Some examples of nontrivial homotopy groups of modules

The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, π¯n(A,B), and the projective theory, π¯n(A,B). They are dual, but not isomorphic. In this paper, we deliver and carry out the precise calculation of the first known nontrivial examples of absolute homotopy groups of modules, namely, π¯n(ℚ/ℤ,ℚ/ℤ),  π¯n(ℤ,ℚ/ℤ), and π¯n(ℤ,ℤ), where ℚ/ℤ and ℤ are regarded as ℤCk-modules with trivial action. One interesting phenomenon of the results is the periodicity of these homotopy groups, just as for the Ext groups

Looking for ultralight dark matter near supermassive black holes

Measurements of the dynamical environment of supermassive black holes (SMBHs) are becoming abundant and precise. We use such measurements to look for ultralight dark matter (ULDM), which is predicted to form dense cores ("solitons") in the centre of galactic halos. We search for the gravitational imprint of an ULDM soliton on stellar orbits near Sgr A* and by combining stellar velocity measurements with Event Horizon Telescope imaging of M87*. Finding no positive evidence, we set limits on the soliton mass for different values of the ULDM particle mass $m$. The constraints we derive exclude the solitons predicted by a naive extrapolation of the soliton-halo relation, found in DM-only numerical simulations, for $2\times10^{-20}~{\rm eV}\lesssim m\lesssim8\times10^{-19}~{\rm eV}$ (from Sgr A*) and $m\lesssim4\times10^{-22}~{\rm eV}$ (from M87*). However, we present theoretical arguments suggesting that an extrapolation of the soliton-halo relation may not be adequate: in some regions of the parameter space, the dynamical effect of the SMBH could cause this extrapolation to over-predict the soliton mass by orders of magnitude.Comment: 9 pages + appendices, 5 + 2 figures. v2: some clarifications and references added; conclusions unchanged; version published in JCAP. v3: few typos correcte

Algebraic functional equations and completely faithful Selmer groups

Let E be an elliptic curve—defined over a number field K—without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses this gives an algebraic func- tional equation of the conjectured p-adic L-function. As an application we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power divi- sion points of another non-CM elliptic curve A

Looking for ultralight dark matter near supermassive black holes

Measurements of the dynamical environment of supermassive black holes (SMBHs) are becoming abundant and precise. We use such measurements to look for ultralight dark matter (ULDM), which is predicted to form dense cores ("solitons") in the centre of galactic halos. We search for the gravitational imprint of an ULDM soliton on stellar orbits near Sgr A∗ and by combining stellar velocity measurements with Event Horizon Telescope imaging of M87∗. Finding no positive evidence, we set limits on the soliton mass for different values of the ULDM particle mass m. The constraints we derive exclude the solitons predicted by a naive extrapolation of the soliton-halo relation, found in DM-only numerical simulations, for 2×10-20 eV≲ m≲8×10-19 eV (from Sgr A∗) and m≲4×10-22 eV (from M87∗). However, we present theoretical arguments suggesting that an extrapolation of the soliton-halo relation may not be adequate: in some regions of the parameter space, the dynamical effect of the SMBH could cause this extrapolation to over-predict the soliton mass by orders of magnitude

Fabrication and characterisation of 45º and Ex 45º:tilted fibre gratings and their applications in fibre lasers and sensors

In this thesis, I present the studies on fabrication, spectral and polarisation characterisation of fibre gratings with tilted structures at 45º and > 45º (namely 45º- TFGs and ex 45º-TFGs throughout this thesis) and a range of novel applications with these two types of grating. One of the major contributions made in this thesis is the systematic investigation of the grating structures, inscription analysis and spectral and polarisation properties of both types of TFGs. I have inscribed 45º-TFGs in standard telecom and polarisation maintaining (PM) fibres. Two wavelength regions of interest have been explored including 1.55 µm and 1.06 µm. Detailed analysis on fabrication and characterisation of 45º-TFGs on PM fibres have also been carried out for the first time. For ex 45º- TFGs, fabrication has been investigated only on low-cost standard telecom fibre. Furthermore, thermal responses have been measured and analysed showing that both types of TFG have low responsivity to temperature change. More importantly, their refractive index (RI) responses have been characterised to verify the high responsivity to surrounding medium. Based on the unique polarisation properties, both types of TFG have been applied in fibre laser systems to improve the laser performance, which forms another major contribution of the research presented in this thesis. The integration of a 45º-TFG to the Erbium doped fibre laser (EDFL) enables single polarisation laser output at a single wavelength. When combing with ex 45º-TFGs, the EDFL can be transformed to a multi-wavelength switchable laser with single polarisation output. Furthermore, by utilising the polarisation property of the TFGs, a 45º-TFG based mode locked fibre laser is implemented. This laser can produce laser pulses at femtosecond scale and is the first application of TFG in the field of nonlinear optics. Another important contribution from the studies is the development of TFG based passive and active optical sensor systems. An ex 45º-TFG has been successfully developed into a liquid level sensor showing high sensitivity to water based solvents. Strain and twist sensors have been demonstrated via a fibre laser system using both 45°- and ex 45º-TFG with capability identifying not just the twist rate but also the direction. The sensor systems have shown the added advantage of low cost signal demodulation. In addition, load sensor applications have been demonstrated using the 45º-TFG based single polarisation EDFL and the experimental results show good agreement with the theoretical simulation

Fabrication and characterisation of 45º and Ex 45º : tilted fibre gratings and their applications in fibre lasers and sensors

In this thesis, I present the studies on fabrication, spectral and polarisation characterisation of fibre gratings with tilted structures at 45º and > 45º (namely 45º- TFGs and ex 45º-TFGs throughout this thesis) and a range of novel applications with these two types of grating. One of the major contributions made in this thesis is the systematic investigation of the grating structures, inscription analysis and spectral and polarisation properties of both types of TFGs. I have inscribed 45º-TFGs in standard telecom and polarisation maintaining (PM) fibres. Two wavelength regions of interest have been explored including 1.55 µm and 1.06 µm. Detailed analysis on fabrication and characterisation of 45º-TFGs on PM fibres have also been carried out for the first time. For ex 45º- TFGs, fabrication has been investigated only on low-cost standard telecom fibre. Furthermore, thermal responses have been measured and analysed showing that both types of TFG have low responsivity to temperature change. More importantly, their refractive index (RI) responses have been characterised to verify the high responsivity to surrounding medium. Based on the unique polarisation properties, both types of TFG have been applied in fibre laser systems to improve the laser performance, which forms another major contribution of the research presented in this thesis. The integration of a 45º-TFG to the Erbium doped fibre laser (EDFL) enables single polarisation laser output at a single wavelength. When combing with ex 45º-TFGs, the EDFL can be transformed to a multi-wavelength switchable laser with single polarisation output. Furthermore, by utilising the polarisation property of the TFGs, a 45º-TFG based mode locked fibre laser is implemented. This laser can produce laser pulses at femtosecond scale and is the first application of TFG in the field of nonlinear optics. Another important contribution from the studies is the development of TFG based passive and active optical sensor systems. An ex 45º-TFG has been successfully developed into a liquid level sensor showing high sensitivity to water based solvents. Strain and twist sensors have been demonstrated via a fibre laser system using both 45°- and ex 45º-TFG with capability identifying not just the twist rate but also the direction. The sensor systems have shown the added advantage of low cost signal demodulation. In addition, load sensor applications have been demonstrated using the 45º-TFG based single polarisation EDFL and the experimental results show good agreement with the theoretical simulation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

CUSP FORMS FOR LOCALLY SYMMETRIC SPACES OF INFINITE VOLUME

Let G be a real simple linear connected Lie group of real rank one. Then, X := G/K is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, X is a real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We define C(Г\G) on Г\G for torsion-free geometrically finite subgroups Г of G. We show that it has a Fréchet space structure, that the space of compactly supported smooth functions is dense in this space, that it is contained in L^2(Г\G) and that the right translation by elements of G defines a representation on C(Г\G). Moreover, we define the space of cusp forms °C(Г\G) on Г\G, which is a geometrically defined subspace of C(Г\G). It consists of the Schwartz functions which have vanishing ''constant term'' along the ordinary set Ω ⊂ ∂X and along every cusp. We show that these two constant terms are in fact related by a limit formula if the cusp is of smaller rank (not of full rank). The main result of this thesis consists in proving a direct sum decomposition of the closure of the space of cusp forms in L^2(Г\G) which respects the Plancherel decomposition in the case where Г is convex-cococompact and noncocompact. For technical reasons, we exclude here that X is the Cayley hyperbolic plane