifold we show that the moduli space of Ricci flat metrics on splits homeomorphically into a product of the moduli space of Ricci flat metrics on and the moduli of sectional curvature flat metrics on the torus
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Hecke Actions on Loops and Periods of Iterated Shimura Integrals
In this paper we show that the action of the classical Hecke operators T_N,
N>0, act on the free abelian groups generated by the conjugacy classes of the
modular group SL_2(Z) and the conjugacy classes of its profinite completion. We
show that this action induces a dual action on the ring of class functions of a
certain relative unipotent completion of the modular group. This ring contains
all iterated integrals of modular forms that are constant on conjugacy classes.
It possesses a natural mixed Hodge structure and, after tensoring with Q_ell$,
a natural action of the absolute Galois group. Each Hecke operator preserves
this mixed Hodge structure and commutes with the action of the absolute Galois
group. Unlike in the classical case, the algebra generated by these Hecke
operators is not commutative. The appendix by Pham Tiep is not included. It can
be found at arXiv:2303.02807.Comment: 92 pages; this version has an appendix by Pham Tiep. It is not
included and can be found at arXiv:2303.02807. In addition, there are various
corrections and improvements, as well as some new material in Section 1
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