ERC FUNDED PROJECTS

The complexity of constraint satisfaction problems (CSPs) is a field in rapid development, and involves central questions in graph homomorphisms, finite model theory, reasoning in artificial intelligence, and, last but not least, universal algebra. In previous work, it was shown that a substantial part of the results and tools for the study of the computational complexity of CSPs can be generalised to infinite domains when the constraints are definable over a homogeneous structure. There are many computational problems, in particular in temporal and spatial reasoning, that can be modelled in this way, but not over finite domains. Also in finite model theory and descriptive complexity, CSPs over infinite domains arise systematically as problems in monotone fragments of existential second-order logic. In this project, we will advance in three directions: (a) Further develop the universal-algebraic approach for CSPs over homogeneous structures. E.g., provide evidence for a universal-algebraic tractability conjecture for such CSPs. (b) Apply the universal-algebraic approach. In particular, classify the complexity of all problems in guarded monotone SNP, a logic discovered independently in finite model theory and ontology-based data-access. (c) Investigate the complexity of CSPs over those infinite domains that are most relevant in computer science, namely the integers, the rationals, and the reals. Can we adapt the universal-algebraic approach to this setting?

1 416 250 €

Start date: 2016-10-01, End date: 2021-09-30

Antibodies secreted by B cells of the adaptive immune system establish an essential barrier against bacteria and viruses and their presence is the hallmark of protective vaccinations. B cells are licensed for their tasks during germinal center (GC) reactions and differentiation into antibody-secreting plasma cells. Unfortunately, B cell-derived autoantibodies and proinflammatory cytokines can cause or contribute to autoimmune diseases. While major transcription factor networks regulating protective (or pathogenic) GCB cell responses have been identified and characterized, little is known about the post-transcriptional regulation by RNA-binding proteins (RBP), whose number rivals that of transcription factors. We postulate that RBPs exercise critical post-transcriptional control over germinal center B (GCB) and plasmacytic cell physiology and we aim to identify and molecularly characterize these regulatory mechanisms. To this end, we will complement sophisticated genetic mouse models with novel cell culture systems. We will monitor RBP activity with fluorescent sensors and use proteomics to reveal RBPs regulating the protein abundance of critical mediators of GCB and plasmacytic cell fates. In addition, we will conduct genetic screens to uncover relevant functions of a short list of 40 RBPs, whose protein expression we found to differ significantly between GCB and mantle zone B cells. Ultimately, we will use cellular immunology and RNA biochemistry to elucidate how these RBPs exert their post-transcriptional control. Through the integrated power of our multi-disciplinary approach we will thus pinpoint and investigate the functions of key RBPs regulating the biology of GCB and plasmacytic cells. GCB-PRID promises to uncover profoundly new insights into post-transcriptional regulation of adaptive immunity. Thereby, this groundbreaking research aims to reveal novel molecular targets for the treatment of autoimmune diseases, whose incidence is steadily on the rise.

1 998 066 €

Start date: 2016-09-01, End date: 2023-02-28

Eversince, the study of symmetry in mathematics and mathematical physics has been fundamental to a thourough understanding of most of the fundamental notions. Group theory in all its forms is the theory of symmetry and thus an indispensible tool in many of the basic theoretical sciences. The study of infinite symmetry groups is especially challenging, since most of the tools from the sophisticated theory of finite groups break down and new global methods of study have to be found. In that respect, the interaction of group theory and the study of group rings with methods from ring theory, probability, Riemannian geometry, functional analyis, and the theory of dynamical systems has been extremely fruitful in a variety of situations. In this proposal, I want to extend this line of approach and introduce novel approaches to longstanding and fundamental problems. There are four main interacting themes that I want to pursue: (i) Groups and their study using ergodic theory of group actions (ii) Approximation theorems for totally disconnected groups (iii) Kaplansky’s Direct Finiteness Conjecture and p-adic analysis (iv) Kervaire-Laudenbach Conjecture and topological methods in combinatorial group theory The theory of `2-homology and `2-torsion of groups has provided a fruitful context to study global properties of infinite groups. The relationship of these homological invariants with ergodic theory of group actions will be part of the content of Part (i). In Part (ii) we seek for generalizations of `2-methods to a context of locally compact groups and study the asymptotic invariants of sequences of lattices (or more generally invariant random subgroups). Part (iii) tries to lay the foundation of a padic analogue of the `2-theory, where we study novel aspects of p-adic functional analysis which help to clarify the approximation properties of (Z/pZ)-Betti numbers. Finally, in Part (iv), we try to attack various longstanding combinatorial problems in group theory with tools from algebraic topology and p-local homotopy theory.

2 000 000 €

Start date: 2016-10-01, End date: 2021-09-30

Algebraic geometry deals with algebraic varieties, that is, systems of polynomial equations and their geometric interpretation. Its ultimate goal is the classification of all algebraic varieties. For a detailed understanding, one has to construct their moduli spaces, and eventually study them over the integers, that is, in the arithmetic situation. So far, the best results are available for curves and Abelian varieties. To go beyond the aforementioned classes, I want to study arithmetic moduli spaces of the only other classes that are currently within reach, namely, K3 surfaces, Enriques surfaces, and Hyperkähler varieties. I expect this study to lead to finer invariants, to new stratifications of moduli spaces, and to open new research areas in arithmetic algebraic geometry. Next, I propose a systematic study of supersingular varieties, which are the most mysterious class of varieties in positive characteristic. Again, a good theory is available only for Abelian varieties, but recently, I established a general framework via deformations controlled by formal group laws. I expect to extend this also to constructions in complex geometry, such as twistor space, which would link so far completely unrelated fields of research. I want to accompany these projects by developing a general theory of period maps and period domains for F-crystals, with an emphasis on the supersingular ones to start with. This will be the framework for Torelli theorems that translate the geometry and moduli of K3 surfaces, Enriques surfaces, and Hyperkähler varieties into explicit linear algebra problems, thereby establishing new tools in algebraic geometry.

1 328 710 €

Start date: 2016-10-01, End date: 2021-09-30