ERC FUNDED PROJECTS

The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967. My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behavior of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic? My motivation is two-fold: - Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious. - In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood. The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behavior.

1 065 500 €

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

Ultracold quantum gases have exceptional properties and offer an ideal test-bed to elucidate intriguing phenomena of modern quantum physics. My project proposes to use a new exotic element to study strong dipolar effects in quantum gases. For its appealing properties, we choose erbium (Er), a rare-earth metal that has hardly been explored until now. This species is strongly magnetic and comparatively heavy. Due to these characteristics, we expect the quantum system to be of extreme dipolar character and to exhibit a large number of magnetic Feshbach resonances, necessary to manipulate the low-energy scattering properties. Moreover, this element has a very rich energy level spectrum, which could open up the way to establish novel laser cooling schemes, and it has numerous isotopes, one of them having a fermionic character. Remarkably, none of the species so far used in ultracold quantum gas experiments offers such a unique combination of properties! By using Erbium, we will be in an optimal position to produce a strongly dipolar atomic gases of bosons and fermions with tunable contact interaction. First important goals of the ERBIUM project include: Extensive study of Er scattering properties, realization of the first Bose-Einstein condensates and degenerate Fermi gases of erbium atoms, study of dipolar effects in atomic system, production of strongly polar weakly-bound Er molecules and study their properties in a two-dimensional trapping environment. We also have a long-term vision for the ERBIUM project: we will mix heavy erbium atoms with much lighter lithium atoms to produce atomic mixtures with extreme mass imbalance.

1 076 442 €

Start date: 2011-01-01, End date: 2015-12-31

"Atoms, as building blocks of nature, consist of an atomic nucleus and the electron shell. Both systems are governed by similar laws and forces. However, the required energies to create changes in the nucleus or the electron shell differ by many orders of magnitudes. This reflects in largely different tools and methods used for their investigation: atomic physics probes the electron shell mainly by means of lasers. Nuclear physicists create excitations at high energies using particle accelerators such as CERN. The radio isotope 229Thorium is the only atom with the potential to bridge the gap between atomic and nuclear physics. It provides an unnaturally low-energy nuclear excited state, accessible to atomic physics tools, most notably laser excitation. It is the aim of the proposed research project to identify the optical nuclear transition and make it usable for fundamental investigations and applications. Currently, our second is defined as 9.192.631.770 oscillations of a light wave that leads to a specific excitation in the electron shell of the Cesium atom. Using the nuclear excited state of 229Thorium instead would increase the time standard accuracy by many orders of magnitudes, at the same time reducing the experimental complexity considerably. Building such a nuclear clock is the main goal of the research proposal. This will directly lead to improved accuracy in satellite based navigation (GPS) and enhanced bandwidth in communication networks. Furthermore, vomparing a nuclear atomic clock to standard time standards will hence allow addressing one of the most fundamental questions in physics: ""are nature s constants really constant?""."

1 245 884 €

Start date: 2010-12-01, End date: 2015-11-30