The last decade has seen remarkable progress in the study of growth in infinite families of groups. The main approach has its roots in additive combinatorics, but has truly given fruit in a non-commutative context. It is becoming clear that the central role is played not by groups in isolation, but by actions of groups. It is from this perspective that my plan addresses, at the same time, questions on growth in groups as such and hard problems in analytic number theory. While this line of research on growth started with the study of matrix groups, it has now given strong results on permutation groups as well. Two outstanding matters are the control of dependence on rank in matrix groups, and the removal of the need for the Classification Theorem in permutation groups. Going beyond these questions on diameter and expansion, there are at least three new directions I propose to follow: towards algorithms, towards geometric group theory, and towards number theory. Some of the main recent results in the area take the form of diameter bounds. Bounding a diameter amounts to showing that one can express any element of a group as a short product of generators. One of the main algorithmic questions consists in actually finding such an expression, and doing so rapidly. Links between geometric group theory (which studies growth in infinite groups) and the new combinatorial techniques ought to become stronger. Sofic and hyperlinear groups -- which arose in part from geometric group theory -- seem to invite a combinatorial approach. Additive combinatorics has already shown its relevance to exponential sums, a key subject in analytic number theory. Can a newer perspective based on actions of groups give more general results? Short Kloosterman sums, which are particularly hard to bound, can be framed as a test case. I also plan to pursue related interests in automorphic forms - which are a classical example of the relevance of group actions to number theory - and model theory.

SoundKnowledge aims to rethink, for the first time, music in terms of the procedural knowledge inherent in and specific to music-making by exploring music-making as knowledge practices in Micronesia, Western Pacific Island world. This knowledge, formed in the performance of musical practice, may prove to be key to survival in the complex postcolonial predicament of Micronesia. I will address the issues of climate change, social alienation and postcolonial trauma in specific parts of Micronesia by fleshing out the nature and dynamics of that knowledge both conceptually and ethnographically. The systematic analysis of music as knowledge will allow me to identify strategies to foster resilience in the face of these urgent crises. At the same time, it will offer a first-of-its-kind theorization of the procedural knowledge inherent in and specific to music-making. The knowledge of music is self-referential and forms multilayered connections and ruptures with pasts, presents and futures and surrounding orders of knowledge. SoundKnowledge asks what Western Pacific musical practices know and how do they know it, how music-making makes this knowledge operable and how humans mobilize upon this knowledge in coping with their life-world through music. The project, therefore, explores how music functions as a distinct epistemic form that is often referred to as the proverbial power of music. Music research has the tools to unlock this power, and SoundKnowledge intends to plough a path here. SoundKnowledge provides insights into the specific knowledge of Western Pacific music in its entanglement with pressing cultural and social issues of the early 21st century. In contributing to the theoretical debate on the knowledge of music, the project probes vital questions of knowledge resources and human futures. SoundKnowledge will also instigate change: In collaboration with local institutions, the research results will be used towards the development of community action strategies.

Chirality, describing the “handedness” of a system, is a fundamental property of physical entities such as magnets, spins, the DNA double-helix, or circularly and elliptically polarized light. This proposal aims at employing intense ultrashort laser fields to control the chirality of two physical entities: photons of extreme-UV high harmonics, and free-flying electrons in a coherent pulsed beam. Specifically, the proposal “UCHIRAL” entails two main objectives: First, we develop a bright ultrafast source of circularly polarized extreme-UV high harmonics with optically controlled helicity (left vs. right) by combining the applicant's expertise with chiral harmonics with the host's efficient source. We will then use these chiral extreme-UV pulses for imaging of nanoscale magnetic features in ferromagnetic metals, which is currently inaccessible for high harmonics. The difference between images acquired with left- vs. right-circularly polarized extreme-UV radiation can isolate magnetic effects from a non-magnetic background. In a second system, we will coherently convert the chirality of optical near-fields into orbital angular momentum of a beam of free electrons. This interaction will be mediated by designed nanostructures exhibiting surface plasmon-polariton excitations with optically-controlled chirality. The envisaged mechanism relies on optical phase control of free electron wave functions in an ultrafast transmission electron microscope, which was recently established in the host institute. Both aspects of the proposal address fundamental interactions of chiral light with matter, currently at different levels of maturity, with expected scientific and potentially industrial applications. In particular, ultrafast chiral microscopy based on photons (first objective) and electrons (second objective) may enable future technologies by tracking chiral dynamics in nanosystems with intrinsic handedness, ranging from magnetic storage materials to biomolecules.

Universal paradigmatic gaps are gaps in linguistic paradigms that appear across languages and across users of a given language. As of yet, only very few gaps of this kind have been discussed in the literature. The one gap that has received substantial discussion concerns the universal absence of a lexicalized negated form for the quantifiers all, every or always: There appears to be no language in the world that exhibits a single word (or lexical item) that means ‘not all’, ‘not every’ or ‘not always’, an observation dating back to Thomas Aquinas (1225-1274). UNPAG will show that the landscape of universal paradigmatic gaps is in fact much richer and more varied than generally thought of. It is deeply enigmatic that such words do not exist across languages and cultures. Clearly, any theory seeking to explain such missing lexicalizations, i.e. any theory of universal paradigmatic gaps, should be able to make clear predictions about what may or may not be lexicalized, and why that is the case. Such a theory has thus far not been developed. It is also without question that such a theory should have a broad empirical foundation. To date, the pool of data has been heavily slanted toward well-studied, Western, Indo-European, adult spoken language, and negative quantifiers therein. No existing study has thus far come even close to targeting a richer empirical base, even though many more universal gaps can be observed! UNPAG will be the long-overdue filler of this gap. Understanding the nature, distribution and behaviour of universal paradigmatic gaps will have several profound implications for our understanding of human cognition, language and communication. UNPAG is the first panoramic study of universal paradigmatic gaps. Why is it that we cannot always say what is thinkable? UNPAG will provide an answer to this question and show when, how and why universal paradigmatic gaps may emerge in the languages that we speak or sign.