ERC FUNDED PROJECTS

The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods. Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject. We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small. We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration. Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further. Much of this is joint work with K Soundararajan of Stanford University.

2 011 742 €

Start date: 2015-08-01, End date: 2020-07-31

Topological insulators constitute a novel class of materials where the topological details of the bulk band structure induce a robust surface state on the edges of the material. While transport data for 2-dimensional topological insulators have recently become available, experiments on their 3-dimensional counterparts are mainly limited to photoelectron spectroscopy. At the same time, a plethora of interesting novel physical phenomena have been predicted to occur in such systems. In this proposal, we sketch an approach to tackle the transport and magnetic properties of the surface states in these materials. This starts with high quality layer growth, using molecular beam epitaxy, of bulk layers of HgTe, Bi2Se3 and Bi2Te3, which are the prime candidates to show the novel physics expected in this field. The existence of the relevant surface states will be assessed spectroscopically, but from there on research will focus on fabricating and characterizing nanostructures designed to elucidate the transport and magnetic properties of the topological surfaces using electrical, optical and scanning probe techniques. Apart from a general characterization of the Dirac band structure of the surface states, research will focus on the predicted magnetic monopole-like response of the system to an electrical test charge. In addition, much effort will be devoted to contacting the surface state with superconducting and magnetic top layers, with the final aim of demonstrating Majorana fermion behavior. As a final benefit, growth of thin high quality thin Bi2Se3 or Bi2Te3 layers could allow for a demonstration of the (2-dimensional) quantum spin Hall effect at room temperature - offering a road map to dissipation-less transport for the semiconductor industry.

2 419 590 €

Start date: 2011-04-01, End date: 2016-03-31

"Understanding structure-property relationships across lengthscales is key to the design of functional and structural materials and devices. Moreover, the complexity of modern devices extends to three dimensions and as such 3D characterization is required across those lengthscales to provide a complete understanding and enable improvement in the material’s physical and chemical behaviour. 3D imaging and analysis from the atomic scale through to granular microstructure is proposed through the development of electron tomography using (S)TEM, and ‘dual beam’ SEM-FIB, techniques offering complementary approaches to 3D imaging across lengthscales stretching over 5 orders of magnitude. We propose to extend tomography to include novel methods to determine atom positions in 3D with approaches incorporating new reconstruction algorithms, image processing and complementary nano-diffraction techniques. At the nanoscale, true 3D nano-metrology of morphology and composition is a key objective of the project, minimizing reconstruction and visualization artefacts. Mapping strain and optical properties in 3D are ambitious and exciting challenges that will yield new information at the nanoscale. Using the SEM-FIB, 3D ‘mesoscale’ structures will be revealed: morphology, crystallography and composition can be mapped simultaneously, with ~5nm resolution and over volumes too large to tackle by (S)TEM and too small for most x-ray techniques. In parallel, we will apply 3D imaging to a wide variety of key materials including heterogeneous catalysts, aerospace alloys, biomaterials, photovoltaic materials, and novel semiconductors. We will collaborate with many departments in Cambridge and institutes worldwide. The personnel on the proposal will cover all aspects of the tomography proposed using high-end TEMs, including an aberration-corrected Titan, and a Helios dual beam. Importantly, a postdoc is dedicated to developing new algorithms for reconstruction, image and spectral processing."

2 337 330 €

Start date: 2012-01-01, End date: 2017-12-31

Topological materials have developed rapidly in recent years, with my previous ERC-AG project 3-TOP playing a major role in this development. While so far no bulk topological superconductor has been unambiguously demonstrated, their properties can be studied in a very flexible manner by inducing superconductivity through the proximity effect into the surface or edge states of a topological insulator. In 4-TOPS we will explore the possibilities of this approach in full, and conduct a thorough study of induced superconductivity in both two and three dimensional HgTe based topological insulators. The 4 avenues we will follow are: -SQUID based devices to investigate full phase dependent spectroscopy of the gapless Andreev bound state by studying their Josephson radiation and current-phase relationships. -Experiments aimed at providing unambiguous proof of localized Majorana states in TI junctions by studying tunnelling transport into such states. -Attempts to induce superconductivity in Quantum Hall states with the aim of creating a chiral topological superconductor. These chiral superconductors host Majorana fermions at their edges, which, at least in the case of a single QH edge mode, follow non-Abelian statistics and are therefore promising for explorations in topological quantum computing. -Studies of induced superconductivity in Weyl semimetals, a completely unexplored state of matter. Taken together, these four sets of experiments will greatly enhance our understanding of topological superconductivity, which is not only a subject of great academic interest as it constitutes the study of new phases of matter, but also has potential application in the field of quantum information processing.

2 497 567 €

Start date: 2017-06-01, End date: 2022-05-31