Applications of representation theory and K-theory to problems in algebraic number theory

In this thesis we consider three main problems: the Galois module structure of rings of integers in wildly ramified extensions of Q; Leopoldt's conjecture; and non-commutative Fitting ideals and the non-abelian Brumer-Stark conjecture. For each of these problems, which correspond to each main chapter, we will review and use tools from representation theory and algebraic K-theory. In the first main chapter, we will prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers OK to be free over its associated order in the rational group algebra Q[G]. In the second main chapter, we will work on Leopoldt's conjecture. Let p be a rational prime and let L/K be a Galois extension of number fields with Galois group G. Under certain hypotheses, we show that Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L; a crucial tool will be the theory of norm relations in Q[G]. We also consider relations between the Leopoldt defects at p of intermediate extensions of L/K. Finally, we will investigate new properties of (non-commutative) Fitting ideals in integral group rings, with the general idea of reducing to simpler abstract groups (such as abelian groups) that can emerge as subquotients. As an application we will provide a direct proof of the (non-abelian) Brumer-Stark conjecture in certain cases, by reducing to the abelian case as recently proved by Dasgupta and Kakde. The direct approach avoids use of technical machinery such as the equivariant Tamagawa number conjecture

VGF changes during the estrous cycle: a novel endocrine role for TLQP peptides?

Although the VGF derived peptide TLQP-21 stimulates gonadotropin-releasing hormone (GnRH) and gonadotropin secretion, available data on VGF peptides and reproduction are limited. We used antibodies specific for the two ends of the VGF precursor, and for two VGF derived peptides namely TLQP and PGH, to be used in immunohistochemistry and enzyme-linked immunosorbent assay complemented with gel chromatography. In cycling female rats, VGF C-/N-terminus and PGH peptide antibodies selectively labelled neurones containing either GnRH, or kisspeptin (VGF N-terminus only), pituitary gonadotrophs and lactotrophs, or oocytes (PGH peptides only). Conversely, TLQP peptides were restricted to somatostatin neurones, gonadotrophs, and ovarian granulosa, interstitial and theca cells. TLQP levels were highest, especially in plasma and ovary, with several molecular forms shown in chromatography including one compatible with TLQP-21. Among the cycle phases, TLQP levels were higher during metestrus-diestrus in median eminence and pituitary, while increased in the ovary and decreased in plasma during proestrus. VGF N- and C-terminus peptides also showed modulations over the estrous cycle, in median eminence, pituitary and plasma, while PGH peptides did not. In ovariectomised rats, plasmatic TLQP peptide levels showed distinct reduction suggestive of a major origin from the ovary, while the estrogen-progesterone treatment modulated VGF C-terminus and TLQP peptides in the hypothalamus-pituitary complex. In in vitro hypothalamus, TLQP-21 stimulated release of growth hormone releasing hormone but not of somatostatin. In conclusion, various VGF peptides may regulate the hypothalamus-pituitary complex via specific neuroendocrine mechanisms while TLQP peptides may act at further, multiple levels via endocrine mechanisms involving the ovary

The CMS Electromagnetic Calorimeter Pre-calibration with Cosmic Rays and Test Beam Electrons

The electromagnetic calorimeter of the CMS experiment at the new CERN proton-proton Collider (LHC) is at an advanced stage of construction. A necessary condition for its optimal performance is a precise channel-to-channel calibration. The use of cosmic rays allows the pre-calibration of all the channels at the level of 2% before the final installation in CMS and provides an extensive functionality test, essential for the commissioning of the detector. On the other hand, a beam of electrons permits extremely precise (better than 0.5%) pre-calibration coefficients to be obtained on a fraction of the calorimeter, that can also be used as a reference for the in situ calibration procedures that will rely on physics data

Interplay of disorder and geometrical frustration in doped Gadolinium Gallium Garnet

The geometrically-frustrated, triangular antiferromagnet GGG exhibits a rich mix of short-range order and isolated quantum states. We investigate the effects of up to 1% Neodymium substitution for Gallium on the ac magnetic response at temperatures below 1 K in both the linear and nonlinear regimes. Substitutional disorder actually drives the system towards a more perfectly frustrated state, apparently compensating for the effect of imperfect Gadolinium/Gallium stoichiometry, while at the same time more closely demarcating the boundaries of isolated, coherent clusters composed of hundreds of spins. Optical measurements of the local Nd environment substantiate the picture of an increased frustration index with doping.Comment: 5 pages, 5 figure

Backscattering Differential Ghost Imaging in Turbid Media

In this Letter we present experimental results concerning the retrieval of images of absorbing objects immersed in turbid media via differential ghost imaging (DGI) in a backscattering configuration. The method has been applied, for the first time to our knowledge, to the imaging of small thin black objects located at different depths inside a turbid solution of polystyrene nanospheres and its performances assessed via comparison with standard imaging techniques. A simple theoretical model capable of describing the basic optics of DGI in turbid media is proposed.Comment: 5 pages, 6 figure

Combined Effects of the North Atlantic Oscillation and the Arctic Oscillation on Sea Surface Temperature in the Alborán Sea

We explored the possible effects of the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO) on interannual sea surface temperature (SST) variations in the Albora´n Sea, both separately and combined. The probability of observing mean annual SST values higher than average was related to NAO and AO values of the previous year. The effect of NAO on SST was negative, while that of AO was positive. The pure effects of NAO and AO on SST are obscuring each other, due to the positive correlation between them. When decomposing SST, NAO and AO in seasonal values, we found that variation in mean annual SST and mean winter SST was significantly related to the mean autumn NAO of the previous year, while mean summer SST was related to mean autumn AO of the previous year. The one year delay in the effect of the NAO and AO on the SST could be partially related to the amount of accumulated snow, as we found a significant correlation between the total snow in the North Albora´n watershed for a year with the annual average SST of the subsequent year. A positive AO implies a colder atmosphere in the Polar Regions, which could favour occasional cold waves over the Iberian Peninsula which, when coupled with precipitations favoured by a negative NAO, may result in snow precipitation. This snow may be accumulated in the high peaks and melt down in spring-summer of the following year, which consequently increases the runoff of freshwater to the sea, which in turn causes a diminution of sea surface salinity and density, and blocks the local upwelling of colder water, resulting in a higher SST.CGL2009-11316 (Ministerio de Ciencia e Innovación, Spain, and FEDER