Stratifying quotient stacks and moduli stacks

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.Comment: 25 pages, submitted to the Proceedings of the Abel Symposium 201

A solid state light-matter interface at the single photon level

Coherent and reversible mapping of quantum information between light and matter is an important experimental challenge in quantum information science. In particular, it is a decisive milestone for the implementation of quantum networks and quantum repeaters. So far, quantum interfaces between light and atoms have been demonstrated with atomic gases, and with single trapped atoms in cavities. Here we demonstrate the coherent and reversible mapping of a light field with less than one photon per pulse onto an ensemble of 10 millions atoms naturally trapped in a solid. This is achieved by coherently absorbing the light field in a suitably prepared solid state atomic medium. The state of the light is mapped onto collective atomic excitations on an optical transition and stored for a pre-programmed time up of to 1 mu s before being released in a well defined spatio-temporal mode as a result of a collective interference. The coherence of the process is verified by performing an interference experiment with two stored weak pulses with a variable phase relation. Visibilities of more than 95% are obtained, which demonstrates the high coherence of the mapping process at the single photon level. In addition, we show experimentally that our interface allows one to store and retrieve light fields in multiple temporal modes. Our results represent the first observation of collective enhancement at the single photon level in a solid and open the way to multimode solid state quantum memories as a promising alternative to atomic gases.Comment: 5 pages, 5 figures, version submitted on June 27 200

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

Double hadron leptoproduction in the nuclear medium

First measurement of double-hadron production in deep-inelastic scattering has been measured with the HERMES spectrometer at HERA using a 27.6 GeV positron beam with deuterium, nitrogen, krypton and xenon targets. The influence of the nuclear medium on the ratio of double-hadron to single-hadron yields has been investigated. Nuclear effects are clearly observed but with substantially smaller magnitude and reduced $A$-dependence compared to previously measured single-hadron multiplicity ratios. The data are in fair agreement with models based on partonic or pre-hadronic energy loss, while they seem to rule out a pure absorptive treatment of the final state interactions. Thus, the double-hadron ratio provides an additional tool for studying modifications of hadronization in nuclear matter

Spectral hole burning: examples from photosynthesis

The optical spectra of photosynthetic pigment–protein complexes usually show broad absorption bands, often consisting of a number of overlapping, ‘hidden’ bands belonging to different species. Spectral hole burning is an ideal technique to unravel the optical and dynamic properties of such hidden species. Here, the principles of spectral hole burning (HB) and the experimental set-up used in its continuous wave (CW) and time-resolved versions are described. Examples from photosynthesis studied with hole burning, obtained in our laboratory, are then presented. These examples have been classified into three groups according to the parameters that were measured: (1) hole widths as a function of temperature, (2) hole widths as a function of delay time and (3) hole depths as a function of wavelength. Two examples from light-harvesting (LH) 2 complexes of purple bacteria are given within the first group: (a) the determination of energy-transfer times from the chromophores in the B800 ring to the B850 ring, and (b) optical dephasing in the B850 absorption band. One example from photosystem II (PSII) sub-core complexes of higher plants is given within the second group: it shows that the size of the complex determines the amount of spectral diffusion measured. Within the third group, two examples from (green) plants and purple bacteria have been chosen for: (a) the identification of ‘traps’ for energy transfer in PSII sub-core complexes of green plants, and (b) the uncovering of the lowest k = 0 exciton-state distribution within the B850 band of LH2 complexes of purple bacteria. The results prove the potential of spectral hole burning measurements for getting quantitative insight into dynamic processes in photosynthetic systems at low temperature, in particular, when individual bands are hidden within broad absorption bands. Because of its high-resolution wavelength selectivity, HB is a technique that is complementary to ultrafast pump–probe methods. In this review, we have provided an extensive bibliography for the benefit of scientists who plan to make use of this valuable technique in their future research

Orbit Closures and Invariants

The first author would like to thank Sebastian Herpel for the conversations we had which led to the first iteration of some of the ideas in this paper, and also Stephen Donkin for some very helpful nudges towards the right literature. All three authors acknowledge the funding of EPSRC grant EP/L005328/1. We would like to thank the anonymous referee for their very insightful comments and for pointing out a subtle gap in the proof of Theorem 1.1.Peer reviewedPublisher PD